Abstracts of the Fifth Spanish Meeting on Game …mbilbao/pdffiles/booksmgt.pdfAbstracts of the Fifth Spanish Meeting on Game Theory and Applications SEVILLA, 2002 J. M. BILBAO F.R

Abstracts of the Fifth SpanishMeeting on Game Theory and

Applications

SEVILLA, 2002

J. M. BILBAO

F.R. FERNÁNDEZ

Editors

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PREFACE

The number of Spanish game theorists has increased considerably in recentyears. In 1994 it was decided to hold a Spanish Meeting on Game Theoryevery two years. The meetings in Bilbao (1994), Santiago de Compostela (1996)Barcelona (1998) and Valencia (2000), were a great success, with more than100 presentations in each of them. This shows the good health of the manySpanish research groups in Game Theory and their growing international links.The Fifth Spanish Meeting on Game Theory and Applications will take place inSeville on July 1-3, 2002. The 5th SMGT continues the series of the alternateItalian and Spanish conferences on Game Theory.

The conclusion of the Conference of Representatives of the Governmentsof the Member States opens the way for the enlargement of the EuropeanUnion. With ratification of the Treaty of Nice, the European Union wouldcomplete the necessary institutional changes for the accession of new MemberStates. During the 5th SMGT the workshop European Voting Games devotedto European Union decision-making, will take place.

Scientific Committee

• Salvador Barberà (Universitat Autònoma de Barcelona)http://hotelling.uab.es/~Salvador/

• Pierpaolo Battigalli (Università Bocconi di Milano)[email protected]

• Jesús Mario Bilbao (Universidad de Sevilla)[email protected]

• Steven Brams (New York University)[email protected]

• Francisco R. Fernández (Universidad de Sevilla)[email protected]

• Gianfranco Gambarelli (Università di Bergamo)http://www.unibg.it/dmsia/staff/gambar.html

• Ignacio García Jurado (Universidad de Santiago de Compostela)[email protected]

• Carmen Herrero (Universidad de Alicante)http://merlin.fae.ua.es/carmen/

i

ii Fifth Spanish Meeting on Game Theory

• Elena Iñarra (Universidad del País Vasco)[email protected]

• Andreu Mas-Colell (Universitat Pompeu Fabra)http://www.econ.upf.es/crei/mcolell.htm

• Jacqueline Morgan (Università di Napoli “Federico II”)http://cds.unina.it/~morgan/

• Guillermo Owen (Naval Postgraduate School)[email protected]

• Fioravante Patrone (Università di Genova)http://www.dima.unige.it/~patrone/

• Carles Rafels (Universitat de Barcelona)[email protected]

• Stef Tijs (Tilburg University)http://cwis.kub.nl/~few5/center/staff/tijs/

• Amparo Urbano (Universitat de València)[email protected]

• Federico Valenciano (Universidad del País Vasco)[email protected]

We announce that there is a special issue of Annals of Operations Researchdevoted to the 5th SMGT. The title is Contributions to the Theory of Games,the Editor-in-Chief is Peter L. Hammer (RUTCOR, Rutgers University) andthe Guest Editors are Jesús Mario Bilbao (University of Seville), JacquelineMorgan (Universitá di Napoli “Federico II”) and Hans Peters (University ofMaastricht).

This volume features the plenary lectures and the contributed abstracts pre-sented at the Fifth Spanish Meeting on Game Theory and Applications. Theinformation about the conference and the special issue can be found at

http://www.esi2.us.es/~mbilbao/game2002.htm

We are grateful to several colleagues of the Local Organizing Committe, inparticular, to Encarnación Algaba, Julio Fernández, Miguel A. Hinojosa, NievesJiménez, Andrés Jiménez Losada, Esperanza Lebrón, Jorge López, Amparo M.Mármol, Luisa Monroy and Justo Puerto. Finally we gratefully acknowledgegrants from the Fundación El Monte and the University of Seville.

Seville, April 2002Jesús Mario Bilbao

Francisco R. Fernández

CONTENTS

I PLENARY LECTURES 1

1 Values in Game Theory 3

2 Forming Stable Coalitions: The Process Matters 7

3 The Not-Quite Non-Atomic Games: Values and Cores of LargeGames 10

4 Approximate solutions and Well-posedness in MulticriteriaGames 13

II CONTRIBUTED ABSTRACTS 17

III Coalition Formation 19

5 The Absorbing Sets Solution in a Public Goods Economy 19

6 Case-Based Coalition Formation 21

7 Gradual Coalition Formation in Games With Externalities 22

IV Cost Sharing Models 25

8 Optimal Sharing of Surgical Costs in the Presence of Queues 25

9 A procedure to share recycling costs 27

10 Network Formation and Coordination: Bargaining the Divi-sion of Link Costs 28

V Coalition Structures 30

11 Outside options in coalition structures 30

12 Coalition Structures and Proportional Share Analysis for TUCooperative Games 31

13 An extension of the τ-value to games with coalition structures 32

VI European Voting Games 33

14 Apportionment Strategies for the European Parliament 33

15 On the number of swings in the Nice voting game 37

16 Strategic Power Revisited 40

VII Bankruptcy Games I 42

17 The TAL-family of rules for bankruptcy problems 42

18 An Experiment on Bankruptcy 44

19 Additivity properties in bankruptcy problems and other re-lated problems 45

VIII Social Networks 46

20 Social Interaction in Anti-Coordination Games 46

21 Bidding for the surplus: Realizing efficient outcomes in generaleconomic environments 48

22 Sequential formation of coalitions through bilateral agreements 49

IX Cooperative Games I 51

23 Efficiency in Uncertain Cooperative Games 51

24 The max-Core of Cooperative TU Games 52

25 Values of interior games 54

X Dynamic Games 58

26 Competition and cooperation in commercial fishing with adap-tive expectations 58

27 An Evolutionary Game Approach to Fundamentalism and Con-flict 61

28 A Model of Evolutionary Drift 62

XI Games with Information 66

29 Market Power and Information Revelation in Dynamic Trad-ing 66

30 The Principal-Agent Matching Market 67

31 Achieving Efficient Outcomes in Economies with Externalities 69

XII NTU Games 72

32 A Consistent Value for NTU Games with Coalition Structure 72

33 Subbalancedness and the bargaining set 73

34 The Procedural Value 74

XIII Auctions I 75

35 Auctions with Incumbents and Newcomers 75

36 Contest Architecture 76

37 Revenue Equivalence in Asymmetric Auctions 77

XIV Social Choice 78

38 Voting by Committees under Constraints 78

39 Monotonicity of Social Optima with respect to ParticipationConstraints 80

40 Agglomeration of Spatial Clubs 82

41 Fixed Agenda Solutions for Weak Tournaments 83

XV Voting Situations 85

42 Maximal domains for strategy-proof rules in the candidatesselection problem 85

43 Condorcet Consistency of the Citizen-Candidate Approach 87

44 Positive and normative assessment of voting situations 88

XVI Bargaining Models 90

45 Stakeholders, Bargaining and Strikes 90

46 Conflict as a part of the Bargaining Process: Theory and Evi-dence 93

47 Mediation: Incomplete information bargaining with filteredcommunication 96

XVII Cooperative Games II 98

48 Stability of Cartels and the Incentive of Merger in OligopolySituations without Transfer 98

49 Compensation rules in IC situations: continuity and computa-tional aspects 99

50 Partial ordered cooperative games 101

XVIII Political Competition 102

51 Funding of Political Parties with Electoral Cheques 102

52 Governability and Representativeness: Simulation of ConcreteVoting Situations 105

53 Spatial Competition Between Two Candidates of Different Qual-ity 107

XIX Cournot Games 108

54 The Value of Public Information in a Cournot Duopoly 108

55 Information Acquisition: A (enhanced) two-stage approach 109

56 Inventories and Price Volatility in a Dynamic Cournot Duopoly110

XX Matching Markets 111

57 A Generalized Assignment Game 111

58 Many-to-one matching when the colleagues do matter 112

59 Matching, Search, and Intermediation with Two-Sided Hetero-geneity 113

XXI Production Games 114

60 LP Games with Committee Control: behavior of the core un-der direct market replication 114

61 Competition and cooperation in non-centralized linear produc-tion games 116

62 Profit Sharing in Enterprises 117

XXII Simple Games 118

63 Dimension of complete simple games with minimum 118

64 On the convenience to form coalitions or partnerships in simplegames 120

65 Structure of voting simple games with several levels of ap-proval 121

XXIII TU Games I 123

66 On coalitional semivalues 123

67 On the serial cost sharing 124

XXIV Decision Theory 125

68 Dynamic Consistency in Extensive Form Decision Problems 125

69 Systemic Game Theory: Werterian Approach 126

XXV Games in Extensive Form 128

70 Modeling Deception in Extensive Form Games 128

71 Games in Code Form 130

XXVI Bankruptcy Games II 131

72 The constrained equal award rule for bankruptcy problemswith a priori unions 131

73 Bankruptcy situations with references 132

XXVII Learning Models, Contracts 133

74 Equilibrium Properties of Reinforcement Learning by Imita-tion 133

75 Bidding in a Pool with Contract Market 134

76 Mixed Equilibria in Games of Strategic Complements are Un-stable 135

XXVIII Repeated Games 136

77 The Repentance Strategy 136

78 The economics of information transmission 137

79 Repeated Games, Automata, and Growing Strategy Space 138

XXIX Voting Procedures, Game Models in Ethics 139

80 Candidate Stability and Probabilistic Voting Procedures 139

81 The Impossibility of Ethics 140

XXX Utility and Subjective Probability 141

82 A Theory of Choice over Time 141

83 Forward Induction and the Minimum Revision Principle 143

XXXI Strategic Equilibrium 145

84 Cooperation with Strategy-Dependent Uncertainty Attitude 145

85 Nash equilibria in normal form games 147

XXXII Values of Cooperative Games 148

86 Potential, Value and Probability 148

87 The Hamiache value for union stable cooperation structures 150

88 Axiomatizations of the Shapley value for cooperative games onantimatroids 153

XXXIII Sequencing Games 155

89 On the Balancedness of Semi-infinite Sequencing Games 155

90 On Games corresponding to Sequencing Situations with Prece-dence Relations 156

91 Sequencing games without initial order 158

XXXIV Multicriteria Problems 159

92 Multicriteria Linear Bargaining Games 159

93 Solutions for Multiple Scenario Cooperative Games 160

94 Pareto Optimality in Multicriteria Bargaining 161

XXXV Environmental and Experimental Games 162

95 Application of Game Theory in a Groundwater Conflict inMexico 162

96 Professionals Play Minimax 164

97 Game Theory Applied to Water Problems 167

XXXVI Semivalues and Power Indices 168

98 Commutation Games and Multilinear Extension 168

99 Several ways of taking decisions by voting 169

100Characterization of Binomial Semivalues and their Modifica-tions 170

XXXVII Cooperative Games III 171

101Buyer-seller exactness in the assignment game 171

102 Inventory Games with Discrimination 173

103A model of competition in inventory games 175

XXXVIII Price theory and market structure 176

104On Auctions and Posted prices 176

105Who searches? 178

106About trading mechanisms in a model of endogenous money179

XXXIX Noncooperative Games 180

107On Nash Equivalence Classes of Normal Form Games 180

108A Communication Protocol for Random Consensus in the Co-ordinated Attack Problem 182

109The Coordination Paradox in the Electronic Mail Game 183

XL Communication Networks 184

110On Economies of Scale and Cooperation in Some Communi-cation Networks 184

111Semivalues and Centrality in Social Networks 186

112Stochastic Minimum Spanning Tree Games 187

113Multicriteria Minimum Spanning Tree Games 188

XLI Market Models 189

114Matching, Heterogeneity, and Noisy Observable Types 189

115The Inner Core of Concave Exchange Economies 190

116 Competitive equilibrium as the outcome of a capacity-pricegame 191

117Market Equilibria via Integer Programming 193

XLII Conflicts, Collective Decision 194

118An Amendment to Final-Offer Arbitration 194

119Bargaining with Commitment under an Uncertain Deadline 195

120Political Competition as a War of Attrition 197

121On Efficiency and Sustainability in Collective Decision Prob-lems with Heterogeneous Agents 199

XLIII Solution Concepts 200

122The Not-quite Non-atomic game: the Role of Differentiabilityfor the Core of Production Games 200

123Serial cost sharing: Discrete and continuous case 202

124The Nucleolus as a Power Index 203

125A survey on the potential approach in cooperative game the-ory 205

XLIV Preferences, Rationality 206

126Hold up and Intergenerational transmission of Preferences 206

127Rationalizable Foresight Dynamics: Evolution and Rational-izability 208

128Robust models for Nash equilibrium 209

129Rational expectation can preclude trades 210

XLV Effectivity Functions, Partition Functions, . . . 212

130Continuous representations of effectivity functions 212

131Bilateral Coalescence and Superadditivity of Partition Func-tions 213

132 Aggregation and Disaggregation of European Players in theIMF Compound Games 214

133From Cases to Rules 216

XLVI Auctions II, Equilibrium 217

134Auctions Versus Negotiations Revisited 217

135An Optimal Auction with Externalities 219

136Necessary and sufficient condition for two-player subgame per-fect implementation 220

137An Anti Folk Theorem? Existence and Evolution of PropertyRights as a Unique Equilibrium 223

XLVII TU Games II 224

138 An Adaptive Model of Demand Adjustment in CooperativeGames 224

139On Games corresponding to a new class of Parallel SequencingSituations 225

140Games with homogeneous groups and convexity 226

141New rules of amortization 227

PLENARY LECTURES

University of Seville, July 1-3, 2002 3

1 Values in Game Theory

Stef TijsTilburg University, The Netherlands

[email protected]

A century ago the pioneers in game theory John von Neumann (1903—1957)and Oskar Morgenstern (1902—1978) were born. At the age of 23 (December 7,1926) John von Neumann presented for the Mathematical Society in Göttingenhis paper ‘Zur Theorie der Gesellschaftspiele’, which appeared in 1928 ([1]; foran English translation, see [2]). This paper, inspired by parlour games (cf. [3]),contained already the outline of the book [4], by von Neumann and Morgensternin 1944. Present were here in the germ already games in extensive, normal andcoalitional form. The main theorem was the existence theorem of values formixed extensions of matrix games. Using this theorem games in coalitional formcould be defined. Now 75 years later we can look back in great admiration tothis paper, which revolutionarised social sciences so much.

About 350 years ago a similar impact on science had games of chance in thehands of Blaise Pascal, which made probability theory and statistics flourishing.

Common in the contributions of von Neumann and Pascal is the startingquestion of the value of a position of a player in a situation, where natureand/or other players play a role in the final result. This lecture deals withvalues in all kind of interactive situations.

Let us look first at zero-sum games with arbitrary action spaces. Here valuesnot necessarily exist (cf. Wald [5]) but we can define upper and lower values. Ahuge literature on ‘minimax’ theorems exists, where sufficient conditions on ac-tion spaces and the payoff function are given to guarantee that the upper valueis equal to the lower value. Also extensions to stochastic games (cf. Shap-ley, [6]) to vector payoffs (cf. Blackwell, [7]) and to ‘zero-sum like’ arbitrationgames (cf. [8]) can be mentioned.

An axiomatic characterization of the value for matrix games is given byVilkas [9] (cf. Vorobev [10], p. 95). Here the properties monotonicity, domi-nance, symmetry and objectivity play a role. Inspired by this beautiful paperextensions to a class of infinite zero-sum games [11], to stochastic games [12]and to linear programming problems [13] were deduced.

Attempts to characterize value sets for non-zero sum games were undertakenby Vilkas [14] and Jansen and Tijs [15].

For ‘generic’ extensive form games with complete information (and also forother classes of non-cooperative games with a unique Nash equilibrium) a valuevector is uniquely determined.

In the spirit of von Neumann, for n-person zero-sum games, where the cor-responding two-person games have not necessarily a value, one can associate a

4 Fifth Spanish Meeting on Game Theory

game in coalitional form, where the values of the coalitions are closed intervalswith the lower and upper value as end points. Such cooperative games deservesome attention in a future study.

Let us turn now to bargaining problems. What is the value of a playerto be in such a bargaining situation? This question posed by J. Nash [16]led to his now famous bargaining solution, where the axiomatic method washelpful. Another approach was taken by Raiffa [17], who took as one of hissolutions the feasible compromise between the disagreement vector and theutopia vector. An axiomatic characterization of this solution was given byKalai and Smorodinsky [18], and the solutions is nowadays called the Raiffa-Kalai-Smorodinsky solution.

For TU-games, via an axiomatic approach, Shapley [19] found his famousvalue. Variants were introduced by Myerson [20], Borm, Owen and Tijs [21]and others. Another approach was undertaken by Tijs [22], who considered asvalue vector for a TU-game the feasible compromise of an upper and a lowervector (cf. [23]). An extension to NTU-games was given in [24]. For a surveyof compromise values, see [25].

We will discuss in the lecture more extensively the following recent contri-butions.

(i) The value of information in information collecting situations, [26].

(ii) The compromise value for games on matroids, [27].

(iii) Hypercubes and compromise values for cooperative fuzzy games, [28].

(iv) Stable and additive values on perfect cones of cooperative games, [29].

References

[1] J. von Neumann (1928) Zur Theorie der Gesellschaftsspiele, Math. Ann.100, 295—320.

[2] J. von Neumann (1959) On the theory of games of strategy, Annals ofMathematics Studies 40, Princeton University Press, 13—42.

[3] D. B. Gillies, J. P. Mayberry, and J. von Neumann (1953) Two variantsof poker, Annals of Mathematics Studies 28, Princeton University Press,13—50.

[4] J. von Neumann and O. Morgenstern (1944) Theory of Games and Eco-nomic Behavior, Princeton University Press, Princeton.

[5] A. Wald (1945) Generalization of a theorem by von Neumann concerningzero sum two person games, Annals of Mathematics 46, 281—286.


[6] L. S. Shapley (1953) Stochastic games, Proc. Nat. Acad. Sci. U.S.A. 39,1095—1100.

[7] D. Blackwell (1956) An analog of the minimax theorem for vector payoffs,Pacific J. Math. 6, 1—8.

[8] S. H. Tijs and M. J. M. Jansen (1982) On the existence of values for arbi-tration games, Int. J. of Game Theory 11, 87—104.

[9] E. J. Vilkas (1963) Axiomatic definition of the value of a matrix game,Theory of Probability and Its Applications 8, 304—307.

[10] N. N. Vorobev (1970) The present state of the theory of games, RussianMathematical Surveys 25, 77—136.

[11] S. H. Tijs (1981) A characterization of the value of zero-sum two-persongames, Naval Research Logistics Quarterly 28, 153—156.

[12] S. H. Tijs and O. J. Vrieze (1981) Characterizing properties of the valuefunction of stochastic games, Journal of Optimization Theory and Appli-cations 33, 145—150.

[13] S. H. Tijs (1981) A characterization of the value of linear programs, SimonStevin 55, 141—148.

[14] E. I. Vilkas (1968) The axiomatic definition of equilibrium points and thevalue of a non-coalitional n-person game, Theory of Probability and ItsApplications 13, 523—527.

[15] M. J. M. Jansen and S. H. Tijs (1981) On characterizing properties of thevalue sets and the equilibrium point sets of non-cooperative two-persongames, Math. Operationsforsch. Stat. , Ser. Optimization 12, 263—270.

[16] J. Nash (1960) The bargaining problem, Econometrica 18, 155—162.

[17] H. Raiffa (1953) Arbitration schemes for generalized two-person games,Annals of Mathematics Studies 28, Princeton University Press, 361—387.

[18] E. Kalai and M. Smorodinsky (1975) Other solutions to Nash’s bargainingproblem, Econometrica 43, 513—518.

[19] L. S. Shapley (1953) A value for n-person games, Annals of MathematicsStudies 28, Princeton University Press, 307—317.

[20] R. B. Myerson (1977) Graphs and cooperation in games, Mathematics ofOperations Research 2, 225—229.


[21] P. Borm, G. Owen, and S. H. Tijs (1992) On the position value for com-munication situations, SIAM J. Disc. Math. 5, 305—320.

[22] S. H. Tijs (1981) Bounds for the core and the τ -value. In: Game Theoryand Mathematical Economics (Eds. O. Moeschlin and D. Pallaschke)North Holland, 123—132.

[23] S. H. Tijs and F. Lipperts (1982) The hypercube and the core cover ofn-person cooperative games, Cahiers du Centre d’Etudes de RechercheOpérationnelle 24, 27—37.

[24] P. Borm, H. Keiding, R. P. McLean, S. Oortwijn, and S. H. Tijs (1992) Thecompromise value for NTU-games, Int. J. of Game Theory 21, 175—189.

[25] S. H. Tijs and G.-J. Otten (1993) Compromise values in cooperative gametheory, Top 1, 1—51.

[26] R. Brânzei, S. H. Tijs, and J. Timmer (2001) Compensations in informa-tion collecting situations, CentER DP 2001—02, Tilburg University.

[27] J. M. Bilbao, E. Lebrón, A. Jiménez-Losada, and S. H. Tijs (2002) Theτ -value for games on matroids, Top 10 (to appear).

[28] R. Brânzei, D. Dimitrov, and S. H. Tijs (2002) Hypercubes and compro-mise values for cooperative fuzzy games, CentER DP 2002—14, TilburgUniversity.

[29] R. Brânzei and S. H. Tijs (2001) Additive stable solutions on perfect conesof cooperative games, CentER DP 2001—105, Tilburg University.


2 Forming Stable Coalitions: The Process Matters

Steven J. Brams, Michael A. Jones and D. Marc KilgourNew York [email protected]

Coalitions are collections of players. Their stability is usually defined interms of outcomes and the incentives coalition members have to sustain them.In this paper, we show that the process by which the players come togetherand form coalitions also may critically affect how enduring coalitions will be.

To determine which coalitions are likely to form and be stable, we assumethat each player ranks all other players as coalition partners. A coalition ofk players, or k-coalition, is stable if no member would prefer to be in anotherk-coalition.

It is apparent that there is always at least one stable coalition –the grandcoalition, or n-coalition, that comprises all n players– because there is no othern-coalition. But below the grand coalition, what coalitions will form, and howstable they will be, is unclear. The coalition-formation processes we postulateclarify this question and also enable us to distinguish two levels of stability.

We rule out strategic issues that arise because of differences in player size orcapability. (For example, two ideologically distant players might join togetherif it would enable them to win, but neither would join a smaller more centrallylocated player if the resulting coalition were not winning.) Instead, we assumethat (i) all players are of equal weight (as in a legislature in which each mem-ber has one vote) and (ii) winning coalitions are those with at least a simplemajority, m, of members. Coalitions form according to two processes:

• Fallback (FB): Players seek coalition partners by descending lower andlower in their preference rankings until some majority coalition, all ofwhose members consider each other mutually acceptable, forms.

• Build-Up (BU): Same descent as FB, except only majorities whose mem-bers rank each other highest form coalitions.

BU seems most applicable to studying coalition formation in bodies likethe US Congress, in which small subsets of members coalesce and build up toa majority, all of whose members rank each other highest and are thereforestable. FB probably better describes the formation of a governing coalitionsin parliamentary democracies, wherein disconnected coalitions sometimes form.Because parties in such coalitions rank some parties outside the coalition higherthan parties in it, these coalitions are at best semi-stable.


We begin with notation and definitions in section 2. In section 3 we showthat several FB majority coalitions with k = m members may form simulta-neously. Call the set of FB coalitions that form first (i.e., at the lowest valueof k) FB1. The analogous set, BU1, comprises a unique coalition, which isstable. If the preferences of the players are single-peaked, FB1 coalitions maybe disconnected, but the BU1 coalition is always connected.

BU1 contains all coalitions in FB1, which may have fewer members thanBU1. This raises the question of which majority coalition is most likely to form-smaller FB1 majority coalitions, in which some members may prefer playersoutside the coalition to players inside, or the BU1 coalition, in which thiscannot happen? We call smaller FB1 majority coalitions semi-stable becauseat least some of their members are attracted to outside players.

Semi-stable coalitions, which may be disconnected if preferences are single-peaked (i.e., they may not include all adjacent players along a line), tend notto last. According to Riker’s (1962) size principle, some of their members growdisaffected and leave if there are insufficient resources to reward them in anoversized coalition.

In section 4 we show that semi-stable FB1 coalitions are strongly manipu-lable in that a player, by announcing a false preference ranking, can induce amajority coalition that it prefers. By contrast, BU1 is weakly strategyproof:A manipulator may be able to induce a smaller majority coalition with a falseannouncement, but this coalition will not necessarily be preferred. The reasonis that the larger BU1 coalition, which forms when the manipulator is truthful,must contain at least one member the manipulator prefers to some player inthe smaller majority coalition –so the manipulator will not assuredly preferthe smaller coalition.

In section 5 we investigate the properties of stable coalitions. BU1 may bea simple-majority coalition, the grand coalition, or any size in between. Moregenerally, stable coalitions of any size between m and n − 1 may or may notexist. Two stable coalitions (of any size) are either disjoint, or one containsthe other. A bandwagon strategy may enable a player to be a member of awinning coalition sooner than it would be otherwise, but it will not necessarilybe a winning coalition that it prefers.

In section 6 we show that if all player preferences are equally likely, theprobability that a randomly chosen majority coalition is stable first decreasesto some minimum between m and n − 1, then increases to 1 when the grandcoalition forms. This finding holds for the distribution of first-forming majoritycoalitions when preferences, whether single-peaked or not, are randomly cho-sen. We discuss the possibility of coalitions based on indirect as well as directlinkages, but these may lump together implausible coalition partners.

Empirical data on the size of US Supreme Court majorities that we present insection 7 show the distribution to be bimodal, with most being either minimal


(5-person) or maximal (9-person) majorities. The size of government coalitionsin some parliamentary democracies (Germany is an example used later) alsoshows there to be a paucity of majority coalitions in the middle range. Ifstability can be measured by durability, then our models may provide insightinto why parliamentary coalitions in a country like Italy are less durable thanthose in the Scandinavian countries, whose government coalitions sometimesdo not include even a simple majority of members.

The models might also enhance our understanding of the stability of coali-tions in other arenas, including international relations. Some internationalalliances like NATO have been long-lasting, others ephemeral. Is the processthat led to the former more BU-like, the latter more FB-like?

We conclude that FB and BU mirror different real-life coalition-formationprocesses. BU yields larger and less manipulable majority coalitions, comparedto the more wieldy but vulnerable coalitions of FB. Together these modelsshow how the stability of outcomes is inextricably linked to the processes thatgenerate them.


3 The Not-Quite Non-Atomic Games: Values and Cores of LargeGames

Guillermo OwenDepartment of Mathematics

Naval Postgraduate School, Monterey, [email protected]

Non-atomic games were first introduced in several articles by Aumann andShapley, which eventually culminated in their definitive book [1974, here de-noted A/S]. We note that this was not the first treatment of games with acontinuum of players; other articles, notably by Shapley et al., had done so. Itis in A/S, however, that a rigorous definition of such games, and their meaning,first appears.

Essentially, a perfect market requires that any single trader’s actions havenegligible effect on the payoffs of other traders. For example, if one dairy farmerincreases or decreases his production of milk, this will not (presumably) affectthe other farmers, who will still be selling their production at the same price.Nor will it affect consumers, who will be able to buy their normal quantity ofmilk at an unchanged price. This can only happen because the single farmer’scapacity is negligible by comparison with the size of the market.

In a similar way, in a general election (in a large voting constituency) anysingle voter’s choice of candidate (or, for that matter, the voter’s possible deci-sion not to vote) will presumably have a negligible effect on the outcome of theelection. Again, this can only happen if there is a very large number of voters(and no voter has more than a small number of votes).

In reality, of course, non-atomic games (whether market games or votinggames) do not exist. By a simple induction argument, it is clear that, if nosingle player has a non-negligible effect, then no finite set of players has anon-negligible effect.

We can therefore think of non-atomic games only as limiting cases of verylarge games. Properties of non-atomic games are to be thought of as propertiesthat hold, in the limit, for large games.

Under the circumstances, it is important to know the exact way in whichthese properties are obtained as the number of players increases (withoutbound). In general, it is clear that mere numbers are not enough: the wayin which new players are brought into the game is very important.

As an example, we can consider one of the most interesting results of theA/S book: non-atomic production games. In these games, each player is givenan endowment (a vector in m-space) and a production function that transformsm-vectors into utility. Coalitions can act by re-allocating their endowments so


as to maximize total production. A/S proves that, under rather broad and veryreasonable conditions, these games (1) have a non-empty core, consisting (2) ofa single point, which (3) coincides with the value of the game. Moreover, (4)this core can be expressed in terms of a vector of equilibrium prices, which areobtained (5) by an integration technique (the so-called diagonal process).

Now, under what conditions will large games have these properties (evenapproximately)? For example, property (3) seems to say that, for large eco-nomic/production/market games, the Shapley value will be close to the core.There is, in fact, a theorem (the value equivalence theorem) that states so.However, things are not as simple as this: certain conditions must be put onthe players, their endowments, or their technology. We can no more say that–for arbitrary games– the Shapley value converges to the core, than probabil-ity theory can say that an arbitrary sum of many random variables convergesto a normal variable.

In general, we have to think of these properties as holding, if at all, inthe limit for large games, i.e., as the number of players in the game increaseswithout bound. The simplest is to think of an infinite universe Ω of players;finite games grow in size as new players, all taken from Ω, are brought into thegame. Conditions can be put on Ω. (For example, all members of Ω have atleast 100, and not more than 1000, units of some good in their endowment, orthey all like one good more than another, etc.)

We can then talk about properties that hold in the limit either in a strongsense –they will hold so long as new players are all taken from Ω– or in aweaker sense –they hold if new players are taken from Ω only in a specific way.As an example of this weaker sense, we have the idea of market replication inwhich, starting from an original set N of n players, replicas of these playersare brought into the game so that we only consider games with kn players, kof whom are identical (in their endowment, preferences, etc.) to each of theplayers in the original set N . (This is then known as the k-fold replicated game.)

As a very simple but popular example of this difference, we consider the shoegame (also known as the glove game in other authors’ articles). The universalset Ω contains infinitely many players, each given an endowment consisting ofeither one right shoe or one left shoe. Utility is created by forming pairs ofshoes: each pair can be sold for $1. Thus the game has characteristic function

v(S) = min |S ∩ L| , |S ∩R| ,

where L and R are those subsets (of Ω) consisting of left-shoe and right-shoeholders, respectively.

Suppose we wish to study property (3) above –coincidence of the core andthe value. For finite games, this corresponds to saying that the value convergesto the core, or (a slightly weaker statement) that the distance between the valueand the core (measured, say, by the L∞ norm) converges to 0.


Now, it is easily proved that, starting from a specific set N of players (N ⊂Ω), replication of the game will indeed give us the desired result, which we canput in the following terms:

Let r and l be, respectively, the number of left-hand and right-hand shoes in the original set N . Assume r < l; the core thenconsists of a single imputation x, given by xi = 1 for each i ∈ R,and 0 for each i ∈ L.Then, for any ε > 0, there exists an integer h such that, for anyk > h, the Shapley value ϕ of the k-fold replicated game satisfiesϕi > 1− ε for all i ∈ R, and ϕi < ε for each i ∈ L.

Thus, the convergence here is very much dependent on the way in which newplayers are brought into the game. If new players are introduced in a randommanner, there is no guarantee of any type of convergence of the value.

Similar considerations will affect each of the five properties discussed above.In a series of papers dating to the early 1970’s, we have looked at these prop-erties.

1. Is the core of large production games non-empty? Is there an ε-core withsmall positive ε? Non-emptiness of the core will not in general hold for fi-nite production games unless the production functions are concave. In thenon-concave case, an article by Flåm/Owen (2001) shows the existenceof near-core imputations under reasonably broad conditions (broader,i.e., than replication of market); in the differentiable case, Saboyá/Owen(2002) gives a stronger result.

2. Will the core (at least generically) reduce to a single point? The core,when non-empty, does not always reduce to a single point. An article[Owen 1975] discusses conditions for both finite and infinite convergenceof the core to a single point. This depends on non-degeneracy of thefunctions. Essentially, this article makes use of the replication technique.

3. The same article [Owen 1975] discusses conditions for the core to con-verge to imputations that correspond to equilibrium prices. Once againreplication is necessary.

4. In [Owen 1972], the diagonal integration technique was developed as atechnique for calculation of the Shapley value of finite games. For othertypes of games (e.g., weighted voting games), other authors have obtainedsimilar results: in large games, a voter’s power (Shapley value) is propor-tional to his weight (number of votes). Once again, some strong regularityconditions are necessary for this to hold.


4 Approximate solutions and Well-posedness in MulticriteriaGames

Jacqueline MorganDipartimento di Matematica e Statistica

Università di Napoli “Federico II”, 80126 [email protected]

For long time, real valued functions have played a central role in game the-ory. More recently, motivated by applications to real-world situations, muchattention has been attracted to multicriteria games, i.e. games with vectorpayoffs. Different concepts of solutions have been introduced and existence ofsuch solutions has been investigated by various authors, mostly in the settingof finite dimensional spaces (cf. [1], [3], [5], [6], [15], [16], [18], [19], [20]). Formotivations and applications see, for example, [12], [13], [17], [21]. In the firstpart of the lecture I will present, together with illustrative examples, some ex-istence results for different concepts of solutions under sufficient conditions ofminimal character, also in the case of pseudo-games where the players can affectthe feasible sets of the other players. The results obtained in reflexive Banachspaces will allow to consider differential non-zero sum multicriteria games.

On the contrary, no attention has been paid to well-posedness of multicriteriagames, an interesting question from theoretical and numerical point of view.In Scalar Optimization, the two standard notions of well-posedness are basedeither on the behaviour of minimizing (or maximizing) sequences (Tychonovwell-posedness) or on the dependence of the optimal solutions from the dataof the optimization problem (Hadamard well-posedness). These notions forScalar Optimization have been extensively investigated in many papers (cf. forexample, [4] in which a good list of references can be found).

Different extensions have been introduced in Vector Optimization, BilevelOptimization, Zero and Non-Zero Sum Games and Cooperative Games (cf.among the others [2], [7], [8], an interesting survey on well-posedness in VectorOptimization, [9], [10], [11], [14]) but, differently from what happens in ScalarOptimization, the notion of approximate solution, which plays a central rolefor Tychonov well-posedness, is far from being univoquely determined. So,different definitions of Tychonov well-posedness for the same problem can beconsidered.

Clearly, the situation is the same for the more complex problems definedby multicriteria games so, in the second part of the lecture, various notions ofapproximate solutions for multicriteria will be presented together with suitableconcepts of well-posedness which can be useful for numerical purposes. Moreprecisely, I will introduce notions of approximating sequences for multicriteria


games and I will determine classes of problems which guarantee the convergence(in a sense to be precised) of such sequences to a solution of the original prob-lem. Therefore, any algorithm or method which produces such approximatingsequences allows to approach a solution.

Finally, I will present and discuss some results on Hadamard well-posednessfor multicriteria games.

References

[1] Borm, P., van Hegen, F., Tijs, S. H. (1999) A perfectness concept for mul-ticriteria games, Math. Methods Oper. Res. 49, 401—412.

[2] Cavazzuti, E. and Morgan, J. (1983) Well-posed saddle point problems,Lecture Notes in Pure and Applied Mathematics 86, Marcel Dekker, NewYork, 61—76.

[3] Corley, H. W. (1985) Games with Vector Payoffs, Journal of OptimizationTheory and Applications 42, 491—498.

[4] Dontchev, A. L. and Zolezzi, T. (1993) Well-posed optimization problems,Lecture Notes in Mathematics 1543, Springer-Verlag.

[5] Ghose, D. B. (1991) A Necessary and Sufficient Condition for Pareto-Optimal Security Strategies in Multicriteria Matrix Games, Journal ofOptimization Theory and Applications 68, 468—480.

[6] Ghose, D. B. and Prasad, U. R. (1989) Solution Concepts in Two-personsMulticriteria Games, Journal of Optimization Theory and Applications63, 167—188.

[7] Lignola, M. B. and Morgan, J. (2002) Approximate Solutions and α-well-posedness for Variational Inequalities and Nash Equilibria, Decision andControl in Management Science, Kluwer Academic Publishers, 367—378.

[8] Loridan, P. (1995) Well-posedness in vector optimization. Recent Develop-ments in well-posed variational problems, Eds. Lucchetti, R. and Reval-ski, J., Kluwer Academic Publishers, 171—192.

[9] Loridan, P. and Morgan, J. (2000) Convergence of approximate solutionsand values in parametric vector optimization. Vector Variational In-equalities and Vector Equilibria, Eds. Giannessi F., Kluwer AcademicPublishers, 335—350.

[10] Lucchetti, R., Patrone, F., Tijs, S. H. and Torre, A. (1987) ContinuityProperties of Solution Concepts for Cooperative Games, OR Spektrum 9,101—107.


[11] Margiocco, M., Patrone, F. and Pusillo Chicco, L. (1997) Anew approachto Tikhonov well-posedness for Nash equilibria, Optimization 40, 385—400.

[12] Monroy, L. and Mármol, A. (1999) Aplicaciones económicas. Avances enteoría de juegos con aplicaciones económicas y sociales, Eds. Bilbao, J.M. and Fernandez, F. R., Universidad de Sevilla, 73—88.

[13] Monroy, L. and Puerto, J. (1999) Juegos matriciales vectoriales, Avancesen teoría de juegos con aplicaciones económicas y sociales, Eds. Bilbao,J. M. and Fernandez, F. R., Universidad de Sevilla, 27—53.

[14] Morgan, J. (1989) Constrained well-posed two-level optimization problems.Nonsmooth Optimization and related topics, Eds. Clarke F., Demianov V.and Giannessi F., Ettore Majorana International Sciences Series, PlenumPress, New York, 307—326.

[15] Puerto, J. and Fernandez, F. R. (1995) Solution concepts based on SecurityLevels in Constrained Multicriteria Convex Games, Opsearch 32, 16—30.

[16] Singh, C., Rueda, N. (1994) Constrained Vector Valued Games and Mul-tiobjective Minmax Programming, Opsearch 31, 144—154.

[17] Szidarovszky, F. Gershon, M. E. and Duckstein, L. (1986) Techniques forMultiobjective Decision Making in Systems Management, Elsevier, Ams-terdam.

[18] Wang, S.Y . (1993) Existence of a Pareto Equilibrium, Journal of Opti-mization Theory and Applications 79, 373—386.

[19] Yu, P. L. (1979) Second-order Game Problems: Decision Dynamics inGaming Phenomena, Journal of Optimization Theory and Applications27, 147—166.

[20] Yuan, X. Z. and Tarafdar, E. (1996) Non-compact Pareto Equilibria forMultiobjective games, Journal of Math. Analysis and Applications 204,156—163.

[21] Zeleny, M. (1976) Games with multiple payoffs, Intern. Journal of GameTheory 4, 179—191.

CONTRIBUTED ABSTRACTS


Coalition Formation

5 The Absorbing Sets Solution in a Public Goods Economy

Norma Olaizola, Elena Iñarra and Jeroen KuipersUniversity of The Basque Country

[email protected]

The formation of coalitions to provide a public good creates positive exter-nalities on nonmember players. Players which sign an agreement to providea public good increase their total contributions to the provision of the good.Therefore some players may prefer to free-ride, that is to benefit from the in-creased supply of the public good without increasing their own contributions.This free-riding effect induces a high instability in the agreements, since, once acoalition has been formed, members of the coalition obtain a lower profit thanoutsiders and thus have an incentive to break the agreement.

There are some works in the literature on noncooperative theory that ana-lyze the stability of coalition structures. Some of them are Bloch [1995, 1996],Iñarra, Kuipers and Olaizola [2000], Ray and Vohra [2001], and Yi [1996]. Al-though this works share a common objective of analyzing equilibrium coalitionstructures, each adopts a different notion of stability.

In this work, we apply the framework developed in Iñarra, Kuipers andOlaizola [2000] to a problem of international environmental agreements betweencountries introduced by Barrett [1994]. The aim of the paper is to provide acomplete characterization of the equilibrium coalition structures for this model.Let us briefly explain our process of coalition formation: We allow for the for-mation of several coalitions, which give rise to coalition structures (partitionsof the set of countries). Given a coalition structure, some countries may besatisfied but others may tray to force transition to another coalition structure.We assume that the transition from one coalition structure to another is mo-tivated by the payoffs that countries can obtain in them. These payoffs areprovided by a profit function derived from the model. In the paper we describeprecisely how transitions take place. Moreover we consider that players are my-opic, that is when considering a movement they do not wonder about furtherconsequences that this movement can provoke. And we do not consider anystopping criteria to end the transitional process from one coalition structure toanother. Therefore, countries may move successively in discrete steps unlessthey converge upon some coalition structures which are stable.

The stability concept we use is the absorbing sets solution. Each absorbingset coincides with the elementary dynamic solution, Shenoy [1979]. And theabsorbing sets solution is the collection of all the absorbing sets. The notion


of stability lying on the absorbing sets solution may be understood as follows.Suppose that at some point in time an alternative in an absorbing set is reached,then alternatives in the absorbing set will be visited an infinite number of times,while no alternative outside the set will ever be visited again.

Our main result demonstrates that for the environmental agreement modelthere is a unique absorbing set and this set contains the grand coalition. Wepay special attention to stability of the grand coalition since it is the uniqueefficient coalition structure. Moreover we identify the unique absorbing set. Theequilibrium coalition structures belonging to the absorbing set are described asfollows: Only one group of countries will decide to sing an agreement whilethe others free-ride. But whenever the number of coalitions is smaller than acertain critical number at least one country will find it profitable to break theagreement, while up to this critical number all countries agree on signing theagreement for the grand coalition.


6 Case-Based Coalition Formation

Xavier Vilá and Ivanna FerdinandovaUniversitat Autònoma de Barcelona

[email protected]

We approach in this paper the analysis of coalition formation in a situationin which individual players have incomplete information with regards to theeffective value that belonging to a specific group of people reports to him. Forinstance, firms involved in a merging project do not know in reality how theirprofits will be affected until the very moment in which the merging actuallytakes place. Also, countries discussing trading agreements do not actually knowhow their wealth might improved until a treaty is implemented. To our knowl-edge, this approach has not been found in the literature on the topic and wethink that it deserves some attention.

To start with, we focus our attention on hedonic games, although somecomments on future directions of research including more general frameworksare discussed in the conclusions. To deal with the fact that players do notknow how valuable is to belong to a given coalition until the very momentsuch coalition is formed we use a derivative of Case-Based Theory by Gilboaand Schmeidler suited to our framework. Losely speaking, Case-Based Theoryassumes that when evaluating unknown situations, players take into accountthe situations that they know, how “similar” are they to the unknown situation,and how “valuable” were. Our framework is a very simple dynamic coalitionformation model. Starting with a given situation (configuration of coalitions),any player can decide at any moment whether to remain in the same grouphe currently belongs to, or joint one of the other coalitions that exist. If someof these “other coalitions that exist” is a group to which the player has neverbelonged to, he will evaluate the group by comparing it to “similar” groupshe has belonged to at some point and, hence, knows the true value that suchgroup had for him. In some sense, this model is close to the analysis by Konishiand Ray, with the only difference that they allow for coalitional moves (i.e., notonly individual players can move at some point, but also existing coalitions mayto merge or split), and that in their model players have perfect informationon the value that each possible coalition has for them. Another key featurein our approach is that players are farsighted in a very week sense that willbe specified. Briefly, if a player is considering to move to a coalition he hadbelonged to in the past and so happenned that such coalition was not stable(in the sense that some people departed from it or other people joined it), thenthe player will take this fact into consideration in his evaluation process. Wemodel the underlying strategic situation as a stochastic game and characterizethe set of Markov Perfect Equilibria of it.


7 Gradual Coalition Formation in Games With Externalities

Christian KjaerUniversity of Aarhus, Denmark

[email protected]

I study a non-cooperative multilateral bargaining game of coalition forma-tion, based on underlying TU games in partition form, in which coalitions areallowed to renegotiate agreements. Special attention is given to the strategicconsiderations induced by the possibility of renegotiation and its consequencesfor the efficiency of bargaining. In negative externality games, I find that rene-gotiation promotes the strategic formation of subcoalitions. On the contrary,in positive externality games, renegotiation promotes efficiency by motivatingplayers to form the grand coalition immediately. Thus, the possibility of rene-gotiation has very different strategic effects in the bargaining game dependingon characteristics of the underlying game.

Introduction

I consider a multilateral bargaining situation described by an n-person gamein partition form with transferable utility. The bargaining game is explicitlymodeled as an infinite-length extensive form game with perfect information.This puts the paper in line with Chatterjee et al.(RES. 60, 1993, 463-477),Bloch (GEB. 14, 1996, 90-123), Ray & Vohra (GEB. 26, 1999, 286-336), andothers where the bargaining process is explicitly modeled and in contrast to themore abstract notions found in Chwe (JET. 63, 1994, 299-325), Xue (ET. 11,1998, 603-627), Konishi & Ray (WP., 2001), and others with little structure onthe bargaining process.

Despite the recent interest in non-cooperative coalition formation with exten-sive form bargaining Chatterjee et al. (1993), Okada (GEB. 16, 1996, 97-108),Bloch (1996), Montero (WP., 1999), Ray & Vohra (1999), and others, littleattention has been given to possible renegotiation of coalitional agreements.Exceptions are Seidmann & Winter (RES. 65, 1998, 793-815) and Okada (JER.51, 2000, 34-50) where the possibility of renegotiation plays an important role.Both papers however restrict attention to environments where the underlyinggame is set in coalitional form. Even though lots of interesting issues canbe analyzed in a coalitional form setting, the assumption is very restrictiveand reduces the applicability of the results considerably. In addition, as thepresent paper will show, by restricting attention to games in coalitional form,we miss out on interesting strategic issues present in coalitional bargaininggames with externalities. I will argue, that by disallowing renegotiation weimplicitly assume implausible abilities to commit to coalitional agreements. To


briefly illustrate the main arguments of the paper, consider the following ruleof bargaining originally due to Okada (1996) and applied to games in partitionform in Montero (1999).

In the first round a proposer is randomly selected among the n players withequal probability. This player proposes a coalition and a payoff vector de-scribing the division of coalitional worth among the members of the proposedcoalition. The proposal is either accepted or rejected sequentially by all otherplayers in the coalition in some predefined order. If all accept, the coalition isformed and the game continues in similar fashion among the remaining play-ers. Otherwise, the game goes on to the next round where the same process isrepeated. The game ends when there are no remaining players. It is assumedthat future payoffs are discounted. In Okada (1996) it is shown, that underthis bargaining rule, the grand coalition is formed only if this is the coalitionwith the highest average payoff. Chatterjee et al. (1993) shows that under asimilar bargaining rule, where the initial proposer is selected according to afixed protocol and the first rejecter becomes the next proposer, the same resultholds true. In what seems natural and simple bargaining procedures, strict con-ditions on the underlying game is thus needed to get efficient outcomes. I willargue however that these results hinges crucially on the large degree of com-mitment allowed in the process of coalition formation. Particularly, in positiveexternality games inefficient coalitional structures are formed basically becauseplayers are able to commit not to join any coalition, knowing that coalitionswill eventually be formed among the remaining players. Without this commit-ment ability, the grand coalition will form immediately. On the other hand, innegative externality games, the grand coalition is formed because players arenot allowed to renegotiate the coalitional agreements. With this possibility,strategic subcoalitions will form in order to increase future bargaining powerthereby delaying the formation of the efficient coalition.

In this paper I aim to identify the strategic issues induced by the possibilityof successive renegotiation in games with externalities and determine the con-sequences for the efficiency of bargaining. The framework is a variant of therandom proposer bargaining model of Okada (1996) described above in whichplayers are allowed to renegotiate coalitional agreements with current agree-ments as threat-points of negotiations. This procedure is closely related to theone employed in Okada (2000) where similar issues are treated for games incoalitional form.

The main results of the paper are as follows. First, the possibility of rene-gotiation leads to the eventual formation of the efficient coalitional structurewithout delay. Second, if the underlying game has a positive externality, the an-ticipation of renegotiation induces immediate formation of the grand coalitionwhen the discount factor of future payoffs is sufficiently large. On the contrary,in negative externality games, the possibility of renegotiation promotes strate-


gic formation of subcoalition to increase future bargaining power. The strategicconsiderations induced by the possibility of renegotiation thus crucially dependupon the nature of the underlying game. Moreover, since the overall welfareis maximized when the grand coalition is formed immediately, allowing forrenegotiation can improve efficiency when the underlying game has a positiveexternality, whereas this reduces efficiency in negative externality games.

Seidmann & Winter (1998) and Okada (2000) both consider closely relatedbargaining models. Both papers find that renegotiation eventually will leadto efficient agreements. The fixed bargaining protocol employed in Seidmann& Winter (1998) may cause delay in agreements whereas no delay occurs inthe random proposer model of Okada (2000). In Seidmann (1998) it is shownthat the grand coalition will form gradually if the underlying game has anempty core. Okada (2000) shows that the prospect of renegotiation itself maylead to the gradual formation of the grand coalition. In contrast, the presentpaper shows, that the possibility of renegotiation can lead to both immediateand gradual coalition formation depending upon the nature of the underlyinggame.


Cost Sharing Models

8 Optimal Sharing of Surgical Costs in the Presence of Queues

Paula González and Carmen HerreroUniversity of [email protected]

The widespread access to Public Health Care in western European coun-tries is placing the system at a point in which optimal allocation of resourcesbecomes a major management problem. Citizens are particularly sensitive toany phenomena related to the provision of health services and among thesephenomena, one of the most relevant is the severe congestion suffered by publichealth services. These congestion problems have as one of their most directconsequences the enormous length of waiting lists for surgical treatment. Theformation of waiting lists to get elective surgery, can be framed as a queueingsystem. Assuming that the agents are served respecting their arrival order, theonly control variable is the capacity to install. Consequently, the theory canhelp us to take decisions concerning that capacity, taking into account thatthe higher the capacity the higher the associated costs, but the shorter theexpected queues. The queueing system arising in surgical treatment has somespecific characteristics:

1. There are two sources for the formation of waiting lists. On the one hand,the capacity of the operation theatre, and, on the other hand, the bedcapacity of the hospital;

2. Several medical procedures share both servers, namely, customers fromdifferent treatments need to use both the operation theatre and the beds;

3. Each of those procedures have their own rate of arrival;

4. Not all medical procedures are considered as equally urgent, in the sensethat the average waiting time politically considered as adequate differsamong procedures.

In the managing of such a situation, a cost allocation problem arises: Sincedifferent procedures share both the operation theatre and the hospital beds, wehave to design a cost allocation rule in order to share the joint costs. This isthe main purpose of this paper. In order to construct a cost allocation rule, weuse a game theoretical perspective, designing a cost allocation game.


In the first part of the paper we concentrate ourselves on the costs associatedto the operating theatre. Then, we construct a game by confronting two situa-tions: one in which each medical procedure has its own operating theatre, andanother one in which there is a unique theatre that serves all the diseases. Weshow that sharing the use of the operating theatre to treat the patients of thedifferent medical procedures, leads to a cost reduction. Then, we construct acost-sharing game and, given the characteristics of the game, we suggest a cost-sharing rule that recommends the Shapley value allocation of the cost-sharinggame. Thus, our optimal tariff has all the nice properties of the Shapley value.

The fact that this cooperative solution can be computed easily, is certainly animportant property in a practical environment. The cost-sharing game emerg-ing among the treatments is the sum of an additive game plus an “airportgame”, where the different landing track capacities are translated in our modelto the capacity required by the operating theatre in order to satisfy its demand,according to the maximum average waiting time guarantee. Up to this point,only the direct costs derived from surgical interventions were considered. How-ever, we have to take into account that an operation generates also other costs,more precisely the costs incurred during the patients’ hospitalization time forrecovering.

Then, we introduce in the model the post-operative costs and we study howthey are affected by the cooperation among medical procedures. Treating thebeds as servers, we may model the hospitalization stage also as a queueingsystem. Then, the number of servers (beds) required to guarantee the service,can be computed in different scenarios. Nonetheless, there is no possibility ofarriving at general results, due to the lack of analytical solvability of the model.In spite of that, something general can be said about the average number ofbeds. By so doing, we show that sharing the use of the operating theatre hasan ambiguous effect on average post-operative costs. If the medical procedurewith the highest priority level, has a higher recovering time than the averagehospitalization time of the rest of the pathologies, we can ensure that in averageterms cooperation leads to post-operative cost savings.

Finally, a numerical example with real data is analyzed. In this example,we compute the distribution of surgical costs, applying the theoretical resultsobtained previously. As for the number of beds required, we also compute them,under different scenarios. Also, we estimate the distribution of bed costs amongthe procedures, provided that an upper bound of .1 is set on the probability ofwaiting after the intervention.


9 A procedure to share recycling costs

José Alcalde and José A. SilvaUniversity of [email protected]

This paper examines a situation in which the production activities of dif-ferent agents, in a common geographical location, create waste products thatare either of a similar biological or chemical composition or offer commerciallycompatible combinations. What we propose here, therefore, is a cost-sharingmodel for the of recycling of their waste products. We concentrate, however,on the specific case in which the agents’ activities are heterogeneous.

We first examine, from a normative point of view, the cost-sharing rule,which we shall call the multi-commodity serial (MCS) rule. We introduce aproperty, that we call Cost-Based Equal Treatment, and we demonstrate thatthe unique rule verifying the Serial Principle and this property is the MCS rule.We then deal with the analysis of the agents’ strategic behavior when they areallowed to select their own production levels, in which case the total cost is thensplit, in accordance with the MCS rule. We show that there is only one Nashequilibrium, which is obtained from an interactive elimination of dominatedstrategies.

The paper is available at

http://merlin.fae.ua.es/alcalde/papers/recycling.pdf


10 Network Formation and Coordination: Bargaining the Divisionof Link Costs

Miguel Ángel Meléndez Jiménez,Universidad de Málaga y Universidad de Alicante

[email protected]

This paper presents a model of network formation in which links are costly.We endogeneize the part of the cost supported by each of the players involvedin a bilateral link. In this sense we consider that these sharings result frombargaining. We study this process in a context of coordination games. We showthat, if this cost is not too high, players coordinate either in the risk-dominantaction or in the efficient one: if costs of forming links are higher than therisk-dominance premium the efficient action is selected; meanwhile, if they arelower, the risk-dominant action prevails.

There are social and economic situations in which the existence of some kindof connections between the agents is necessary to interact. We can think forexample on information transmission: agents need some way of communicationin order to be able to exchange information. In many cases the establishmentand maintenance of these connections is costly. We model a situation in whichthe benefits from interacting is related to coordination, that is, any two playerswho establish a link benefit if they are coordinated in the same action. The mainfeature of our model is how the agents who form a link share the cost it involves.We propose that this division results from bargaining; in this sense, we makethe agents’ shares of the link cost endogenous. The model deals with the choiceof a standard in a population (e.g. PC vs. Mackintosh) and with the networkformation, that is, given the choices on standards, each agent decides who shewants to interact with (i.e. to form links). The earnings of the interactionbetween two agents (i.e. of forming a link) are represented by the payoffs ofa 2 × 2 symmetric coordination game in which we identify the actions withthe standards chosen. This game is characterized by two pure strategy Nashequilibria, one efficient and the other risk-dominant. The formation of a linkis costly and we consider that the part of the cost each of the involved agentsupports results from bargaining. Thus, in this model each player will firstdecide a standard and then, each possible pair of players enters in a bargainingprocess in which they have to agree on how to share the cost (and form a link),or reach the outside option of not forming the link.

We propose the Nash Bargaining solution to distribute the cost. We findthat this game presents multiplicity of equilibria. Therefore we introduce adynamics in which, from any initial state, each period players receive revisionopportunities. We assume a best-response adjustment to update strategies.


We analyze the set of limit states of this process and we get that the initialmultiplicity persists. To deal with the equilibrium selection we use stochasticstability techniques. We find a treshold for the link cost that coincides withthe risk-dominance premium. If the cost of the link is lower than this tresholdwe get that all the population coordinates in the (inefficient) risk-dominantstandard. If the cost of the link is higher, efficiency is achieved, provided thatthe cost is not so high that no link can be profitable for both players involved.

The study of networks has been increasingly considered in the literaturein the last years. Specially relevant in this field is the work by Jackson andWolinsky (1996), who study the stability and efficiency of social and economicnetworks; in their work they do not formally model the procedure through whicha graph is formed. There are also studies that explicitly analyze the dynamicprocess of network formation. Consider for example Bala and Goyal (2000),Jackson and Watts (1998), and Watts (2001). In these models agents onlydecide about link formation and there are no other actions that influence theirpayoffs. Bala and Goyal (2000) develop a noncooperative model of networkformation considering both directed and non-directed networks. They showconvergence to strict Nash networks.

Specifically, if we consider models in which the formation of links is costly,in the literature we find two ways to tackle with the link cost: (i) the one-sidedlinks models, that are characterized by the fact that the agent who proposesto form a link will completely cover the cost; (ii) the two-sided links models,in which each of the two agents involved in the link will share the cost inequal amount. In the setup of social coordination games (where our researchfits) Goyal and Vega-Redondo (2000) is framed in the first kind of models;among the second ones we find Jackson and Watts (1999) and Droste, Gillesand Johnson (2000), in which a spatial location of agents is introduced. Bothkind of models (one-sided and two sided) seem to be questionable since it isreasonable to argue that when two players have the possibility to form (ormaintain) a link, the one who will get a greater payoff from it will be willingto cover a higher part of the cost it involves. We propose the Nash solutionto distribute the cost of a link in this setup of bilateral coordination games.This endogeneization of the distribution of the cost provides two importantadvantages to our model over the former ones: (i) now, whenever a link isprofitable (the sum of the link-payoffs of the two agents is higher than the link-cost) the link will be formed, which in fact will result in a higher connectivity;and (ii) the cost supported by each player in a certain link will depend on therelative payoff of both agents involved in it. We get that our results are relatedto the first kind of models.


Coalition Structures

11 Outside options in coalition structures

Harald WieseUniversity of Leipzig

[email protected]

The paper presents and axiomatizes two coalition-structure values that aremeant to capture outside options of players in a cooperative game. A coalitionstructure is a partition on the set of players; the sets making up the partition arecalled components. Building on the Shapley value, several values for coalitionstructures have been presented in the literature, most notably by Aumannand Dreze (1974) and by Owen (1977); they are alluded to as AD-value andO-value, respectively. While the AD value assumes component efficiency, theO-value obeys overall efficiency. However, with both values, the bargainingprocess inside the components is dictated by the Shapley axioms. This impliesthat differing outside options of players within a component do not bear on thepayoffs. In contrast, the two values presented in our paper deviate from thenull-player axiom in order to reflect outside options. Close to the AD-approach,these values obey component efficiency (and symmetry and additivity).

Consider the following game defined by the player set N = 1, 2, 3 andthe transferable-utility coalition function v : 2N → R which ascribes worthsv(1, 2) = v(2, 3) = 60, v(N) = 72 and vanishes elsewhere. The Shapley valueof this game is (14, 44, 14) reflecting player 2’s prominence.

Consider also the coalition structure 1, 2 , 3. It leads to the AD-value(30, 30, 0) and the O-value (33, 33, 6). The two outside-option values generatethe payoffs (20, 40, 0) and (15, 45, 0), respectively. They attribute a higherpayoff to player 2 than to player 1, thus reflecting the outside opportunities ofplayer 2 (v(2, 3) = 60 > 0 = v(1, 3)).

Outside option values are especially suited to analyze voting games. Whilethe power is distributed within a government coalition (component efficiency),the possibilities of a party included in the government coalition to form po-tential coalitions with parties outside is to be a major factor determining thisparty’s weigth.


12 Coalition Structures and Proportional Share Analysis for TUCooperative Games

Cori Vilella and Carles RafelsUniversitat Rovira i Virgili and Universitat de Barcelona

[email protected]

Selten in 1972 developed a new theory called “equal share analysis”. Thistheory was expressed in the form of three hypothesis about the outcome ofcharacteristic function games with restricted coalition structures.

First hypothesis says that there will be a strong tendency to form an “ex-haustive coalition structure” which means that in the final coalition structurethere are no coalitions who have incentives to merge.

The second hypothesis asserts that the payoff vector will have a strong ten-dency to be in the equal division core, which is a new game theoretic conceptintroduced by Selten. A payoff vector is in the equal division core, if no coali-tion can divide its value equally among its members and in this way give moreto each of the members than they receive in the payoff vector.

After giving an order of strength within the players, the third hypothesissays that within a coalition a stronger player will not receive less than a weakerplayer.

The result of the play is a configuration, that consists of a coalition struc-ture which is a partition of N into permissible coalitions and a payoff vector.A configuration that satisfies the three hypothesis above is called regular con-figuration. Selten showed that a regular configuration exists for any game.

In this paper we extend Selten’s result assuming that every player has aninitial weight αi > 0 i = 1, . . . , n. We introduce the concept of the proportionaldivision core which depends on the vector α and is very related to the equaldivision core. We say that a payoff vector is in the proportional division core ifno coalition can divide it’s value proportionally to α among it’s members andin this way give more to all its members than they receive in the payoff vectordivided by αi.

On the other hand we introduce a new order of strength with respect to αwithin the players, and we consider that a coalition structure and a payoff vectoris a proportional regular configuration if the coalition structure is exhaustive,the payoff vector is in the proportional division core and it preserves the orderof strength with respect to α between the players in the same coalition.

The main result in the paper is that for any cooperative TU game and forany vector α > 0 we can always guarantee the existence of a proportional regularconfiguration.


13 An extension of the τ -value to games with coalition structures

Balbina Casas, Ignacio García-Jurado, Anne van den Nouweland andMargarita Vázquez

Universidad de Santiago de [email protected]

We introduce the coalitional τ -value, which is an extension of the τ -valuefor TU-games to games with a priori unions. We identify a class of TU-gamesthat satisfy the property that for every game in this class and every system of apriori unions of its player set it holds that the coalitional τ -value can be definedfor the corresponding game with a priori unions. We study properties of thecoalitional τ -value and provide an axiomatic characterization of this allocationrule. We use the coalitional τ -value to study bankruptcy problems and thedetermination of aircraft landing fees.


European Voting Games

14 Apportionment Strategies for the European Parliament

Gianfranco Gambarelli, Cesarino Bertini and Izabella Stach-JanasUniversity of Bergamo, Italy

[email protected]

A solution is proposed to the problem of assigning seats to new countriesapplying for membership of the European Parliament. This solution is obtainedby weighting the populations and Gross Domestic Products of all members. Astrategy of optimization for each single country is also studied.

The apportionment of seats to incoming members to the European parlia-ment has always been a source of discussion, as no fixed rule has so far beenestablished. The trend in the past has been to take into account the size of pop-ulation, in an attempt to guarantee representation for the major political partiesof each country (cf. Luxembourg) and avoid any reduction in the number ofseats held by existing members. The innovation at the Nice European Councilwas the application of a large number of countries having weaker economiesthan those of existing members. This involves a considerable increase in thetotal number of Euro MBPs, many of whom could have influenced decisionsof a specifically economic nature to the disadvantage of current members. Asolution to this impasse could be to restructure the distribution of seats forall countries using a formula, which takes into account both populations andGross Domestic Products (GDP). The most direct method consists of adequateweighting of this data. For instance, let populations and GDP percentages ofthe i-th country shown by Pi and Gi . Suppose that we decide to weight thepopulation with 30% and GDP with 70%. In this case the seat percentages Siof the i-th country would be Si = 0.3Pi + 0.7Gi. Generally speaking, if a isthe weighting we wish to assign to the population (0 ≤ a ≤ 1), the resultingseat percentages are Si = aPi + (1− a)Gi. To transform the seat percentagesinto real ones, a suitable rounding method can be used (for instance Hondt’sproportional system, or Hamilton’s Great Divisors, or Gambarelli’s minimaxapportionment [1999] and so on).

Once this method of apportionment has been accepted, it is a question offixing the value of a to weight the populations and GDP. The value of a stronglycharacterizes the seat distribution. In fact if a = 0, the seats are assignedproportionally on the basis of the countries’ economic powers, without takinginto account the size of population at all. Viceversa if a = 1. Then, a discussionon this can be expected between countries with strong economies and those withweak economies. From an initial examination, it would seem in the interestsof countries with higher GDP percentages than their population percentages(Denmark, Finland etc.), to have lower values of a (possibly 0). Viceversa, for


countries with lower GDP percentages (Poland, Romania etc.) it would seemin their interests to have high values of a (possibly 1). However, this rule doesnot always apply.

In this presentation a paradox on the matter is illustrated. The work pro-ceeds with an application of these ideas to the new Parliament size, examinedin the Nice Council of 2000. The program quoted in [Bilbao, 2000] has beenused for our computations. The seat distribution percentages and the relatedpower indices are studied with parameter a varying from 0 to 1 in 10% steps.Other studies are presented in reference to different majority quotas, optimumweight intervals, overcrossing between countries and so on. Here, we limit our-selves to quote that the Netherlands and Ireland, although they have a GDPpercentage higher than their population percentage, they do not have the max-imum advantage if only the GDP is taken into account (i.e. if a = 0). Forthese countries the maximum advantage (in terms of Martin-Banzhaf-Colemanindex) is achieved with a balanced division (a = 0.3).References

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Banzhaf, J.F. (1965) “Weighted Voting doesn’t Work: a Mathematical Anal-ysis” Rutgers Law Review, 19, 317-343.

Berg, S., J.E. Lane and R. Maeland (1996) “Voting Power Under the EUConstitution” in Andersen, S. and Elliassen, K. A. eds., The EuropeanUnion: How Democratic Is It? SAGE Publications, London.

Bilbao, J.M. (2000) “Nice European Council” fromhttp://hercules.us.es/~mbilbao/niza.htm

Bilbao, J.M., J.R. Fernández, A. Jiménez Losada, and J.J. López (2000) “Gen-erating Functions for Computing Power Indices Efficiently” Top 8, 2, 191-213, or http://hercules.us.es/ ~mbilbao/pdffiles/generat.pdf

Brams, S.J. (1976) Paradoxes in Politics: An Introduction to the Nonobviousin Political Science. Free Press, New York.

Coleman, J.S. (1971) “Control of Collectivities and the Power of Collectivityto Act” Social Choice, (B. Liberman ed.), Gordon and Breach, New York.

European Commission (2000) European Economy - Directorate General forEconomic and Financial Affairs, n. 70.

Felsenthal, D.S. and M. Machover (2000) Enlargement of the EU andWeightedVoting in its Council of Ministers. London School of Economics and


Political Science, Centre for Philosophy of Natural and Social Science,VPP 01/00.

–— (2001) “The Treaty of Nice and Qualified Majority Voting” fromhttp://www.lse.ac.uk

Freixas, J. and G. Gambarelli (1997) “Common Properties among Power In-dices” Control and Cybernetics, 4, 591-603.

Gambarelli, G. (1983) “Common Behavior of Power Indices” InternationalJournal of Game Theory, 12, 4, 237-244.

–— (1990) “A New Approach for Evaluating the Shapley Value” Optimization21, 3, 445-452.

–— (1994) “Power Indices for Political and Financial Decision Making: aReview” Annals of O.R. 51, 165-173.

–— (1996) “Takeover Algorithms” Modelling Techniques for Financial Mar-kets and Bank Management: Proceedings of the 16-th and 17-th EuroWorking Group of Financial Modelling Meetings (M. Bertocchi, E. Cav-alli and S. Komlosi eds.), Physica Verlag, Heidelberg, 212-222.

–— (1999) “Minimax Apportionments” Group Decision and Negotiation, 8,6, 441-461.

Gambarelli, G. and G. Owen (1994a) “Indirect Control of Corporations” In-ternational Journal of Game Theory, 23, 4, 287-302.

–— (1994b) “50 Years of Game Theory” Journal of European Business Edu-cation, 4, 1, 30-45.

–— (2000) “Power in Political and Business Structures” Jahrbuch für NeuePolitische Ökonomie, 20 (forthcoming).

Gambarelli, G. and J. Holubiec (1990) “Power Indices and Democratic Appor-tionments” Proceedings of the 8-th Italian-Polish Symposium on SystemsAnalysis and Decision Support in Economics and Technology (M. Fedrizziand J. Kacprzyk eds.). Omnitech Press, Warsaw, 240-255.

Garrett, G. and G. Tsebelis (1999) “Why Resist the Temptation to ApplyPower Indices to the EU” Journal of Theoretical Politics, 11, 321-331.

Herne, K. and H. Nurmi (1993) “The Distribution of a Priori Voting Power inthe EC Council of Ministers and the European Parliament” ScandinavianPolitical Studies, 16, 269-284.


Holler, M.J. (1978) “A Priory Party Power and Government Formation”Munich Social Science Review, 1, 25-41 (reprinted in M.J. Holler (ed.)(1982), Power, Voting, and Voting Power. Physica-Verlag, Würzburgand Vienna).

Holler, M.J, and M. Widgrén (1999) “Why Power Indices for Assessing EUDecision Making?” Journal of Theoretical Politics, 11, 291-308.

IMF (2000) International Financial Statistics Yearbook - International Mon-etary Fund Vol. LIII, IMF Statistics Department.

Laurelle A. and M. Widgrén (1998) “Is the Allocation of Voting Power AmongEU states fair?” Public Choice, 94, 317-339.

Nurmi, H., T. Meskanen and A. Pajala (1996) “Calculus of Consent in theEU Council of Ministers” Department of Political Science, University ofTurku, Finland, Working Paper.

Owen, G. (1995) (III ed.) Game Theory. Academic Press, San Diego.

Owen G. and L.S. Shapley (1989) “Optimal Location of Candidates in Ideo-logical Space” International Journal of Game Theory, 18, 3, 339-356.

Riker, W. (1986). “The First Power Index” Soc. Choice and Welfare 3, 293-295.

Saari, D. G. and K.K. Sieberg (2000) “Some Surprising Properties of PowerIndices”, Games and Economic Behavior, 36, 241-263.

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Shapley, L.S. (1953) “A Value for n-Person Games” Contributions to the The-ory of Games, 2, 39 (H.W. Kuhn, and W. Tucker, Eds.) Princeton Uni-versity Press, Princeton, NJ, 307-317.

Shapley, L.S. and M. Shubik (1954) “A Method for Evaluating the Distri-butions of Power in a Committee System” American Political ScienceReview 48, 787-792.

Turnovec, F. (1996) “Weights and Votes in European Union: Extension andInstitutional Reform” Prague Economic Papers, V, 161-174.

Widgrén, M. (1996) “A Voting Power Analysis of Supranational and NationalInfluence in the EU” Centre for Economic Policy Research, London, Dis-cussion Paper 1288.


15 On the number of swings in the Nice voting game

J. M. Bilbao, J. R. Fernández and J. J. LópezUniversidad de Sevilla

[email protected]

We present a new method to compute the number of swings in weighted votinggames as well as weighted double and triple majority games. We calculate theswings in an exact way by generating functions, with a significant decrease inthe computational complexity. Moreover, these swings are calculated for thedecision rules approved in the Nice summit meeting, which will be used in theEuropean Union enlarged to 27 countries. We show that the triple majority ruleadopted is quasi equivalent to a weighted voting game. Therefore, the requiredpopulation quota to approve a decision does not change the voting power of theEuropean countries.

A weighted voting game is represented by [q; w1, . . . , wn] . Here, there are nplayers, wi represents the voting weight of player i with 0 < wi < q, for all i,and q is the quota needed to win. We shall assume that q > 1

2

∑i∈N

wi. In suchvoting games the characteristic function is defined by v(S) = 1 if w(S) ≥ q andv(S) = 0 otherwise, where w(S) =

∑i∈S

wi. We suppose that all the weightsand the quota are positive integers. A weighted m-majority game is the gamev1∧· · ·∧vm where vt = [qt; wt

1, . . . , wtn], 1 ≤ t ≤ m are weighted voting games.

Then

(v1 ∧ · · · ∧ vm) (S) =

1, if

∑i∈S

wti ≥ qt, 1 ≤ t ≤ m,

0, otherwise.

A swing for player i is a pair of coalitions (S∪ i, S) such that S∪ i is winningand S is not. For each i ∈ N, we denote by ηi(v) the number of swings for i ingame v, that is, the number of winning coalitions in which player i is critical.The total number of swings is η(v) =

∑i∈N

ηi(v) and the normalized Banzhafindex is the vector β(v) = (β1(v), . . . , βn(v)) where

βi(v) =ηi(v)

η(v), 1 ≤ i ≤ n.

The generating function of the swings for player i in a weighted m-majoritygame is given by

Bi(x1, . . . , xm) =n∏

j=1, j =i

(1 + x

w1

j

1· · ·x

wmj

m

).


The Shapley-Shubik index for a simple game v : 2N → 0, 1 is the vectorΦ(v) = (Φ1(v), . . . ,Φn(v)) defined by

Φi(v) =∑

S/∈W:S∪i∈W

s! (n− s− 1)!

n!=

n−1∑j=0

j!(n− j − 1)!

n!dij ,

where each dij is the number of swings of player i in coalitions of size j. Similarto what happens with the Banzhaf index it is also possible to obtain, usinggenerating functions, an analogous result for the calculation of the Shapley-Shubik power index. If v = v1 ∧ · · · ∧ vm is a weighted m-majority game wherevt = [qt; wt

1, . . . , wtn], 1 ≤ t ≤ m, then the generating function of the swings of

player i in coalitions of size j is

Si(x1, . . . , xm, z) =n∏

j=1, j =i

(1 + z x

w1

j

1· · ·x

wmj

m

).

We next present some results concerning to the European Union enlarged to27 countries. We compute the number of swings in the voting games definedby the decision rule adopted in the Nice European Union summit, held inDecember, 2000. The players are:

Germany, United Kingdom, France, Italy, Spain, Poland, Romania,

The Netherlands, Greece, Czech Republic, Belgium, Hungary, Portugal,

Sweden, Bulgaria, Austria, Slovak Republic, Denmark, Finland, Ireland,

Lithuania, Latvia, Slovenia, Estonia, Cyprus, Luxembourg, Malta.

The decision rule is the weighted triple majority game v1∧v2∧v3, where thethree weighted voting games corresponding to votes, countries and population,are the following:

v1 = [255; 29,29,29,29,27,27,14,13,12,12,12,12,12,10,10,10,7,7,7,7,7,4,4,4,4,4,3],

v2 = [14; 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1],

v3 = [620; 170,123,122,120,82,80,47,33,22,21,21,21,21,18,17,17,11,11,11,8,8,5,4,3,2,1,1].

The game v3 is defined assigning to every country, a number of votes equalto the rate per thousand of its population over the total population and thequota represents the 62% of the total population. So, a voting will be favorableif it counts on the support of 14 countries with at least 255 votes, and with atleast the 62% of the population.


In a game with 27 players, the total number of coalitions which can beformed is 227 = 134217728. For example, for the game v1, with q = 255,which is called Game1, the total number of swings is 28186428; and for theweighted triple majority game v1 ∧ v2 ∧ v3 (votes, majority of countries andmore than 62% of the population) which is named Game3a, the number ofswings is 28186324. The difference is insignificant, only 104 swings over a totalsuperior to 28 millions. The following table shows the number of swings forevery country in both games.

Countries Game 1 Game 3a Difference

Germany 2193664 2193654 −10

The United Kingdom 2193664 2193650 −14

France 2193664 2193650 −14

Italy 2193664 2193650 −14

Spain 2091380 2091358 −22

Poland 2091380 2091358 −22

Romania 1200504 1200482 −22

The Netherlands 1120138 1120116 −22

Greece 1038492 1038476 −16

Czech Republic 1038492 1038476 −16

Belgium 1038492 1038476 −16

Hungary 1038492 1038476 −16

Portugal 1038492 1038476 −16

Sweden 871654 871654 0

Bulgaria 871654 871654 0

Austria 871654 871654 0

Slovak Republic 614702 614712 10

Denmark 614702 614712 10

Finland 614702 614712 10

Ireland 614702 614712 10

Lithuania 614702 614712 10

Latvia 352374 352384 10

Slovenia 352374 352384 10

Estonia 352374 352384 10

Cyprus 352374 352384 10

Luxembourg 352374 352384 10

Malta 265568 265584 16

Number of swings in the Nice voting game

Germany has as much power as The United Kingdom, France and Italy. Inthe weighted triple majority game v1 ∧ v2 ∧ v3 the difference is only 4 swingswith respect to 28 millions.


16 Strategic Power Revisited

Mika Widgren, and Stefan NapelTurku School of Economics, Finland

[email protected]

A lively debate about the adequate tool for measuring decision power inreal-life institutions such as the European Union has been sparked by Garretand Tsebelis (1999, G&T). Their verdict that power indices exclude variablesthat ought to be in a political analysis (institutions and strategies) and in-clude variables that ought to be left out (computational formulas and hiddenassumptions) has motivated many authors to defend traditional power indicesby clarifying the supposedly hidden assumptions underlying power formulas,and giving reasons for not taking institutions and strategies corresponding toexplicit decision procedures and rational, preference-driven agents into account.

Earlier attempts to take players’ preferences into power index calculationsinclude the Shapley-Owen modification of the Shapley-Shubik index (Shapley1977, Owen and Shapley 1989). It, however, still neglects the procedural as-pects of decision making. So, motivated by the recent debate, Steunenberg,Schmidtchen, and Koboldt (1999, SS&K) have proposed a new strategic powerindex (StPI) which explicitly takes both decision procedures and players’ pref-erences into account.

We disagree with G&T’s critique on many points and in particular see animportant role for a priori power analysis that does not rely on detailed knowl-edge of preferences on this and that policy issue. But, like SS&K, we see aneed to have adequate tools for measuring power if information on preferencesand procedures is available. Unfortunately, SS&K’s measure is not the rightanswer. It can yield extremely counterintuitive indications in rather basic ex-amples and, for instance, give a player negative power despite normalization.The latter makes sense once the P for power in the StPI is replaced by an Sfor success. Namely, on closer inspection, SS&K’s StPI turns out to be a usefulmeasure of average success, however not one of power.

We propose an alternative unified approach. It takes up G&K’s critique butstill allows for –sometimes merely convenient, sometimes desirable– a prioripower measurement. Our approach rests on a player’s marginal contributionor marginal impact on the collective decision taken a posteriori as the prim-itive of power, comparing an actual outcome with a shadow outcome whichalternatively could have been brought about by the considered player. Thisgeneralizes the concept of swings or pivot positions from cooperative gamesto non-cooperative games, and yields a measure of a posteriori power which issuited to take decision procedures and preferences into account. Having defineda meaningful measure of a posteriori power, meaningful a priori measures can


be constructed. A priori-ness of different degrees can be considered, concerningplayers’ preference-based or unmotivated actions, decision procedures, or both.By making corresponding distribution assumptions, one obtains traditional apriori power indices such as the Penrose index or Shapley-Shubik index as aspecial case. More generally, our measure defines power as a player’s expectedmarginal impact, where expectation is taken with respect to an appropriateprobability measure on (power-relevant) states of the world. Applications ofour measure to the European Union are then discussed.


Bankruptcy Games I

17 The TAL-family of rules for bankruptcy problems

Juan D. Moreno-Ternero and Antonio VillarUniversity of [email protected]

This paper analyzes a family of solutions to bankruptcy problems that gener-alizes the Talmud rule (T) and encompasses both the constrained equal-awardsrule (A) and the constrained equal-losses rule (L). We study the structural prop-erties of this family of rules and provide a characterization result that allowsus to identify the distinctive features of these three reference rules, T,A,L.

A bankruptcy problem describes a situation in which an arbitrator has toallocate a given amount E > 0 of a perfectly divisible commodity, referred toas the estate, among a group N of agents, when the available amount is notenough to satisfy all their claims (ci)i∈N . That is,

E ≤∑i∈N

ci.

A solution to a bankruptcy problem is a procedure or “rule” that exhibitssome desirable properties and determines, for each specific problem, an alloca-tion that satisfies two elementary restrictions: (i) No agent gets more than sheclaims nor less than zero; and (ii) The whole estate is distributed. Note thatmost rationing problems can be given this form. Since this is a well knownproblem and examples of these situations abound, we shall not insist on itsrelevance. The reader is referred to the works of Young (1994, ch. 4), Thomson(1995) and Moulin (2001) for a review of this literature.

There are four classical solutions to the bankruptcy problem: the propor-tional solution, usually associated with Aristotle, the constrained equal-awardsrule and the constrained equal-losses rule, that can be traced back to Mai-monides, and the Talmud rule, that extends the ancient “contested garmentprinciple” [see Herrero & Villar (2001) for a comparative analysis of these solu-tions]. The first three of those rules implement the idea of equal division, withdifferent reference variables (ratios, awards and losses, respectively). The Tal-mud rule is an alternative procedure that combines the principles that identifythe former three rules. It can be interpreted as implementing a protective crite-rion that ensures that all agents suffer a rationing that is “of the same sort” ofthat experienced by the whole society. The distribution procedure depends onwhether the estate E is above or below one half of the aggregate claim

∑i∈N

ci.It can be justified on the psychological principle of “more than half is like the


whole, whereas less than a half is like nothing”. Thus, this rule looks at the sizeof the awards when they are below half of the aggregate claim (E ≤ 1

2

∑i∈N

ci)and at the size of the losses above that magnitude (E ≥ 1

2

∑i∈N

ci).The TAL-family generalizes this idea by applying exactly the same principle

to all possible shares of the estate in the aggregate claim1 . That is, for any givenvalue of the parameter θ ∈ [0, 1], a rule in this family takes into account whetherE is above or below θ

∑i∈N

ci, and distributes the estate correspondingly. Therule associated with θ = 1

2is precisely the Talmud rule, as expected, and the

extreme values θ = 1 and θ = 0 correspond to the constrained equal awards ruleand the constrained equal losses rule, respectively. The proportional solution,however, is not part of this family.

We show that the TAL-family is made exactly of those rules which are con-sistent, continuous, claims monotonic (both in gains and losses), and satisfyeither independence of claims truncation or composition from minimal rights.In the firtst case we have θ ∈

[1

2, 1], whereas in the second one θ ∈

[0, 1

2

].

Moreover, the characterization result is tight. We also show that these rulesexhibit a precise duality relationship: the dual of the rule associated with theparameter θ is that rule associated with the parameter (1− θ).

1See Hokari & Thomson (2000) for a different extension of the Talmud rule.


18 An Experiment on Bankruptcy

Giovanni Ponti, Carmen Herrero and Juan D. Moreno-TerneroUniversity of [email protected]

We test the experimental properties of three well known solutions for bankruptcyproblems, that is, the constrained equal-awards, the proportional and the con-strained equal-losses rule.

To do this, we first let subjects play three games designed such that the“obvious” way to play coincides with one of these three rules. Moreover, we alsolet subjects play an additional game, that has the property that all (and only)strategy profiles in which players unanimously agree on the same rule constitutea strict Nash equilibrium. While in the first three games subjects’ play easilyconverges to the unique equilibrium rule, in the last game the proportional ruleoverwhelmingly prevails as a coordination device.


19 Additivity properties in bankruptcy problems and other relatedproblems

Gustavo Bergantiños and Juan José VidalUniversidade de [email protected]

In this paper we characterize the set of rules satisfying additivity on theestate and additivity on the estate and the claims in bankruptcy problems andother related problems. We study the discrete and the continuous problem.Moreover, new characterizations of the equal award, equal loss, and propor-tional rule are provided using these additivity properties.


Social Networks

20 Social Interaction in Anti-Coordination Games

D. López-Pintado, Y. Bramoullé, S. Goyal and F. Vega-RedondoUniversity of [email protected]

We study a setting in which individual players choose their partners as wellas a mode of behavior in 2 × 2 anti-coordination games. We characterize thenature of equilibrium network structures as well as study the effects of networkstructures on individual behavior. Our analysis shows that the structure of theequilibrium networks depends on the value of the cost of forming links. As thecost grows the structure is less connected and we get stable bipartite graphs.We also show that, the relative magnitude of the individuals doing each actionin the anti-coordination game depends crucially on the cost of forming links.If the costs are high, then there is a wide variety of stable networks, some ofthem being very asymmetric bipartite graphs. However, if they are low, only thecomplete and essential graph with a particular proportion of individuals doingeach action is stable.

In the past few years, there has been an extensive literature on social net-works which shows that the structure of interaction between individuals canbe decisive in determining the nature of the outcomes and, in particular, theplayers’ action choices in an underlying game. In much of this literature, thestructure of interaction is exogenously specified and the nature of the outcomeunder different specifications is examined (see e.g., Ellison, 1993; Goyal, 1996;Morris, 2000).

Recently, interest has grown in understanding the process through whichthe interaction structure itself develops. The earlier part of this literature (e.g.Jackson and Wolinsky (1996) or Bala and Goyal (2000)) has focused on con-texts where players choose links with others and there is no additional strategicdimension (i.e. there is no explicit game being played among connected play-ers). Later contributions, such as Jackson and Watts (2000) or Goyal andVega-Redondo (2000), have studied settings in which each agent plays a gamewith each of her ‘partners’ and therefore (in addition to connecting decisions)has to choose a mode of behavior in the accompanying game. This researchhas focused on a class of games where individuals have an incentive to choosethe same action as their partners; these games are referred to as coordinationgames. In the present paper, we wish to consider the role of network forma-tion in the polar case, where individuals prefer to choose an action unlike theaction chosen by their partners. We shall refer to these interactions as games


of anti-coordination1 . Many interesting situations can be suitably conceived inthis fashion, e.g. when the successful completion of a task requires that theindividuals involved adopt complementary actions (or skills), or when a mean-ingful interaction can only be conducted when the agents adopt different roles(say, buyers and sellers), or when in the contest for a certain resource (say,in a Hawk-Dove game), an optimal response is not to respond with the samebehavior (aggressive or peaceful) as one’s opponent.

We study the following model: We consider a model where individuals canform pair-wise links on their own, i.e., the link formation is one-sided. The play-ers initiating the link pay for the link. In addition to the links, each player alsochooses between two actions that he uses in the interaction with his partners.The payoffs to a players consist of returns in each of the pairwise interactionsless the costs of forming links.

In this setting, we first characterize the Nash equilibria. We find that thenature of the stable networks (complete, bipartite, semi-bipartite,...) dependscrucially on the range of cost values. If the costs are high, then very asym-metric bipartite graphs are stable, whereas if they are low only the completenetworks with a particular proportion of individuals doing each strategy arestable. This variety of equilibria motivates an examination of the dynamic sta-bility of different outcomes. Our analysis however demonstrates that all Nashequilibria are stochastically stable, i.e., they are robust to small but persistentperturbations.

Finally, we also study the efficiency of the different network structures. Weobtain that when costs are low, the equilibrium and efficient networks coincideonly under very specific payoffs configurations of the anti-coordination game.Notwithstanding, in general, equilibrium and efficient profiles are two disjointsets. When costs are sufficiently high we observe that, in general, there aresome efficient networks which are also equilibrium, but there is a wide range ofequilibria highly inefficient.

It is interesting to relate these results to the findings concerning coordinationgames. Goyal and Vega-Redondo (2000) show that the unique stochasticallystable interaction structure is the complete network (with every pair of playerslinked). Moreover, social conformism obtains, with everyone doing the sameaction. When costs are low, the risk dominant action is obtained in the stochas-tically stable outcome, whereas when costs are high it is the efficient action theone that arises in a stochastically stable state. The results of this paper suggestthat anti-coordination games have very different properties: a wide variety ofnetwork structures can arise and they have widely varying levels of efficiency.

1Bramoullé (2001) analyzes anti-coordination games played on a fixed structure. He showsthat anti-coordination games lead to much richer outcomes than coordination games. No-tably, equilibria are more sensitive to the structure.


21 Bidding for the surplus: Realizing efficient outcomes in generaleconomic environments

David Wettstein, Suresh Mutuswami and David Pérez-CastrilloBen-Gurion University of the Negev

[email protected]

In this paper, we consider two classes of economic environments. In thefirst type, agents are faced with the task of providing local public goods thatwill benefit some or all of them. In the second type, economic activity takesplace via formation of links. Agents need to both form a network and to decidehow to share the output generated. For both scenarios, we suggest a biddingmechanism, whereby agents bid for the right to decide upon the organizationof economic activity. The subgame perfect equilibria of this game generateefficient outcomes.


22 Sequential formation of coalitions through bilateral agreements

Ines Macho-Stadler, David Pérez-Castrillo and Nicolas PorteiroUniversitat Autonoma de Barcelona

[email protected]

In many cases, it is very difficult for big coalitions to form and the playersonly have the possibility of forming small coalitions. However, this does notmean that once some coalitions are formed, they cannot decide to continuewith the process of forming bigger entities. The banking sector provides severalexamples of such a sequential process of mergers. In Spain, the now called SCHis the outcome of the merger of the Banco de Santander with the Banco CentralHispano, which in turn was issued of the merger of banks Central and Hispano.Similarly, banks Bilbao and Vizcaya first merged to form the BBV and thenthis new entity merged with the Banco Argentaria to form the BBVA.

We model the formation of coalitions as a sequential process in which, ateach moment in time, only two existing coalitions can decide to merge. Westudy the subgame perfect equilibria of such a game. In particular, we considera market where identical firms facing linear demand compete a la Cournot. Ateach period, firms take decisions on quantity. To concentrate our analysis onthe incentives to form coalitions, we assume that production is a short-termdecision. Also, at each period, two randomly chosen coalitions in the existingpartition can merge. A merger means forming a cartel where the decision onthe total level of production by the partners is made jointly. The decision onthe merger is made taking into account the long-term profits.

As Salant, Switzer, and Reynolds (1983) pointed out, two firms (or coali-tions) will not be interested in merging if they only consider the present periodprofits and there are at least three firms (coalitions) in the industry. Theirresult extends easily to our model: If the firms’ discount rate is low enough,they will not merge at any period in the unique Markov perfect equilibrium ofthe game. Hence, all firms will stay as singlentons.

The situation when firms are forward looking is more interesting. In thiscase, firms may want to merge even if they lose profits in the short run. Weprove that, when the firms’ discount factor is high enough, that is, when firmsare patient enough, and there are enough firms in the industry, then the finaloutcome of any subgame perfect equilibria is the grand coalition. The firmswill form coalitions sequentially so that they end up all together. Moreover,in the process of forming coalitions, the firms will accept some of the mergersand will reject others. We will characterize the paths (the coalition structures)that the firms will follow in order to surely arrive to the grand coalition.


The fact that the grand coalition can result as the equilibrium of a game ofcoalition formation in a linear Cournot model is in contrast with other resultsin the literature of coalition formation. Interestingly, it is the fact that only apair of coalitions can merge at each period (so bigger coalitions cannot form)which allows reaching the grand coalition. Even if this restriction to bilateralagreements is ex-ante bad for the formation of coalitions, since it reduces thechoice set of the firms, it ex-post results in an impulse to the process of merging.The reason is that the bilateral nature of the negotiation induces the coalitionsto be formed gradually, in such a way that the incentives of all the players canbe made simultaneously compatible.


Cooperative Games I

23 Efficiency in Uncertain Cooperative Games

Hans Peters, Walter Bossert and Jean DerksDepartment of Quantitative Economics, University of Maastricht

[email protected]

A contingent contract in a transferable utility game under uncertainty spec-ifies an outcome for each possible state. It is assumed that coalitions evaluatethese contracts by considering the minimal possible excesses. A main questionof the paper concerns the existence and characterization of efficient contracts.It is shown that they exist if and only if the set of possible coalitions contains abalanced subset. Moreover, a characterization of values that result in efficientcontracts in the case of minimally balanced collections is provided.


24 The max-Core of Cooperative TU Games

Francesc Llerena, Josep M. Izquierdo and Carles RafelsUniversitat Rovira i Virgili and Universitat de Barcelona

[email protected]

One of the main goals of cooperative game theory is to describe fair methodsto allocate the join profit arising from cooperation among agents. A preliminarynormative task of the theory is to describe which are the possible allocationvectors according to some criteria related to equity. In this sense, we can men-tion the core, the bargaining sets or the stable sets. All these solutions proposedistributions that may be accepted simultaneously by all players. Neverthe-less, sometimes decisions regarding payoff allocations are not taken as a singledecision but a sequence of decisions. In this work we introduce the idea ofsequential payoffs in set-solutions concepts defined for TU cooperative games.

The concept of a sequential payoff scheme has been already used in the anal-ysis of point-solutions concepts for TU games. First of all, recall the classicalinterpretation of the Shapley value as the vector of expected marginal contri-bution of players. As it is known, this value is the average of all marginalworth vectors associated to different ordering of players. Each of these vectorscan be interpreted as a sequential assignment decision where each player re-ceives his marginal contribution to the coalition formed by his predecessors inthe ordering and by himself. At this point it is interesting to cite the paperof Nuñez and Rafels (1998) introducing the so called reduced marginal worthvectors where the marginal contribution of players is evaluated with respect tosuccessive reduced games. The second reference to sequentiality could be foundin the analysis of priority for rationing problems. In this context there is a re-cent work by Moulin (2000) where priority rules are analyzed and axiomaticallycharacterized. Priority implies a sequential assignment process where no unitis assigned to a player until the claims of players with higher priority are metin full. Potters and Sudholter (1999) have analyzed point-solution concepts forthe class of airport games. Their analysis is based on an axiom called first-player consistency and the sequentiality appears implicitely as the criterion fordetermining the payoff to the first player and the application of a sequentialconsistency determine iteratively the whole payoff vector.

In the above mentioned works, thet authors have mainly paid attention topoint-solution concepts. In this paper we will focus on the sequentiality appliedto set-solution concepts. Our starting point will be the core of a game.

To set ideas, assume that (N, v) is an arbitrary game and fix an ordering σof N . Take a distribution x in the core of the game. By core‘s definition, itholds that the payoff of any player, and in particular the first-player in N withrespect to σ, is in between his individual worth and his marginal contribution


to the grand coalition. Formally,

v(σ(1)) ≤ xσ(1) ≤ v(N)− v(N\σ(1)). (1)

In this sense, we will say that x meets the first-player marginally with respectto σ. As we’ve said before analogous bounds also holds for the rest of players.Nevertheless, this bounds could be sharpened if the payoff for player σ(1) hasbeen announced in advance. If this happens, the relevant reference game forsetting these bounds is not the original game but the so called reduced game.Being fixed the payoff for some players (we say that these players are in factout of the game), reducing the game means that players who are still in thegame reevaluate his worth not only taking into account potential subcoalitionsamong themselves but coalitions that might include players out of the game,and having in mind that these players will claim they have been promised.

The first system of reduction was historically introduced by Davis and Maschler(1965). If we take this reduced game, it is easy to check that, having fixed σ,if (N, v) is a balanced game then any core element x satisfies the followinginequalities:

vFσ

k

x (σ(k)) ≤ xσ(k) ≤ vFσ

k

x (Fσk )− v

Fσ

k

x (Fσk \σ(k)), (2)

where Fσk

:= σ(k), . . . , σ(n), k = 1, . . . , n and vFσ

1

x denotes the reducedgame (a la Davis and Maschler) of the game (N, v) relative to Fσ

kand x, with

vFσ

1

x ≡ v.Notice that the payoff of players and their bounds are obtained sequentially

starting from the original game and following the reduction process as well asplayers are given their payoff. Then, if a vector x verifies the inequalities listedin (2) we say that this vector satisfies the marginality property. This propertycharacterizes the core of the game.

This result allow us to reinterpret the core from the sequential point ofview as the set of those payoff satisfying a suitable sequentiality property andmarginality property at the same time. Following this scheme, we can definea family of set-solution concepts using different reduced games. In this sense,we will use a family of reduced games inspired on the original idea of Davisand Maschler. The subset of vectors meeting this characteristic is called themax-Core set (with respect and specific reduced game). We characterized ax-iomatically the max-Core and study geometric properties. As a particular casewe obtain two new axiomatic characterization of the core.


25 Values of interior games

J. M. Bilbao, A. Jiménez-Losada, C. Chacón and E. LebrónUniversidad de Sevilla

[email protected]

We introduce and study a new family of cooperative games called interiorgames. Interior games are additive games restricted by antimatroids, that allowto focus of the same way interesting problems.

An antimatroid over a finite ground set N is a pair (N,A) , where A ⊆ 2N

is a collection of subsets with ∅ ∈ A that verifies the following two properties:(Accesibility) For any S ∈ A, S = N , there exists an element e ∈ N \ S withS ∪ e ∈ A; (Union) For all S, T ∈ A happen S ∪ T ∈ A. We name feasiblesthe sets in A and we will consider only normal antimatroids, i.e., for all e ∈ Nthere exists S ∈ A such that e ∈ S. With an antimatroid (N,A) we associatethe interior operator int : 2N → A which assigns to any subset S ⊆ N theunique maximal feasible set that the elements of S can form among them

int (S) =⋃

T∈A:T⊆S

T.

Now, we introduce some concepts about antimatroids, which can be found inKorte, Lovász and Schrader (1991). A feasible set S is a path if it has only oneendpoint (e ∈ S such that S \ e ∈ A). If e is this unique endpoint of S weuse e-path. It is known any set is feasible if and only if is the union of paths, inparticular int (S) is the union of the paths contained in S. The set of e-paths in(N,A) is denoted by A(e). Elements e ∈ N such that e ∈ A (A (e) = e)are called atoms. For those e ∈ N that satisfy N \ e ∈ A we will use coatoms(in fact, N \ e is the coatom). Two fundamental classes of antimatroids arecoatomic antimatroid (where all e ∈ N is a coatom) and poset antimatroids(each e ∈ N has only an e-path). The dual structure S ⊆ N : N \ S ∈ A ofan antimatroid A, is a set system called convex geometry. The interior operatorof an antimatroid is the dual operator of the closure operator in its dual convexgeometry, if S ⊆ N we have S = N \ int (N \ S).

Given an antimatroid (N,A), a vector w ∈ RN+ such that w ≥ 0, let us

consider w(S) =∑

e∈S we for all nonempty S ⊆ N , w (∅) = 0. Then weintroduce the interior game (N,wA) as a cooperative game defined by

wA(S) = w(int(S)) for all S ⊆ N.

These game supposes that the best feasible coalition for the players of S is itsinterior. We have proved that interior games are monotonic, superadditive andtotally balanced games, but not convex games in general.


• An interior game (N,wA) is a convex game if and only if we = 0 for alle ∈ N such that |A(e)| ≥ 2.

• Fixed an antimatroid (N,A) , every interior game (N,wA) is convex ifand only if (N,A) is a poset antimatroid.

The core of an interior game (N,wA) can be calculate only considering thefeasible coalitions, that is,

Core (wA) =x ∈ R

N

+ : x (N) = w(N), x(S) ≥ w(S) for all S ∈ A.

There exist interesting examples of interior games: Games on permissionstructures (Gilles, Owen and van den Brink 1992, and van den Brink 1997),Peer-group games (Brânzei, Fragnelli and Tijs 2000), Big boss games (Muto,Nakayama, Potters and Tijs 1988), Clan games (Potters, Poos, Muto and Tijs1989), Information market games (Muto, Potters and Tijs 1989). Also, we in-troduce modified clan games where the clan players can form feasible coalitionsamong them. The dual game of an interior game is a closure game on the dualconvex geometry L, defined by wL (S) = S for all S ⊆ N . An example ofclosure game is given by Meggido(1978).

We show that solutions of any interior games (N,wA) can be expressedin terms of the paths of the antimatroid A. We consider the probabilisticvalues given by Shapley and Banzhaf , and the compromise value given byTijs. Two sets are basic in the structure for a player e ∈ N, P e =

⋃T∈A(e) T

and Pe =⋂T∈A(e) T. The main results about these values are:

• The Shapley and the Banzhaf values satisfy, for every e ∈ N ,

She (N,wA) =∑

e′∈N:e∈Pe′

∑S e′-block:e∈S

λSe′

|S|

we′ ,

Bae (N,wA) =∑

e′∈N:e∈Pe′

∑S e′-block:e∈S

λSe′

2|S|−1

we′ .

We define a e-block as a feasible coalition union of e-paths. The numbers λSefor a e-block are obtained by recurrence

λSe =

me,S∑p=1

(−1)p−1 ne,Sp = 1−∑

TS:T e-block

λTe , where

ne,Sp = |S1, ..., Sp e-paths : S1 ∪ ... ∪ Sp = S| ,

me,S = |e-paths in S| ,


and λSe = 1 for e-paths.

• In particular, if (N,A) is a poset antimatroid then

She (N,wA) =∑

e′∈N:e∈Pe′

we′

|P e′ |;Bae (N,wA)

=∑

e′∈N:e∈Pe′

we′

2|Pe′ |−1

.

The Tijs value is the unique efficient vector of the line determined by theupper and the lower vectors. We have that e = e′ : e ∈ Pe′ .

• The upper and the lower vector of an interior game are, for all e ∈ N :

Mτ,wAe = w

(e

)and mτ,wA

e = max0, we − ce where

ce = minT∈A(e)

∑e′∈T\e

[w(e′

)− we′

].

• The gap function of an interior game is

gwA (S) =

∑e∈S w

(e

)− we S ∈ A,

gwA (int (S)) +Mτ,wA (S \ int (S)) S /∈ A.

• If (N,A) is a coatomic antimatroid then τ (N,wA) = w.

• If for all e with |e| > 1 we have |Pe| = 1, then the interior games aresemiconvex games and

τ e (N,wA) = (1− pA)w(e

)+ pAw (int (e))

where

pA = (w (A)− w (N)) (w (A)− w (a (A)))−1

,

a (A) is the atom set and w (A) =∑

e∈N w(e

).

We understand a e-path as a dependency of e with the rest of the path.Hence, if e′ ∈ P e then e has one dependency with e′, anf if e′ ∈ Pe then e′

controls e. The Shapley and the Banzhaf values consider that a player obtainsprofit by every player with some dependency of him, and the Tijs value onlyconsiders the players controled by him.


References

Algaba, E., Bilbao, J. M., Brink, R. van den, and Jiménez-Losada, A. (2000)Cooperative games on antimatroids, CentER Discussion Paper No. 124,Tilburg University.

Brânzei, R., Fragnelli, V. and Tijs, S. (2000) Tree-connected peer group situa-tions and peer group games, CentER Discussion Paper No. 117, TilburgUniversity.

Brink, R. van den (1997) An axiomatization of the disjuntive permission valuefor games with a permission structure, Inter. J. Game Theory 26, 27—43.

Gilles, R. P., Owen, R. and Brink, R. van den (1992) Games with permissionestructures: the conjuntive approach, Inter. J. Game Theory 20, 277—293.

Korte, B., Lóvasz, L. and Schrader, R. (1991) Greedoids. Springer-Verlag,Berlin.

Meggido, N. (1978) Computational complexity of the game theory approachto cost allocation for a tree, Math. Oper. Res. 3, 189—196.

Muto, S., Potters, J. and Tijs, S. (1989) Information market games, Intern.J. Game Theory 18, 209—226.

Muto, S., Nakayama, M., Potters, J. and Tijs, S. (1988) On big boss games,Economic Studies Quarterly 39, 303—321.

Potters, J., Poos, R., Muto, S. and Tijs, S. (1989) Clan games, Games andEconomic Behavior 1, 275—293.


Dynamic Games

26 Competition and cooperation in commercial fishing with adap-tive expectations

G. I. Bischi, M. Kopel and F. SzidarovszkyIstituto di Scienze Economiche

University of Urbino, I-61029 URBINO (Italy)[email protected]

Since the pioneering work of Gordon (1954), many bioeconomic models forthe description of the commercial exploitation of common property renewableresources, such as fisheries, stressed the problem known as “the tragedy ofthe commons” (Hardin, 1968, see also Clark, 1990). This problem can bebasically identified with a typical prisoner’s dilemma, as “aggressive harvesting”strategies of all the players lead to severe depletion of the resource and, as aconsequence, to lower profits for all the competitors, whereas an agreement ona sustainable exploitation leads to higher profits for all (see e.g. Mesterton-Gibbons, 1993).

Cournot oligopoly games are now a common tool in the description of fisherymanagement, see e.g. Levhari and Mirman, 1982, Okuguchi, 1998, Szidarovszkiand Okuguchi, 1998. In these games, the harvesting decisions of profit maxi-mizing agents are characterized by strategic interactions related not only to theinfluence of total supply on the market prices, but also to cost externalities dueto the dependence of harvesting cost on available fish stock, whose evolution isgoverned by biological laws as well as harvesting activity.

Following Szidarovszky and Okuguchi (1998, 2000) we propose an oligopolymodel where n players (that may represent n different countries fishing in thesame sea) sell the harvested fish in n markets, i.e. each player sells in his homemarket and in (n − 1) foreign markets. Moreover, we assume that agents donot have perfect knowledge of the fish stock. This differs from the currentapproach in resource economics literature, where it is usually assumed thatfishermen, when they decide their optimal harvesting, have precise knowledgeof the environment. Instead, we guess that in the real world this will hardlybe the case and, more realistically, these economic agents have access to acollection of past data about the harvested amounts of fish and some otherindicators of the size of the fish population, from which they try to derive anestimate of the future fish stock. With every new piece of information, theirestimate will be updated and used to determine the future harvesting activities.In our model we assume a simple learning rule called adaptive expectations : atthe end of each period, using past observations about harvests and fish stock,fishermen are able to derive the exact stock of the resource, and then they


make a prediction about the future fish stock by forming a weighted averagebetween the current fish stock and their previous prediction (see Bischi andKopel, 2002).

We consider two different models: in the first one we assume that each playermaximizes his own expected profit, as in a typical non cooperative oligopolisticcompetition; in the second one we assume that the players form a cooperativeventure, where each player determines its harvesting activity such that the jointprofit of all players is maximized. We compare the Nash equilibria obtained inthese two cases, and we examine their stability properties under the assumptionthat the game is played repeatedly by boundedly rational players (due to thelimited knowledge of the fish stock and the related adaptive learning process).In both the models we consider the effect of a new player which enters thegame.

The results obtained confirm that cooperation does not only lead to higherconservation, but the larger stock eventually leads also to a higher total harvest(due to increased growth of the resource) like in the above mentioned prisoner’sdilemma. However, unexpected results are obtained about the global dynamicproperties. In fact, from the point of view of the mathematical structure ofthe dynamic model, when we replace the assumption of fishermen with per-fect foresight about the next period fish stock by the weaker assumption ofadaptive expectations, a one-dimensional dynamical system is replaced by atwo-dimensional one, and this implies that remarkable differences may ariseconcerning the role of the initial conditions, due to more complex structuresof the basins of attraction and to the occurrence of global bifurcations whichchange their topological properties. The results obtained are characterized bycomplex dynamic patterns and bifurcation phenomena, because the interac-tions of human economic decision making with ecological dynamics are highlynon linear (on this point see also Rosser, 2001). Moreover, in our model thecomplexity of the basins of attraction is related to the fact that the dynamicevolution of the game is obtained through the repeated application of a two-dimensional noninvertible map (see Mira et al, 1996, Puu, 2000, Agliari et al.,2002).

The main problems addressed in our work are:

(i) the influence of the market structure and the role of strategic effects on theoligopoly evolution, both in the non cooperative and in the cooperativecase;

(ii) the role of harvesting costs in the problem of extinction or conservationof the resource;

(iii) the influence of errors in predicting the fish stock when determining theharvesting activities.


References

A. Agliari, G. I. Bischi and L. Gardini (2002) Some methods for the GlobalAnalysis of Dynamic Games represented by Noninvertible Maps. Chapter3 in Oligopoly and Complex Dynamics: Tools & Models, T. Puu and I.Sushko editors, Springer Verlag, forthcoming.

G. I. Bischi, M. Kopel (2002) The Role of Competition, Expectations and Har-vesting Costs in Commercial Fishing. Chapter 3 in in Oligopoly and Com-plex Dynamics: Tools & Models, T. Puu and I. Sushko (Eds.), SpringerVerlag, forthcoming.

C. W. Clark (1990), Mathematical Bioeconomics, 2nd edition, Wiley Inter-science.

H. S. Gordon (1954) The economic theory of a common property resource:the fishery, Journal of Political Economy, 62, 124-142.

G. Hardin (1968) The tragedy of the commons, Science 162, 1243-1247.

D. Levhari, L. J. Mirman (1982) The Great Fish War: An Example Using aDynamic Cournot-Nash Solution. In Essays in the Economics of Renew-able Resources, L.J. Mirman and D. F. Spulber (Eds.), North-Holland,243-258.

M. Mesterton-Gibbons (1993) Game-theoretic resource modeling, Natural Re-source Modeling 7, 93-147.

C. Mira, L. Gardini, A. Barugola, J. C. Cathala (1996) Chaotic Dynamics inTwo-Dimensional NonivertibIe Maps, Wold-Scientific, Singapore.

K. Okuguchi (1998) Long-run fish stock and imperfectly competitive interna-tional commercial fishing, Keio Economic Studies, 35, 9-17.

T. Puu (2000) Attractors, Bifurcations, and Chaos, Springer-Verlag.

J. B. Rosser, Jr. (2001) Complex ecologic-economic dynamics and environ-mental policy, Ecological Economics, 23-37.

F. Szidarovszky and K. Okuguchi (1998) An Oligopoly model of commercialfishing, Seoul Journal of Economics, 11, 321-330.

F. Szidarovszky and K. Okuguchi (2000) A Dynamic Model of InternationalFishing, Seoul Journal of Economics 13, 471-476.


27 An Evolutionary Game Approach to Fundamentalism and Con-flict

Daniel G. Arce and Todd SandlerRhodes College and University of Southern California

[email protected]

This paper investigates the evolutionary equilibria of a clash of cultures gamewhere conflict results from failures to share social power in individual pairings.The model is based on a adaptation of Skyrm’s [1996] analysis of the evolutionof social justice within the ultimatum game. Members of a general populationare matched with those of a fundamentalist population, the latter being morecohesive and insistent that its identity traits define the norms for, and outcomesof, social, economic, and political interaction.

The underlying replicator dynamics are found to be a function of in the ini-tial extent of moderate behavior within the general population, and the levelof cohesion and intolerance within the fundamentalist population. Simulationsof the replicator dynamics reveal a tradeoff between the intolerance of funda-mentalism and the likelihood of a takeover. This tradeoff is reversed if fun-damentalism is falsifiable: affording non-fundamentalists the ability to signalfundamentalist traits produces a bandwagon effect.


28 A Model of Evolutionary Drift

José Ramón UriarteUniversidad del País Vasco-Euskal Herriko Unibertsitatea

[email protected]

It is common place to observe that the equilibrium selected by a theorydepends on the manner in which perturbations are handled (see, for example,Selten’s (1975) perfect equilibrium and Myerson’s (1978) proper equilibrium ).Binmore et al.(1995) and Binmore and Samuelson (1999) also emphasize theimportance of perturbations, but they place these in the dynamic process thattakes the players to equilibrium rather than perturbing the game itself (likeSelten or Myerson). It is in modelling such perturbations realistically that thepresent paper is concerned.

The important work of Binmore and Samuelson (1999), –B & S, from nowon–, studies the limiting behaviour of perturbed systems, but little insight isgiven into what perturbations one should expect. The implicit message of B &S is the need of an explicit microeconomic model of psychologically observedhuman choice behaviour that originates the drift that perturbs a selection dy-namics. B & S have opened a possible application for perception theory witthe introduction of a decreasing and Lipschitz continuous drift function in theexpected payoffs differences (Assumption 4). This means that when the pay-offs at stake increase, the player’s efforts for a better perception of the gameare increased and as a consequence of his more careful moves the mistakes willdecrease (the assumption is related to Myerson’s (1978) proper equilibrium).Perception theory is precisely at the heart of our model of drift. The modelderives explicitly from the similarity theory developed first by Tversky (1977)in psychology and later applied to choice theory by Rubinstein (1988) andAizpurua et al. (1993).

The present paper extends the theory developed in Uriarte (1999), by build-ing a similarity compatible choice model valid for a dynamic setting. We shalluse the same methodology as B & S to study the stability properties of Nashequilibria specifying different out-of-equilibrium path appearing in connectedcomponents of stationary states. Thus, we start with a continuous, determin-istic selection dynamic model (in the present paper, the model will be thepopular replicator dynamics model,- RD). Then, for a better approximation toan underlying stochastic strategy-adjustment process that governs the players’behaviour, the selection model will be completed by adding perturbations thatincorporates some of the real-life imperfections or “anomalies” of the humanchoice behaviour which are excluded by the model. These appear in the modelas drift. In the present paper drift is not derived from the behaviour of agentswho misread the game.


We assume a society composed of agents who use different strategy choiceprocedures. Thus, a new type of agents, the Sigma procedural agents, is addedto the population of agents whose behaviour leads to the RD (such as thepopulation of agents whose strategy-adjustment process is governed by, say, theaspiration-based model of learning of Binmore, Gale and Samuelson (1995)).Drift arises from the strategy-adjustment process governing the behaviour of theSigma agents. In that process, the Sigma agents inject continuously strategiesthat are not currently played by the population (this is why we keep using thewords drift and “mistakes” in this paper). Therefore, the Sigma agents willperturb the strategy-adjustment process of the Binmore, Gale and Samuelsonagents (the BGS agents). The resulting perturbed replicator dynamics will havelarge player populations and each population will be composed of two equallylarge subpopulations of agents types, the BGS agent type and Sigma agenttype. We shall see below that the model does not deal with type proportionsor mixtures of population types.

The most important innovation introduced in this paper is related to animportant feature of decision-making procedures excluded by the selection dy-namic model. In contrast to one-shot games, when the frequency with whicha game is played is very high, the experience and learning opportunities andthe information for a decision-maker are increased. As a consequence, decisionscould be taken by considering payoffs to his current strategy and the changes inthe proportions of his own population playing the same current strategy as his.Both variables are indicators of how well one is doing in a game. But, naturalas it seems that an agent may decide by having in mind, among other things,how many agents are playing exactly like him, deterministic selection dynamicmodels do not take into account this feature of decision-making and considerpayoffs as the only indicator considered by the agents to take decisions on thestrategy space. The Sigma agent’s strategy choices and the drift introduced byhim depend on two variables, expected payoffs (as in B & S’s Assumption 4)and population proportions attached to his current strategy. But let us clarifythe role of expected payoffs in the present model of drift. The use of strategyproportions allows us to establish a relationship, crucial for our results, be-tween those proportions and the subjective feeling of how well is one playingthe game.

We assume that a Sigma agent has information about the proportion in hisplayer population playing the strategy he is currently playing. He uses thatinformation to evaluate his play. Even though expected payoffs are relevantfor the “noisy” decision-maker, it happens that the results of the present paperare obtained under extreme sensitivities to the strategy frequencies and henceextreme drift levels. In those cases, Sigma agents’ strategy-adjustments are ledby non-payoff factors. Two extreme playing modes are considered. There areSigma agents who, maybe due to their experience or to the set of values they


are honoring, know with absolute certainty that they are playing the correctstrategy, or the strategy they ought to play. This is the alert playing mode.Under this mode, an agent will keep on playing the same strategy at eachstage. But there are other agents led by a kind of complete confusion derivedfrom the absolute insecurity about how well are playing the game. This isthe absent mode; under this mode, an agent keeps endlessly drifting from onestrategy to another, without taking care about the payoffs obtained at eachstage. Hence, it is clear that in those modes the Sigma agents are not expectedpayoffs maximizers, payoffs do not play any role in their strategy-adjustmentbehaviour and in the selection of equilibria.

This model of drift shows that there exists a trade-off between the stabilityproperties of the components and the sensitivity of drift to either expectedpayoffs (as in B & S) or to strategy frequencies. We show that the less sensitiveis drift to payoffs the stronger are the stability properties of the components.Note, as we mentioned above, that the inclusion of drift is to obtain a morerealistic model of how society evolves and coordinates on some equilibria andnot in others. But we are not arguing that the

∑procedural agents is a better

model of bounded rationality than the RD. Both models complement eachother and what matters are the outcomes obtained by the joint behaviour ofthe BGS and the Sigma agents. With respect to B & S’s results, extensions areobtained in two directions. The introduction of Sigma agents in the selectiondynamics has a great dynamic impact. First, we show that this model of drifthas more cutting power; stability in components of Nash equilibria is achievedindependently of the size of the component. Asymptotic stability is achievedeven when components have empty interior. Second, it is known that the rangeof equilibria that can be obtained with the B & S model does not include thosethat are observed in the experiments of Roth and Erev (1995) with the FullUltimatum Game (see Binmore et al. (1995)).

With the present model, the equilibria not only are consistent with theexperimental data, but the model predicts that they are independent of theobserved initial propensities. Behind these results are the extreme sensitivityof the Sigma agents to strategy frequencies. Highly sensitive agents producepowerful drifts that can dominate the strong selection forces operating outsidea component, pushing the system to its interior and stabilizing any equilibriuminside. From these results, we concludethat the important factor excluded inthe selection dynamics is how each agent reacts to the proportions of his playerpopulation playing his current strategy, rather than the sensitivity of drift toexpected payoff differences.

We show a sufficient condition to stabilize a pure strategy equilibrium in acomponent of any size. The condition says that each equilibrium strategy mustbe played in the alert mode and the rest of strategies should be played in theabsent mode. We show that under this condition, the observed equilibria in the


Full Ultimatum Game (Roth and Erev (1995)) are obtained as global asymp-totic attractors. Thus, the model predicts that those equilibria are independentof the initial propensities observed in the laboratory. To our knowledge, theindependence of the outcomes from the initial play in the Ultimatum Game hasnever been tested. Hence, we propose an experiment to address this questionand test the prediction of the model. From the experiment we would know howrealistic are the assumptions built in the theory of Sigma agents and the roleof drift in the equilibrium selection.


Games with Information

29 Market Power and Information Revelation in Dynamic Trading

Roberto Serrano and Piero GottardiBrown University

[email protected]

We study a strategic model of dynamic trading where agents are asymmetri-cally informed over common value sources of uncertainty. There is a continuumof uninformed buyers and a finite number of sellers, some of them informed.

When there is only one seller, full information revelation never occurs inequilibrium and the only information transmission happens in the first period.

The outcome with n sellers depends both on the structure of sellers’ informa-tion and the intensity of competition among them allowed by the market rules.We show that the latter plays an even more important role. With intense com-petition (absence of clienteles), information is fully and immediately revealedto the buyers in every equilibrium for n large enough, both when all sellersare informed and when only one seller is informed. On the other hand, witha less intense form of competition (presence of clienteles), collusive equilibria,where information is never revealed, also exist, whatever the number of sellers.Moreover, when only one seller is informed, for many parameter configurationsthere are no equilibria with full information revelation, for any n.


30 The Principal-Agent Matching Market

David Pérez-Castrillo and Kaniska DamUniversitat Autonoma de Barcelona

[email protected]

A large set of literature contributing to the theory of incentives analyzesoptimal contracts in principal-agent relationships when there exist asymmetriesof information. When this asymmetry concerns an action, or a decision to bemade by an agent, a moral hazard problem emerges. Several works analyzeoptimal contracts when only one principal and one agent interact, including theseminal works by Pauly (1968), Mirlees (1976), and Harris and Raviv (1978).The principal-agent contracts involve the provision of incentives and typicallylead to inefficiency due to the informational asymmetry.

The main goal of this paper is to propose a useful framework to analyzethe relationship between each principal-agent pair not as an isolated entity butas a part of an entire market where several principals and agents interact. Inthis framework, the utilities obtained by each principal and each agent aredetermined endogeneously in the market. This allows us to improve over theprevious approach where the agents’ utilities are exogeneously given and theprincipals assume all the bargaining power. We consider the simultaneousdetermination of the identity of the parties who meet (i.e., which agent iscontracted by which principal) and the contracts they sign in an environmentwhere each relationship is subject to moral hazard.

We model the principal-agent economy as a two-sided matching game. Anoutcome of this economy is an endogenous matching and a set of contracts,one for each principal-agent pair under the matching. Roughly speaking, anoutcome is said to be stable if there is no individual or no relevant pair ob-jecting the existing outcome. The paper studies the set stable outcomes ofthis principal-agent matching market. In particular, we consider an economywith several identical principals and several agents differentiated only with re-spect to their initial wealth. A pair of individuals, one principal and one agent,can enter into a relationship by signing a contract. This contract specifies thecontingent payments that are to be made by the agent. Also it sets the levelof investment, which together with a non-verifiable effort made by the agent,determines the probability of having a high return from the project the agentoperates on. The initial wealth of the agent may not cover the amount to beinvested and hence, the wealth differences imply differences in liability.

We begin by providing a complete characterization of the set of stable out-comes of the principal-agent economy. The first simple property we prove isthat all the principals earn the same profit in a stable outcome. In particular, if


the principals constitute the long side of the market, their profits are zero. Thesecond feature is that the contracts offered in a stable outcome are Pareto effi-cient, i.e., it is not possible to increase the utility level of the principal withoutmaking the agent strictly worse-off. More interestingly, in a stable outcome,the matching itself is efficient, in the sense that it is the one that maximizesproductive efficiency. For example, if the agents are in the long side of themarket, only the wealthier ones, i.e., the more attractive ones (and all the prin-cipals) are matched. Third, the productive efficiency of a contract signed ina stable outcome increases with the wealth of a matched agent. That is, thericher the agent, closer is his contract to the first-best. The additional surplusgenerated due to this increase in efficiency accrues to the agents. Finally, thecontracts signed in a stable outcome of this economy are more efficient thanprincipal-agent contracts, i.e., the contracts signed when the principals assumeall the bargaining power.

The previous characteristics of the set of stable outcomes have very relevantpolicy implications when applied to particular environments. For example, con-sider an economy where landowners (principals) contract with tenants (agents)who are subject to limited liability. Suppose that the government would liketo improve the situations of the tenants by endowing the agents with some ad-ditional money. Our analysis suggests that the government will be interestedin creating wealth asymmetries among tenants since otherwise, the landownerswould appropriate all the incremental surplus intended to the tenants.

We establish a close relationship between the concept of stability and thatof a competitive equilibrium. From the point of view of matching theory, onecan see our model as a generalization of the assignment game with several buy-ers and sellers described by Shapley and Shubik (1972). We further consolidatestability as a reasonable solution concept for this principal-agent matching mar-ket by proposing a simple mechanism in which each of the agents proposes acontract and each principal chooses an agent. We show that the equilibriumoutcomes of this mechanism coincide with the set of stable outcomes of thematching market.


31 Achieving Efficient Outcomes in Economies with Externalities

Nicolás PorteiroU. de Alicante and U. Autónoma de Barcelona

[email protected]

In real life, when a group of agents negotiate to try agree on a collectivedecision, there are two features that are usually present. One of them is theexistence of externalities in the actions of the players. In many cases, thenegotiations do not only deal with the sharing of a fixed-size prize, but it isprecisely the behavior of each agent what endogenously determines the overallamount of resources available to share. The other common characteristic ofmost negotiation processes is the possibility of renegotiation in case of failure.Usually, when agents start to negotiate they know that if they do not succeed,they can retry again in the future.

In this paper I propose a negotiation protocol that tries to incorporate thesetwo features: the presence of externalities and the possibility of renegotiation.The objective is to design a set of rules such that, when the agents behavenon-cooperatively, the outcome of the negotiation process is efficient and theprofits are splitted in an equitable way.

The outline of the mechanism is the following: the negotiation starts withan auction that allocates the right to have the initiative in the negotiation. Ina posterior stage, the winner proposes an allocation that is implemented if therest of the players unanimously accept it. Finally, in case of a rejection theprocess is started again.

I study the outcome of the Stationary Subgame Perfect Equilibrium of thegame played according to the rules presented above. I show that the uniqueoutcome generates an efficient choice of the good (or goods) with externaleffects, together with a distribution of the profits based on the principle ofequal sharing of the surplus.

The mechanism is proposed for those environments where the agents thatnegotiate do not face a situation of asymmetric information. The reason for thisassumption is two-fold. On the one hand, it is reasonable to deal with economicinteractions among agents who have repeatedly met before in a market. Forexample, the negotiations among the members of the European Union, wherethe countries concerned have been jointly negotiating over several issues in thepast. In such environments, it can be a good approximation to rule out theexistence of private information. On the other hand, by restricting the analysisto situations of symmetric information, we can avoid the problem of strategy-proofness and concentrate on other properties of the mechanism like efficiencyand budget balancedness.


The mechanism I propose has some features that are worth mentioning. Thefirst one is the simplicity of the rules; this adds credibility to the results, spe-cially when one tries to compare them with the outcome of real-life negotiationprocesses. Another characteristic is that the process is completely autonomous:once the players know the rules, the mechanism designer does not need to in-tervene in the negotiation. Moreover, the payoff of the players is not random.Hence, the distributive objectives are attained deterministically and not in ex-pected terms. Finally, the designer does not threat the players to force themto behave efficiently, either with expelling them from the game or by breakingdown the negotiation. In this sense, the protocol has the property of beingrenegotiation-proof.

There exist two possible kinds of economic interactions that require slightmodifications in the mechanism. They differ in the timing of the negotiationwith respect to the production process. In the first type of negotiations, profitsare a flow variable, i.e., the production of the good with externalities andhence the realization of the profits is undertaken every period. This makes theabsence of cooperation the status-quo of the game, since until the players agreeon a cooperative decision, they will be behaving non-cooperatively. This isthe case, for instance, in any pollution abatement negotiation among differentcountries where the countries keep polluting according to the non-cooperativeequilibrium until a consensus is reached.

The other kind of negotiations are undertaken before any productive decisionis taken and hence, payoffs are not realized until the process of negotiation hascome to an end (either successfully or unsuccessfully). In this framework, theabsence of cooperation is an outside option for the negotiation. An example ofthis is a game in which the players have to agree on the implementation of apublic project, or in the future exploitation of a common resource, viz a fishingstock, a forest, a mine, etc.

This difference in the timings affects the process of repeated negotiation andrequire an independent study. When the outcome of no cooperation is thestatus-quo, a process of negotiation will always be followed by another one, asthe worst possible outcome is the absence of agreement. However, when theoutcome of no cooperation is the outside option, a new negotiation process willstart again at the beginning of the next period only if the players agree. Themechanism does not consider the possibility of partial agreements. Therefore,it gives the players the right to break the negotiation process if they think theywill gain more in the non-cooperative scenario than what they would get as aresult of the negotiation.

The outcome of the mechanism consists on a full internalization of the exter-nalities in both environments. However, the distribution of the profits presentssome differences. When non-cooperation is the status-quo of the game, thefact that the agents earn the non-cooperative profits in all the periods until


the agreement is reached, has significant distributional implications. The out-come ensures the non-cooperative profits to each player plus an equal shareof the surplus that cooperation generates. Whereas, when non-cooperation isonly an outside option of the negotiation process, the non-cooperative outcomecan be interpreted as a potential result of the negotiation, the result in caseof failure. As a result, in this scenario, the discount factor plays a crucial rolein the distribution of the profits. The higher the discount factor, the weakerthe effect of the Nash (non-cooperative) profits on the final revenues of theplayers. In particular, if the fully egalitarian solution is individually rationalfor all the agents, there exists a threshold in the discount factor, beyond whichthe fully egalitarian distribution is implemented, eliminating all the impact ofthe non-cooperative profits on the final sharing.


NTU Games

32 A Consistent Value for NTU Games with Coalition Structure

Juan José Vidal and Gustavo BergantiñosUniversidade de Santiago de Compostela

[email protected]

We introduce a new value for NTU games with coalition structure. Thisvalue coincides with the consistent Shapley value for trivial coalition structures,and with the Owen coalitional value for TU games with coalition structure. Fur-thermore, we propose two axiomatizations in terms of balanced contributionsand by a consistency property.

The consistent value is presented by Maschler and Owen (1989, 1992) as ageneralization of the Shapley value for hyperplane games. The main idea behindthis generalization is to maintain (as far as possible) the consistency propertyfrom the Shapley value. It is remarkable that their value arises as the mean ofrandom order marginal contributions of the players. Maschler and Owen (1992)even suggest the name “random order value” instead of “consistent value”.

Later, Hart and Mas-Colell (1996) develop a bargaining mechanism whichimplements the coalitional value. They also characterize this value by meansof balanced contributions.

On the other hand, NTU games with coalition structure are studied byWinter (1992). Winter axiomatizes the Game Coalition Structure (GCS) value,which is a generalization of the Harsanyi value for trivial coalition structuresand the Owen value for TU games.

It was of our interest to know whether the consistent value could be gener-alized the same way to games with coalition structure. Both the Shapley valueand the consistent value are obtained as an average of marginal contributionsdepending on equal-likely orders. Furthermore, the Owen value can be ob-tained by restricting these orders to compatibility with the coalition structure.Thus, it seems reasonable to generalize the consistent value using the same re-strictions. Remarkably, the arising value, however generalizes both consistentand Owen value, misses most of their nice properties; namely, it is not consis-tent, nor satisfies a “balanced contributions” property. We call this value the“random order coalitional value”.

We use the random order coalitional value as a first step in defining ourvalue. We show that the suggested value is consistent (in the sense of Maschlerand Owen (1989) and Winter (1992)) and it satisfies a balanced contributionsproperty. We believe these properties make the value a proper generalizationof the Owen value for NTU games with coalition structure.


33 Subbalancedness and the bargaining set

Chih Chang and Peng-An ChenDepartment of Mathematics

National Tsing Hua UniversityHsinchu, Taiwan

[email protected]

The requirement that a coalition making a counterobjection should have acommon player with the objecting coalition is called the intersection condition.Mas-Colell proposed a version of the bargaining set without the condition. Wewill impose the condition on it and study its existence problem. The con-cept “subbalancedness collection” is introduced and some existence results ofdifferent versions of the bargaining set on NTU games are provided.


34 The Procedural Value

Geoffroy de ClippelCORE, Université Catholique de Louvain

[email protected]

We define a single-valued solution (named the procedural value) that ex-tends both the consistent Shapley value of Maschler and Owen (1989) andRaiffa’s discrete bargaining solution to a large class of NTU games. Thoughnot axiomatized, the solution is motivated via the Nash program.

We consider bargaining procedures GK that are indexed by a positive integerK, related to the size of the tree. At each stage, a player (let’s call himi) is chosen at random within the active coalition (let’s call it S). Player iproposes a feasible allocation for S (let’s call it x). Every member of S thensequentially chooses wether to accept player i’s proposal. If all the membersof S accepts, then x is enforced and the game ends. If somebody rejects,then S \ i becomes the active coalition when S has already been active Ktimes in the past. Otherwise, S remains active. When a player is ejectedfrom the active coalition, he receives a payoff low enough in order to promotecooperation. This payoff could be the disagreement payoff if the former NTU-game is superadditive. The game starts with the grand coalition being active.Each game GK admits a unique subgame perfect equilibrium outcome. Thesequence of subgame perfect equilibrium outcomes converge to the proceduralvalue, as K tends to infinity.

The extensive form gamesGK we consider as relevant bargaining procedures,are closely related to the ones proposed by Hart and Mas-Collel (1996) in orderto support the consistent Shapley value of Maschler and Owen (1992). Intheir framework, the parameter K is replaced by a parameter ρ ∈ (0, 1] thatrepresents the probability for each player to be excluded of the active coalitionand to receive the punishment payoff, when his proposition is rejected by somemember of the active coalition. Their extensive form games can potentiallylast infinitely many periods, although this happens with probability zero. Theythus have to focus on stationary subgame perfect equilibria (SSPE), as folk-liketheorems hold in their model for the larger set of subgame perfect equilibria(SPE). Moreover, they obtain the weaker result that every sequence of SSPEoutcomes indexed by rho converges to some consistent Shapley value, as ρtends to zero. They do not establish the converse. Our approach also showsthe limits of the robustness of their result: a slight variation (out of the class ofvariations defined in their sixth section) of their bargaining procedure may yieldthe Shapley value in the TU-case and something different from the consistentShapley value in the NTU-case.


Auctions I

35 Auctions with Incumbents and Newcomers

M. Angeles de Frutos and Xavier JarqueUniversidad Carlos III de Madrid

[email protected]

We study the performance of the first-price format in auctions with asym-metric common-values. We show that, contrary to the result for second priceauctions, a small advantage for one player translates only to small changes inbidders’ strategies, and the equilibrium remains close to the first-price equilib-rium of the original game. We characterize the equilibrium bidding strategiesand their behavior as the degree of asymmetry increases. Finally, we studythe performance, to enhance the government revenue, of several instruments tofavor designated bidders.


36 Contest Architecture

Aner Sela and Benny MoldovanuBen-Gurion University, [email protected]

A contest architecture specifies how the prize sum is split among severalprizes, and how the contestants (who are privately informed about their abil-ities) are split among several sub-contests. We compare the performance ofsuch schemes to that of grand winner-take-all contest from the point of view ofdesigners that maximize either the expected total effort, or the expected high-est effort. An important explanatory variable is the form of the agents’ costfunctions. The analysis is based on simple but powerful results about variousstochastic dominance relations among order statistics and functions thereof.


37 Revenue Equivalence in Asymmetric Auctions

Arieh Gavious, Gadi Fibich and Aner SelaBen-Gurion University, [email protected]

The Revenue Equivalence Theorem is generalized to the case of asymmetricauctions in which each player’s valuation is independently drawn from a com-mon support according to his/her distribution function as follows. Let e bethe level of asymmetry among the distribution functions and let R(e) be theseller’s expected revenue in equilibrium. Then,

R(e) = R(0) + eR′(0) +O(e2),

where both R(0) and R′(0) are independent of the auction mechanism.


Social Choice

38 Voting by Committees under Constraints

Jordi Massó, Salvador Barberà and Alejandro NemeUniversitat Autònoma de Barcelona and Universidad Nacional de San Luis

[email protected]

Many problems of social choice take the following form. There are n votersand a set K of objects. These objects may be bills considered by a legislature,candidates to some set of positions, or the collection of characteristics whichdistinguish a social alternative from another. The voters must choose a subsetof the set of objects.

Sometimes, any combination of objects is feasible: for example, if we considerthe election of candidates to join a club which is ready to admit as manyof them as the voters choose, or if we are modelling the global results of alegislature, which may pass or reject any number of bills. It is for these casesthat Barberà, Sonnenschein, and Zhou (1991) provided characterizations ofall voting procedures which are strategy-proof and respect voter’s sovereignty(all subsets of object may be chosen) when voters’ preferences are additivelyrepresentable, and also when these are separable. For both of these restricteddomains, voting by committees turns out to be the family of all rules satisfyingthe above requirements. Rules in this class are defined by a collection of familiesof winning coalitions, one for each object; agents vote for sets of objects; to beelected, an object must get the vote of all members of some coalition amongthose that are winning for that object.

Most often, though, some combinations of objects are not feasible, whileothers are: if there are more candidates than positions to be filled, only sets ofsize less than or equal to the available number of slots are feasible; if objects arethe characteristics of an alternative, some collections of characteristics may bemutually incompatible, and others not. Our purpose in this paper is to char-acterize the families of strategy-proof voting procedures when not all possiblesubsets of objects are feasible, and voters’ preferences are separable or addi-tively representable. Our main conclusions are the following. First: rules thatsatisfy strategy-proofness are still voting by committees, with ballots indicatingthe best feasible set of objects. Second: the committees for different objectsmust be interrelated, in precise ways which depend on what families of sets ofobjects are feasible. Third: unlike in Barberà, Sonnenschein, and Zhou (1991),the class of strategy-proof rules when preferences are additively representablecan be substantially larger that the set of rules satisfying the same requirementwhen voters’ preferences are separable.


Our characterization result for separable preferences is quite negative: infea-sibilities quickly turn any non-dictatorial rule into a manipulable one, except forvery limited cases. In contrast, our characterization result for additive prefer-ences can be interpreted as either positive or negative, because it has differentconsequences depending on the exact shape of the range of feasible choices.The contrast between these two characterization results is a striking conclusionof our research, because until now the results regarding strategy-proof mecha-nisms for these two domains had gone hand to hand, even if they are, of course,logically independent.

Notice that here, as in Barberà, Sonnenschein, and Zhou (1991), we couldidentify sets of objects with their characteristic function, and our objects ofchoice as (some of) the vertices of a |K|-dimensional hypercube. Barberà, Gul,and Stacchetti (1993) extended the analysis to cover situations where the ob-jects of choice are Cartesian products of integer intervals, allowing for possiblymore than two values on each dimension. In there and in other contexts ofmultidimensional choice where the range of the social choice rule is a Carte-sian product, strategy-proof rules are necessarily decomposable into rules whichindependently choose a value for each dimension, and are themselves strategy-proof (see Le Breton and Sen (1997) and (1999) for general expressions of thisimportant result, which dates back to the pioneering work of Border and Jordan(1983)).

In Barberà, Massó, and Neme (1997) we considered the consequences of in-troducing feasibility constraints in that larger framework. The range of feasiblechoices is no longer a Cartesian product and this requires a more complex andcareful analysis. All strategy-proof rules are still decomposable, but choices inthe different dimensions must now be coordinated in order to guarantee fea-sibility. While this previous paper makes an important step in understandinghow this coordination is attained for each given shape of the range, it is marredby a strong assumption on the domain of admissible preferences. Specifically,we assume there that each agent’ bliss point is feasible. This assumption isnot always realistic. Moreover, it makes the domain of admissible preferencesdependent on the range of feasible choices.

In the present paper we study the question of voting under constraints fortwo rich and natural sets of admissible preferences: those that are additivelyrepresentable or separable on the power set of K, regardless of the type ofconstraints faced by choosers.


39 Monotonicity of Social Optima with respect to ParticipationConstraints

Lars P. Østerdal, Jens L. Hougaard, and Hans KeidingUniversity of Copenhagen

[email protected]

In this paper we consider the problem of selecting a social optimum satisfyinga participation constraint for any coalition, formulated as a problem of selectingfrom the core. In the theory of allocation rules, a fundamental result is that ofYoung (1985) stating that there is no rule for selecting an element of the coreof all transferable utility (TU) games which satisfies coalitional monotonicity(in the sense that when the worth of a coalition increases, and the worth of allother coalitions remain unchanged, then the payoff given to any member of thecoalition according to the rule should not decrease).

The result shows that there is a certain trade-off between coalitional sta-bility (that is, selecting from the core) and monotonicity. Monotonicity is animportant property in the context of cost sharing rules where its failure meansthat some agents may lack incentives for reducing cost (cf. e.g. Shubik, 1962;Young, 1994). On the other hand, the assumption of coalitional stability isalso a desirable one, and therefore there is a need for investigating the extentto which the two conficting goals may be reconciled. That this is indeed possi-ble is witnessed by several contributions to the literature dealing with convexTU games: The Shapley value is indeed coalitionally monotonic (Shapley, 1971;Sprumont, 1990; Rosenthal, 1990), whereas the nucleolus is not (Hokari, 2000).

In the present paper, we consider a class of core selection rules where thepayoff chosen is found by maximizing a social welfare function on the core ofthe game. When this social welfare function is given, the solution for any givengame will depend only on the core of the game and not on other structuralcharacteristics of the game. This contracts with the situation for, say thenucleolus, which may yield different results for two games with identical cores,a feature which may make it difficult to justify a solution from a social planner’spoint of view.

While we concentrate on a particular class of core selection rules, we putno restrictions on the class of TU games considered (except of course that thecore should be nonempty). Since our basic aim is to investigate the trade-offsbetween coalitional stability and monotonicity, and keeping in mind the generelimpossibility result, we shall be satisfied with monotonicity with respect tothe worth of any particular coalition (in the following called S-monotonicity).It turns out that there are such rules, and moreover, all such rules can becharacterized in terms of the social desirability of giving payoff to the members


of the coalition S. As it turns out, this social desirability has a simple expressionin terms of the marginal social welfare of each individual’s payoff.

While there are S-monotonic core selections on the domain of all TU gameswith nonempty core, this turns out to be a feature which is closely connectedwith the transferability of utility. Indeed, if NTU games are considered, thenthe previously derived results (characterizing S-monotonic core selection rules)may be exploited to yield an impossibility result: Even for any fixed coalitionS there cannot be core selection rules that are monotonic. This shows thatthe non-monotonicity in coalitional worth is a feature which is inherent inany solution satisfying the coalitional stability conditions, that is in any coreselection.

The paper is structured as follows: In Section 2, we give the necessarydefinitions (in the context of TU games); the basic characterization result isgiven as Theorem 1 in Section 3; it exploits the characterization of games withnonempty core by balancedness and consequently needs some details concerningsuitable balanced or non-balanced families of coalitions, which are developedas we proceed. In the following section, we establish the relationship betweenS-monotonicity and the partial derivatives of the social welfare function. Fi-nally, in Section 5 we extend the investigation to NTU games and give theimpossibility result in this context.


40 Agglomeration of Spatial Clubs

Oded HochmanBen-Gurion University, Beer Sheva, Israel

[email protected]

The literature on agglomeration has focused largely on direct attraction ef-fects, that is, entities, usually firms, drawn to one another to enhance the outputof their activity. In this paper we investigate by using an analytical model withgeneral functions, the agglomeration of clubs arising from an indirect effect, themutual attraction between clubs and population which lead both housing andclubs to joint agglomerations. We show that when the population is uniformlydistributed, there is no economic agglomeration of clubs, however, in a firstbest allocation of spatial clubs, the population is never distributed uniformlyover space. To minimize costs, clubs of different types along with populationagglomerate in and around centers. We specify conditions for which the caseof a monocentric city, an agglomeration of all club types to a single point, isa unique solution. We also investigate a case of multiple centers solution andshow that in addition to the agglomeration of some club types in the centerof a complex, the rest of the club types cluster in subcenters which gravitatearound the center.

Keywords: agglomeration, clubs, complex, collective goods, local public goods,indirect attraction


41 Fixed Agenda Solutions for Weak Tournaments

Somdeb LahiriUniversity of Witwatersrand at Johannesburg

[email protected]

Choice theory is concerned with choosing one or more alternatives, froma given set of alternatives. An abiding problem in choice theory has beenone of characterizing those choice functions which are obtained as a result ofsome kind of optimisation. Specifically, the endeavour has concentrated largelyon finding a binary relation (if there be any) whose best elements coincidewith observed choices. An adequate survey of this line of research till the mideighties is available in Moulin (1985). The idea of a function which associateswith a given binary relation a non-empty subset of a given universal set hasa long history whose exact origin is very difficult to specify and in any case isunknown to the author. In Moulin (1986) one finds the rigorous statement of achoice theory which for a fixed universal set of alternatives, associates with eachreflexive, complete and anti-symmetric binary relation, a non-empty subset ofthe given set. Such binary relations are called tournaments. Laslier (1997) canbe found a very exhaustive survey of the theory of tournaments.

In this paper we extend the above set of binary relations to include thosewhich are not necessarily anti-symmetric. Such binary relations which are re-flexive and complete are referred to in a recent paper, by Peris and Subiza(1999), as weak tournaments. Given a weak tournament, a solution is a func-tion, which associates to each binary relation a non-empty subset of the uni-versal set, on the basis of the given weak tournament. Since the set of elementsfrom which choices have to be made is fixed, such problems may be referred toas fixed agenda choice problems.

Lucas (1992) has a discussion of abstract games and related solution con-cepts. The concept of an abstract game is originally due to von Neumann andMorgenstern and they are very similar to the weak tournaments that we studyin this paper. Much of what is discussed in Moulin (1986) carry through intothis framework. An important consequence of both the frameworks is thatoften, a binary relation may fail to have a best element. To circumvent thisproblem the concept of the top cycle set is introduced, which selects from amongthe alternatives only those, which are best with respect to the transitive closureof the given relation. The top cycle set is always non-empty and in this paperwe provide an axiomatic analysis of the top-cycle solution. We also show thatthe top cycle always coincides with the set of sophisticated outcomes of somebinary agenda. The concept of a binary agenda is due to Miller (1977), whoshowed the same equivalence in the case of tournaments. A lucid discussion


of this theory and its implications for social choice can be found in Myerson(1996).

An alternative x is said to cover another alternative y if and only if x ispreferred to y and for every other third element z if (a) y is at least as goodas z, then so is x; (b) if y is preferred to z then so is x. Given any weaktournament, its uncovered set is the set of all elements in the which are notcovered by any other element in the same set. We show that the uncoveredsolution satisfies some desirable properties.

In a final section of this paper we study the Copeland solution. Given a weaktournament and an alternative, its Copeland score is the difference between thenumber of elements it is preferred to and the number of alternatives, which arepreferred to it. The Copeland solution associates with each weak tournamentthose alternatives, which have the highest Copeland score with respect to theweak tournament.

References

J. F. Laslier (1997) Tournament Solutions and Majority Voting, Studies inEconomic Theory 7, Springer-Verlag.

W. F. Lucas (1992) Von Neumann-Morgenstern Stable Sets, Chapter 17 in R.Aumann and S. Hart (ed.) Handbook of Game Theory, Vol. 1, Elsevier,Amsterdam.

N. Miller (1977) A New Solution for tournaments and majority voting, Amer-ican Journal of Political Science 21, 769-803.

H. Moulin (1985) Choice Functions Over A Finite Set: A Summary, SocialChoice and Welfare 2, 147-160.

H. Moulin (1986) Choosing From a Tournament, Social Choice and Welfare3, 271-291.

R. B. Myerson (1996) Fundamentals of Social Choice Theory, Discussion PaperNo. 1162, Center for Mathematical Studies in Economics and Manage-ment Science, Northwestern University.

J. E. Peris and B. Subiza (1999) Condorcet Choice Correspondences for WeakTournaments, Social Choice and Welfare 16, 217-231.


Voting Situations

42 Maximal domains for strategy-proof rules in the candidates se-lection problem

Dolors Berga and Shigehiro SerizawaUniversitat de Girona and Tohoku University

[email protected]

We consider the problem where a set of voters have to elect candidatesamong a set of two alternatives. This is the same framework as in Barberà,Sonnenschein, and Zhou (1991), where they obtain “voting by committees” asthe characterization class of all onto and strategy-proof voting schemes on thedomain of separable preferences. Furthermore, separable preferences turn outto be the maximal rich domain where voting by committees are strategy-proof.In this paper we study maximal domains of preferences for strategy-proofnessof any rule and not only voting by committees. We obtain that the set ofseparable preferences is a maximal domain for the existence of strategy-proofand onto social choice functions satisfying the additional requirement of novetoerness. For more than two alternatives this result is still a conjecture.

Barberà, Sonnenschein, and Zhou (1991) (henceforth, BSZ) considered theproblem where a finite set of voters N must choose, from a finite set K, whichobjects will be adopted. These objects can be bills considered by a legislature,candidates to enter a club, or they can also be interpreted as public goods withtwo feasible levels 0, 1. This last interpretation was given in Barberà, Gul,and Sttacchetti (1993) (henceforth, BGS) where the more general setting ofany number of feasible levels for public goods is considered. BSZ obtained twoimportant results concerning strategy-proofness. On the one hand, they char-acterized voting by committees (which we also call: generalized median voterschemes) as the only onto social choice functions satisfying strategy-proofnesswhen agents’ preferences are separable. On the other hand, they obtain theset of separable preferences as a maximal rich domain for which voting bycommittees with neither veto nor dummy voters are strategy-proof.

Recently, other results improving this necessary condition on preferencesfor which voting by committees are strategy-proof have been obtained. SeeSerizawa (1995) and Barberà, Massó, and Neme (1999) for discrete sets ofalternatives, and Barberà, Massó, and Serizawa (1996) and Berga (2000) forthe case where social alternatives are continuously measured. Although anystrategy-proof rule on separable preferences must be voting by committees, astrategy-proof social choice function may not belong to this class when the rule


is defined outside that domain. Thus, this literature may exclude interestingrules.

From their two results mentioned above, two natural questions arise. Weinitially concentrate in BSZ’s framework with two objects.

The first one is related to the existence of other domains of preferencespreserving voting by committees as the unique class of strategy-proof socialchoice functions. By BSZ, we know that the subdomain of additive preferencespreserves these results. In our paper, we consider “rich domains” and we con-jecture that “a social choice function on a rich domain is strategy-proof if andonly if it is a generalized median voter scheme”.

The second question is how large the domain of preferences can be andstill preserve the existence of strategy-proof rules (not necessarily generalizedmedian voter schemes)? We qualify this question in two different ways. First,we employ the no vetoer condition to rule out trivial rules such as dictators.Second, we require domains to be rich; that is, to contain a minimal varietyof preferences. Then, we conjecture that “the unique maximal rich domainfor strategy-proofness and the no vetoer condition is the domain of separablepreferences”.

Additionally, in this paper we also show how relevant is the rich domaincondition for our results. In Theorem 2 we state our main result (unique forthe moment) which says that “the domain of separable preferences is a max-imal domain for strategy-proofness and the no vetoer condition”. We provideand example of an onto and strategy-proof social choice function satisfying theno vetoer condition which is not a generalized median voter scheme and whichis defined on a non-rich domain. Moreover, we think we can obtain (we con-jecture) that a slightly variation of this domain is maximal for our properties.Thus, the domain of separable preferences is not unique in Theorem 2.

These two questions are in the same line as the ones already answered inBerga and Serizawa (2000) where the authors do not restrict a priori the classof rules to be considered and they study the problem of the provision of a singlepublic good. There, they obtain the domain of convex preferences as the uniquemaximal domain including a minimally rich one allowing for strategy-proof andonto rules satisfying the no vetoer condition. In this paper we restrict to thecase of two public goods (or two objects), however, we believe that our resultcan be generalized to any number of goods. This is still part of current research.


43 Condorcet Consistency of the Citizen-Candidate Approach

Bernardo Moreno and Socorro PuyUniversidad de Málaga

[email protected]

The electoral mechanism in a citizen-candidate approach consist of twostages. In the first stage, each citizen chooses whether or not to run as can-didate, and in the second stage the citizens vote to elect a candidate. In asingle-peaked preference domain, we find that the strategic entry decision ofthe candidates eliminates some of the undesirable properties of Plurality rule,namely to elect a poor candidate in three-candidate elections, since as we show,the Condorcet winner is always elected. It is in four or more candidate electionsthat the Condorcet Loser can be chosen. These main results are obtained fortwo different kinds of voting behavior: sincere voting and strategic voting.


44 Positive and normative assessment of voting situations

Annick Laruelle and Federico ValencianoUniversidad de Alicante y Universidad del País Vasco

[email protected]

This paper is concerned with the analysis of voting situations. By a votingsituation, we mean a situation in which a set of voters faces collective decision-making by means of a procedure that specifies when a proposal is to be acceptedand when is to be rejected after a vote is cast. More specifically we are interestedin the accurate formulation and quantification of several features involved inthese situations. Traditionally voting situations have been modelled by simplesuperadditive games. In this context several measures or ‘power indices’ havebeen proposed in order to assess the voters’ a priori ‘decisiveness’ or ‘power’.This classical approach can be criticized for several reasons. First, the game-theoretical framework and terminology is inadequate and misleading. There isno transferable utility involved, there are not really ‘players’ nor ‘voters’, for thevoting rule only specifies when a proposal will pass: to describe a voting rule oneshould more properly speak just of ‘seats’. There are no ‘coalitions’ -a term thatsuggests cooperation with a purpose where there is only coincidence of vote.There are no ‘marginal contributions’ where there is no cake to share. Moreover,as a rule the axiomatic characterizations of power indices either do not fit thespecificity of the context, or lack compellingness. Second, the existence andmisuse of several indices without a clear interpretation is confusing and doesnot contribute to their credit. Finally, power indices are often criticized onthe basis that the only information they take into account is the voting rule,while the voters’ preferences, which clearly influence their capacity of beingsuccessful, decisive or lucky, are ignored.

In this paper we propose a more general model which includes the two sep-arate ingredients in any real-world voting situation: the voting rule and thevoters. The voting rule, specifies for a given set of seats when a proposal isto be accepted or rejected depending on the resulting vote configuration. Vot-ers, the second ingredient in a voting situation, are included via their votingbehavior, which is summarized by a distribution of probability over the voteconfigurations. This distribution of probability depends on the preferences ofthe actual voters over the issues they will have to decide upon, the likelihoodof these issues being proposed, etc. In each real-world voting situation theseprobabilities must be approximated from the available data. Thus this generalmodel, unlike the traditional one, is apt for positive or descriptive purposes.Within this general framework we re-examine the concepts of ‘succes’, ‘deci-siveness’ and ‘luck’ that in a more or less clear formulation can be traced a long


way back in the literature. This more general setting allows a simple and pre-cise reformulation of these concepts as probabilities which depend on the votingrule and the voters’ voting behavior. Moreover, our formulations extend pre-vious purely normative notions to more general positive/descriptive concepts,which can be particularized into some familiar but not always well-understoodnotions, shedding new light on their meaning and relations.


Bargaining Models

45 Stakeholders, Bargaining and Strikes

Paola Manzini, and Clara PonsatiQueen Mary, University of London

[email protected]

In Marcel Pagnol’s movie “La femme du boulanger”, the baker’s wife runsaway with her lover. But then, the baker bakes bread which his wife sells,so that without her the baker’s enterprise is worthless. So, the baker goes onstrike, sending a clear message to the village: no wife, no bread. The villagegoes up in arm, but there is no convincing the baker. The only option left tothe villagers is to go on the hunt for the baker’s wife, and bring her back toreturn harmony (and bread) to the French village. This amusing story captureseffectively the fact that conflict affecting certain services, whether publicly orprivately provided, does concern, directly or indirectly, the interests of manyparties. Bilateral bargaining in an area of public interest therefore has animpact on third parties, stakeholders, who are interested in the resolution ofthe conflict, yet unable to impose an agreement upon the contending bargainers.In this paper we analyse the effect that stakeholders have on the bargainingoutcome.

The mere fact that the interests affected by the conflict extend well beyondthe two participants is what defines its “public interest” attribute. It is rea-sonable to assume that in many cases the public interest will have much widerconsequences than the issue at stake in the bargaining. For instance, the dis-ruption of essential services like public transport, hospitals, the fire service andthe electricity, gas and water industries has a substantial impact on the pop-ulation at large. Correspondingly, the government’s stake in bilateral conflictswhich are of public concern is of some consequence. That is, one can postulatethat the government’s stake can be quantified as ‘greater’ than the issue bar-gained over. This immediately presents a potential for exploitations from thetwo contenders: as long as the stakeholder has more to lose from disagreement,bargainers should succeed in extracting some resources from the stakeholder.Indeed, especially during the ’70s and the ’80s, Europe was hit by a wave of gen-eral strikes in which workers managed to win (mainly salary) concessions fromthe government of the day. The following two decades have seen an effort ofgovernments across Europe towards a greater flexibilisation of employment lawswhich apply to employment in services of public interest, so as to“harmonise”it with the private sector. These changes in the legislation regulating industrialrelations, which in the mains have been directed at weakening the power oftrade unions, have generally tended towards a decentralisation of bargaining in


the public sector. The effect has been to transform what would have been inessence a bilateral relationship into trilateral negotiations between managementand union in the shadow of possible state . Consequently, the sheer possibil-ity that the stakeholder (i.e. the government) may intervene in negotiationscreates the potential for delays, in the hope to pressurise the government intoconceding extra resources. So why wouldn’t the stakeholder with the power todo so change the “rules of the game” to a more efficient negotiating framework?

These are the type of issues we address in this paper. More precisely, weexplore bilateral bargaining explicitly accounting for the presence of stakehold-ers. We model such bargaining problems as non-cooperative games with threeplayers: two players, the bargainers, have the ability to reach an agreement; thethird player, a stakeholder, can only take (limited) actions that condition thenature of the bilateral bargaining. We start by looking at the strategic incen-tives in a simple bargaining model with perfect and complete information. Ourmain finding confirms our original suspicion that the presence of a stakeholdergenerates delays: stakeholders are usually willing to make contributions to pro-mote agreement, but this willingness may backfire and become the source ofsevere inefficiency. However, and more surprisingly, for a wide range of param-eter values this outcome is better for the stakeholder than a situation in whichhe bargains directly with the union. The intuition for this result is straightfor-ward. Delays are harmful for all of the agents involved in negotiations; however,in trilateral negotiations the management and the stakeholder can join forcesand secure as a coalition an ex ante expected payoff which is greater than whatthey would otherwise obtain as a single negotiator in bilateral bargaining withthe union.

These results concern the situation where all parties are perfectly informedabout the overall resources available. In practice, however, negotiators seldomhave an accurate assessment of the means at the stakeholder’s disposal. Thusin the second part of the paper we analyse the effects of uncertainty by build-ing a simple model where bargainers are unsure about the stakeholder’s stake.Note that besides being more realistic, this type of framework can account forsituations where the stakeholder is genuinely super partes, or even antagonisticto one or both of the other negotiators. We find that, not surprisingly, uncer-tainty does not remove inefficient (i.e. delayed) equilibria. However, as long asthe stake is not too great, there are also equilibria in which the two litigantsreach an efficient agreement with positive probability. This class of equilib-ria are driven by the expectation of the two bargainers that the stakeholderwould not intervene in case of a stalemate, which makes it worthwhile for thebargainers to get to an agreement quickly. Consequently, it is optimal for thestakeholder to dither, thereby pressurising the two bargainers into reaching aspeedy conclusion. The general features of the situation that we model fit manycontingencies beyond industrial relations, and the framework of analysis that


we propose is general enough to model stake holders as “interested’ parties,which may still be genuinely neutral. For example, arbitrating and mediatingefforts by neutral third countries to promote peace settlements to end armedconfrontations are possibly motivated by the assessment that more is at stakethan the welfare of the actors directly involved in the conflict.


46 Conflict as a part of the Bargaining Process: Theory and Evi-dence

Santiago Sánchez-PagésUniversitat Autonoma de Barcelona

[email protected]

Since Nash’s seminal contribution, the bargaining problem has been tradi-tionally summarized by the set of feasible utility possibilities and a threat point.The latter is meant to be the outcome in case of disagreement, a ground floorpayoff they receive in the hypothetical non-cooperative game that would fol-low. Its influence stops here: How the surplus of cooperation is finally sharedis totally independent of the forces that determine the location of the threatpoint. In the last years, several authors have tried to reformulate this ap-proach by giving more structure to agreements by incorporating informationabout how disagreement is actually solved. This is the central issue in Besterand Warneryd (1998), Esteban and Sakovics (2000), Fearon (1996), Grossman(1994), Horowitz (1993) and Powell (1996). These models obtain agreementsin the shadow of conflict (war, redistribution by force...) that are influenced bythe relative power of the parties. With the exception of Esteban and Sakovics(2000), all these papers depict conflict as a costly lottery: Either one party orthe other wins and captures the surplus. Thus, invoking conflict is a game-ending move, an alternative to the bargaining process.

On the contrary, casual observation shows that the main reason behind theend of conflicts is not the total collapse of one of the parties but agreement onstopping hostilities: India and Pakistan do not use nuclear weapons, they onlyengage in skirmishes; Pepsi and Coca-cola do not engage in worldwide pricewars, but only national; family arguments do not necessarily imply divorce...That is, conflict is not mainly a “fight to the finish” but a part of the bargainingprocess.

To illustrate this point consider the following example: Suppose that twoagents have incomplete information on the strength of their opponent in caseof conflict, and that both parties are strong but believe that they are facing aweak opponent. Then, the perceived threat-point will be out of the bargainingset and the result of the negotiation, following the papers above, would be aconflict fought to the finish. However, notice that if parties can engage in aconflict of limited scope that does not entail the end of the game, it may conveyinformation about relative strengths and create a range of possible agreements.

Hence, in this paper, following the pioneering work of Clausewitz (1832)and more recently Wagner (2000), we propose a taxonomy of conflicts (wars,strikes or revolutions) into REAL and ABSOLUTE conflicts and explore theconsequences of incorporating them in a bargaining model with one-sided offers


and one-sided incomplete information. In economically familiar terms, absoluteconflicts have an ex-post interpretation: they are fought when one party lossescompletely the hope of reaching an agreement through ordinary methods. Soit tries to make the opponent defenseless in order to impose its most preferredoutcome without opposition. On the other side, real conflicts (that receivethis name because, as argued in Wagner (2000), few strikes or wars we shouldobserve if their consequences were always the defeat of one of the parties) haveand ex-ante interpretation: they are fought before normal bargaining becausethey can help to improve the position in negotiation by revealing informationabout the true relative power of the parties. Therefore, it is the possibility ofending the conflict in a settlement what differentiates them.

From a formal point of view, revelation of information through conflict in-troduces a novelty in the models of bargaining with incomplete information:In these models, parties have incentives to misrepresent information. This factusually leads to very complex models and multiple equilibria. However, theresult of a conflict is not subject to manipulation because its outcome dependson the true relative strengths of the parties. The only strategic variable is thedecision of invoking it or not, and bluffing the other party becomes a very dif-ficult task. Following Wagner, conflict “is an experiment that allows parties totest competing hypothesis about the outcome of an absolute conflict”.

Our paper has two parts: In the first part we develop a bargaining modelthat takes into account the issues discussed above. In the second, we test theimplications of the model with real data on wars and strikes. In the model wepropose two impatient and risk-neutral players have to decide how to share acake one dollar worth. One of the parties knows the true realization of the rel-ative strength and therefore the outcome of the absolute conflict. Each period,this party makes an offer to the other, who can accept or reject. Agreementmakes the game end but disagreement triggers a “battle”. Although it is wonby one of the parties, this real conflict does not entail the end of the game.However, it has two other effects: possible agreement is delayed one period andit conveys information because the outcome of the battle is related to the real-ization of the relative strengths. Then, each battle makes the uniformed partyto learn about the true disagreement payoffs in case of conflict. We obtain thefollowing results: Agreement may be immediate if the loss of the cake due toabsolute conflict is sufficiently high and players are impatient enough. Then,players may prefer to settle as soon as possible in order to avoid such unfavor-able outcome. On the other side, we also show that the creation of surplus isa necessary and sufficient condition for agreement. That is, once agreement ispossible, the informed party has no incentives to take the chance of improv-ing its position. Therefore, the number of battles (or the duration of the realconflict) depends decisively on the gap between the perceived and the real dis-agreement payoffs, because if the uniformed party is excessively optimistic it


may take several periods to bring the disagreement point inside the bargainingset.

This result has a clear implication regarding real-life conflicts: Given thatthey are a way of learning for the parties, they are more likely to end themore they last. Then, we pursue a hazard rate approach using duration data.The hazard rate is the conditional probability that an event that has lasted tperiod ended at period t + 1. Our theory would predict that this hazard ratemust be increasing in time. We test this hypothesis first with data on Extra-systemic wars from 1816 to 1991. This wars are fought between a state anda non-state entity. They include mainly imperialistic and colonial wars. Theyare appropriated for our study because they are clear asymmetric situations(moreover, these conflicts have not been analyzed in this fashion so far). Weobtain clear evidences that these type of conflicts show increasing hazard ratessupporting our theory. In the second case we deal with U.S. data on strikesand holdouts. A holdout is an impasse; a period of time after the collectivecontract has expired where workers keep working at the same conditions. Theyhave been pointed out as informational devices by Gu and Kuhn (1998) and wepostulate that they are real conflicts whereas strikes can be think as absoluteconflict. Consistently with our hypothesis, strikes are more likely to occur if aholdout has not occurred previously and, if it did, they last more the shorterthe holdout. This gives support to our analysis and suggest that we observeconflict in real world because sometimes they are necessary for obtaining anagreement. To understand why conflicts occur may be helpful to avoid them.


47 Mediation: Incomplete information bargaining with filtered com-munication

Xavier Jarque, Clara Ponsati and Jozsef SakovicsUniversitat Autònoma de Barcelona

[email protected]

In this paper, we present an in-depth analysis of mediation in bargaining.Mediation being a wide-spread practice —from international negotiations to di-vorce proceedings— this is empirically relevant in its own right. Our concurrentintention is more theoretical though: to provide an alternative approach to thestudy of bargaining with incomplete information. The difficulties encounteredby this literature are mostly due to the issues arising from unrestricted —or,arbitrarily restricted— conditional belief structures, which, unfortunately, arisemost naturally in dynamic games of incomplete information. Instead of con-straining the ways in which beliefs can be updated, we work with an extensiveform that exogenously restricts the generation of the information events, whichwould trigger the formation of new posteriors. In particular, we assume thatall offers that do not lead to agreement are unobservable. The easiest way tothink of this situation is to say that it is a dynamic double auction: the playerskeep sending offers to an auctioneer, who only reveals them once they are com-patible, and therefore an agreement has been reached. Within this context, weare able to give a complete characterization, as well as a number of interestinginsights.

We analyze this continuous-time bilateral double auction in the presence oftwo-sided incomplete information and a smallest money unit. A distinguishingfeature of our model is that intermediate concessions are not observable by theadversary: they are only communicated to a passive auctioneer. An alternativeinterpretation is that of mediated bargaining.

We show that an equilibrium (ex post inefficient) using only the extremeagreements always exists (War of Attrition equilibrium) and display the nec-essary and sufficient condition for the existence of (perfect Bayesian) equilibrawhich yield intermediate agreements. In any case, the efficient equilibrium isgiven by a particular solution of a system of differential equations belongingto the stable manifold of a generalized saddle point. Moreover when there isunique possible compromise we give the sufficient condition to ensure that thereis no other ex post inefficient equilibriums.

For the symmetric case with uniform type distribution we numerically cal-culate the equilibria. To do so, we numerically integrate the stable manifold ofthe relevant singular point of the differential equation which define the equilib-rium. The aim of this study is to evaluate the relative efficiency of the equilibria


differing in the number of agreements used and in the support of the type dis-tribution. A priori, it is not clear what is the specific role of the number ofpossible agreements relative to the social welfare. On the one hand, as thisnumber is larger there are more players getting to agreements — that is, ex postefficiency increases— but, on the other hand, these agreements show up later.We find that the equilibrium which does not use compromise agreements is theleast efficient, however, the rest of the equilibria yield the lower social welfarethe higher number of compromise agreements are used.

We have shown that reducing the information flow between negotiators isa powerful tool not only conceptually but technically as well. We were ableto provide a full characterization, and the nature of the multiplicity of ourequilibria —it is directly related to the number of agreements used— makes itvery easy to resolve: the mediator can simply announce the prices he is willingto accept (which should be the ones that maximize social welfare).

Our results build on the combination of a finite set of equilibrium agreementswith a continuous time structure that allows viewing the bargaining processas a game of timing. While the continuous time structure can be relaxed, theassumption that the set of agreements used in equilibrium is finite is technicallyimportant and cannot be dispensed with in our proofs. At the same time, theassumption is quite realistic and, in fact it is reinforced by our results. Recall,that we have shown that the more agreements are used in equilibrium the lessefficient the equilibrium is (and the less likely it is to exist). Therefore, thenegotiators are naturally driven towards employing “few” prices. While thisreduces ex post efficiency, at no point in time could the mediator induce aPareto improvement by allowing more prices.

At the limit, when the set of potential agreements is an interval, things aredifferent. While the equilibria that we have characterized are still supported assuch, equilibria where a continuous set of agreements are attained with positiveprobability may exist as well. Addressing the problem from a point of viewcomplementary to ours, Copic and Ponsati (2001) show that such equilibriado indeed exist. Instead of assuming that concessions must be discrete theyimpose that strategies are differentiable patterns of concessions. Their resultsstand in sharp contrast to ours too: a unique symmetric equilibrium prevails,it is always ex-post efficient and independent of the distribution of types.


Cooperative Games II

48 Stability of Cartels and the Incentive of Merger in OligopolySituations without Transfer

Holger Meinhardt and Theodorus S. H. DriessenUniversity of Karlsruhe

[email protected]

The paper studies the stability of cartels and the incentives for horizontalmergers of the class of oligopoly situations without transferable technologies.When dealing with oligopoly situations the usual prediction is that firms, whichreached an explicit or implicit agreement have an incentive to cheat and todeviate from the agreement. In contrast to the predicted instability of cartelsthere is empirical evidence that firms stick to a cooperative arrangement onprice or output decisions despite of the incentive scheme.

But if we want to know why cartels operate successfully in many situations,we have to look at cooperative solution concepts which give us an answer. Bydoing so, we can present sufficient conditions involving the inverse demand func-tion and the cost functions to establish the convexity for the class of oligopolyTU-games without transferable technologies. Convexity of oligopoly TU-gameswith transferable technologies has been first studied by Zhao (1999) for the caseof a linear inverse demand function and linear cost functions. Zhao has providednecessary and sufficient conditions to establish the convexity property for thesegame types. Although, these games are not convex in general, they are totallybalanced as has been worked out by Norde Do and Tijs (2002). Furthermore, ithas been shown by Norde et al. (2002) that oligopoly TU-games without trans-ferable technologies are convex for the special case of a linear inverse demandfunction and linear cost functions. Following this non-synergy approach we canderive for the most general case economically meaningful conditions concerningthe shape of the individual costs, the average costs for various coalitions andthe individual marginal revenues under which an oligopoly TU-game is convex.

References

Norde, H., Do, K. H. P., and Tijs, S. (2002) Oligopoly games with and withouttransferable technologies, Mathematical Social Sciences 43, 187—207

Zhao, J. (1999) A Necessary and Sufficient Condition for the Convexity inOligopoly Games, Mathematical Social Sciences 37, 189—204.


49 Compensation rules in IC situations: continuity and computa-tional aspects

R. Branzei, F. Scotti, S. Tijs and A. Torre“Alexandru Ioan Cuza” University, Iasi, Romania

[email protected]

In this talk information collecting (IC) situations and games (cf. Branzei etal., 2001 (a, b); Tijs et al., 2001) are central. They model decision-making situ-ations where the outcome of any decision depends on the state of nature and thedecision-maker is imperfectly informed. Collecting information from availableagents who are more informed about the situation creates the potential for bet-ter decisions. More refined information to support the decision-making processyields, in principle, additional reward which is the source of compensating theinformants. Different procedures for collecting information have given rise tonatural compensation rules in the context of cooperative game theory. Relevantfor the class of information collecting games are marginal-based compensationrules and bi-monotonic allocation schemes.

The state space and the action space in an IC-situation can be infinite. Weconsider possibilities of approximating such an IC-situation with a finite IC-situation where the state space and the action space are finite and we relate theori- ginal IC-game with the IC-games of finite approximations. The consideredapproximations turn out to be good in the sense that the corresponding gamesare nearby. So, continuity properties of relevant solutions for the class of IC-games are interesting. We compare relevant solutions such as the core, theset of bi-monotonic allocation schemes and marginal-based allocation rules ofthe approximate game with those of the original game, extending results fromLucchetti et al. (1987).

The procedures for discretizing the state space (going from an infinite statespace to a finite one) and for reducing the dimension of the action space arebasic ingredients for efficient algorithms to compute the compensation sharesof the informants.

A decision support system (DSS) is designed to assist the decision-maker inselecting the procedural approach to collecting information and, consequently,to choose the desired compensation rule to be implemented.


References

Branzei, R., Tijs, S. and Timmer, J. (2001a) Collecting information to improvedecision making, International Game Theory Review 3, 1-12.

Branzei, R., Tijs, S. and Timmer, J. (2001b), Information collecting situ-ations and bi-monotonic allocation schemes, Mathematical Methods ofOperations Research 54, 303-313.

Lucchetti, R., Patrone, F., Tijs, S. and Torre A.(1987), Continuity propertiesof solution concepts for cooperative games, OR Spektrum 9, 101-107.

Tijs, S., Timmer, J. and Branzei, R. (2001), Compensations in informationcollecting situations, CentER DP No. 2001-02, Tilburg University, TheNetherlands.


50 Partial ordered cooperative games

J. Puerto and F. R. FernándezUniversidad de Sevilla

[email protected]

In many real-world situations the only acceptable way to represent the eval-uations of the actions is by means of elements of partial ordered linear spaces(functions of time, random variables, vector valued outcomes, . . . ). In thispaper we analyze cooperative games whose characteristic functions are definedfrom the set of coalitions into a general partial ordered linear space. Theconcept of core is investigated and conditions that ensure nonemptiness aredeveloped. Our results state that the use of any order preserving utility func-tion only gives partial solutions to the general problem. Several examples thatillustrate the general theory are included.


Political Competition

51 Funding of Political Parties with Electoral Cheques

Anaís TarragóUniversidad de [email protected]

The funding of political parties and, in particular, of electoral campaignsis becoming an important political issue for public opinion in all democracies.There exists the belief that the funding system of political parties does notfavor the equality of opportunities of all the voters, in the sense that thosewith a better economic position in society may have a greater capacity toinfluence politicians. The funding system of political parties is not the same inall countries. There are some countries where the major part of the funds arepublic and are related to the number of votes or seats obtained in the election(this is the situation of most European countries), and there are some otherswhere the biggest part of funds comes from individual contributions of voters,firms or interest groups (this is the case of the United States). It is true thatfunding of political parties is not a new issue, nevertheless in our approachwe will try to analyze an issue that has not been studied before: the so called“Economic Democracy”, that is, that each voter’s ballot had the same influencein the political process no matter her economic resources.

As we have said, there exists the opinion that some funding systems ofparties favor those who have more money, since they can “buy” favors frompoliticians. Assuming that money is a very important element for the survivaland reaching of power of the political parties (a very important part of theparties’ budget is dedicated to finance the electoral campaign), these fight toattract the greatest number of contributions. These contributions are, of course,not for free, they are in exchange of the catering of the contributors’ interests.Given that the private funding of parties is an important phenomenon in theUnited States, our paper will focus on the situation in that country. Eventhough, we would like to extend our proposal to other democracies. In theUnited States the private contributions come mainly from voters, corporations,PACs and interest groups. Some authors claim that PAC contributions arenot as important as they are thought to be. On the other hand, campaigncontributions from PACs are often portrayed in the media as the functionalequivalent of bribes. Hence, some authors have claimed that corporate PACcontributions buy legislature. It is not a surprise then, that a recent opinionpoll by the Center of Responsive Politics. Therefore, the idea that a reformis needed is wide extended. Generally, campaign reform plans either seek to“level-up”, by increasing the ability of those shut out of the political system


to participate, or to “level-down” , by decreasing the ability of those withdisproportionate political capital to exercise greater influence over the politicalsystem. A voluntary public financing system, which allows candidates either toaccept public funds or solicit private contributions, is a classic level-up program;it amplifies the voice of the poor but does not limit the influence of the rich. Alaw limiting the amount that individuals or PACs can contribute to a candidateis a classic level-down program. One of the first reform proposals came fromSenator McCain and Senator Feingold who proposed to set some limits on thecontributions given to the parties, candidates and Congressmen. From thisstarting point, there have been several proposals to reform the funding systemof political parties in the United States such as Donnelly, Fine and Miller (1999),Ackerman, Pollett and Phelps. The State of Maine has carried out a reformbased on the establishment of limits to private contributions as a previous stepto a complete public system of funding similar to the European one. Otherstates, like Arkansas, have gone a little bit further and have reformed thefunding system in such a way that are the voters who make the contributionsto parties via taxes (Pollet (1999), Phelps (1999)).

The model presented here is based on the idea of Bruce Ackerman (1999) andRichard Hasen (1998). The dissatisfaction with the current system of fundingprovides an opening for the true political reform, but, what type of reform isbest? This paper argues for a new system of campaign finance that shouldappeal to those on the left and the right: a plan for mandatory campaignfinance vouchers (or also called Patriot Money) that supplants rather thansupplements, the current campaign finance system in the U.S. Under this plan,each voter would have the opportunity to contribute vouchers to candidates orto interest groups in every federal or local election cycle. The interest groupwould use the vouchers to contribute to candidates or to organize independentexpenditures campaigns. Only funds from the voucher system could be spent tosupport or oppose candidates for elected offices. The voucher plan will replacethe plutocratic campaign finance system with a system driven by the intensityof voter support for candidates, their causes and ideologies. The plan does notdepend upon utopian hopes that politics can become less self-regarding or morealtruistic; instead, it reforms the political market with proper incentives. Thelegal detail of this type of reform have already been questioned and analyzedby the authors previously mentioned (Hasen (1996)). Then, our research doesnot question the viability of the plan, but focuses on the effects of the reformon the political process. The voucher plan proposed by Bruce Ackerman andRichard Hasen is a financing system that both levels-up and levels-down. Theplan levels -up in the sense that all voters, even those who never made campaigncontributions before, are given vouchers to contribute to candidates for office.Vouchers facilitate the representation of groups which lack a voice in the currentsystem. In this sense, the proposed plan favors the appearance of new interest


groups. But these vouchers also level down by prohibiting all other sourcesof campaign money; the rich can no longer exercise greater influence throughprivate contributions and independent expenditures.

Our proposal will be modeled in the framework of political competition andfor that we will follow the most important literature in this field and alsothe literature on funding of political parties. We will follow basically the modelproposed by Grossman and Helpman (1996). Voters will be à la Baron (1994) inthe sense that there are some informed and some uninformed voters. There willbe a proportion of voters that are informed about the proposals and ideologiesof the parties and the rest of voters who are not. The latter are the parties’objective: the uninformed or impressionable voters are those who have no prioropinion about politics (see Zaller (1999)). They are susceptible to electioneeringactivities like campaign propaganda. Their vote does not depend on the policiessupported by the parties rather on the political campaign that each party carriesout. Regarding the electoral campaigns, these can be seen both in the role ofpersuasive and informative. Persuasive campaigns allow those parties withmore funds to obtain more votes from impressionable voters. That is, money“buys” votes. On the other hand, informative campaigns inform those voterswho did not have previous information about the parties’ proposals. In thiscase, those parties who are more efficient in transmitting information (or havea better information technology) will obtain more votes (Ortuño-Ortín andSchultz (2000)). In our model the campaigns will be persuasive. That is, moremoney, more votes.

The new issue in our model is the fact that voters receive vouchers froman institution and that they have to allocate them to any of the agents inthe political process. Moreover we also introduce the possibility that votersform interest groups. Hence, the formation of lobbies is endogenous. We willmodel this part following the literature on endogenous formation of institu-tions and groups (Baron(1993), Caplin and Nalebuff (1997), Marhuenda andOrtuño-Ortín (2001) ) and on party activism (Aldrich (1983a,1983b). We willsee how the interest groups and the voters behave and what are the effectson the political process. Considering the fact that both parties are, ex-ante,equally popular we find that an interest group will always form and that it willnever have more than one half of the population. At the same time, we can alsosee that the interest group will always be radical. This last result makes sensewith the popular idea that interest groups cluster the most extremist individ-uals in a society. On the other hand, when we study the situation with twointerest groups we find that interest groups are horizontal and that, if there isno possibility of exclusion, the lobby’s members will always be one half of theinformed voters.


52 Governability and Representativeness: Simulation of ConcreteVoting Situations

Vito Fragnelli, Giovanni Monella and Guido OrtonaDipartimento di Scienze e Tecnologie Avanzate

Universita del Piemonte [email protected]

When we elect a Parliament, we decide not only who will represent us, butmany other things too: first of all, obviously, who will govern us, and some“hows” of that. The result depends crucially on the electoral rule we use. Inother words, the choice of the electoral rule is relevant for many goals. A veryrough survey of (some) literature can count up to seventeen items. Can weproperly evaluate the effect of an electoral system on a goal? In principle, theanswer is yes.

The first step is to elaborate a suitable and measurable index for that goal;next, all what we need is to compute an estimation of the value of the indexfor different systems. This can be done through simulation. What above doesnot allow to pinpoint the “best” electoral system; on the contrary, it goes veryclose to prohibit it. The obvious reason is that the choice of an electoral systemaffects specifically many goals, possibly all; hence, the choice of the “best”electoral system crucially depends on the weight assigned to each specific goal.An electoral system X that performs better than another system Y on all goalsdominates Y , and hence excludes Y from the choice; but X may be consideredthe best one only if it dominates all the other systems. In general this neverhappens, mostly because systems that perform well on goal A usually performpoorly on goal B, and vice versa, as we noticed above. However, this is notthe right approach to find out the best electoral system. The rank of a specificgoal (as measured by a suitable index) is a function of different characteristicsof the choice process; the electoral system is only one of them. Let us consider(not only for sake of simplicity) only two indices, commonly recognized to bethe most relevant ones: the degree of correspondence to the choices of voters(representativeness) and the efficiency in government (governability), denotedby G and R. With a suitable definition of the two indices it is possible, by asimple empirical rule, to choose the best electoral system.

We need three elements: colleges, parties and constituents. Colleges andparties can be identified by a number in the interval [−1, 1], where −1 indi-cates an extreme left position, 1 an extreme right one and 0 a central one.A constituent profile is characterized by the complete sequence of its politicalpreferences, expressed from a sequence of hexadecimal numbers. The programworks on a hypothetical country formed by 100 colleges and produces as outputa parliament. It’s possible, selecting the parties, to define a majority and to


have the relative indices of governability and representativeness. It’s possibleto choose ten different electoral systems.

Finally a new electoral system was implemented: the VAP (Votum A Pos-teriori in Latin) system (Ortona, 2001). It runs through two stages. In thefirst one, the parliament is elected with a proportional system (e.g. perfectproportionality), in order to maximize the representativeness. Note that someparties are labelled as relevant. When a majority forms the Government, itsconfidence vote is as usual, except that from this moment on, the votes of themembers of the relevant parties in the majority have a voting weight larger than1, if they vote accordingly to the Government. These parties are labelled ascrucial. This is a kind of majority premium that is not applied if the membersact against the Government. This electoral system increases the stability as itpunishes defections from the majority.

Main References

Ortona, G. (1998) Come funzionano i sistemi elettorali: un confronto speri-mentale, Stato e Mercato 54.

Ortona, G. (2001) Experimental Assessment of a Suggested Two-stage Elec-toral System, Working Paper.

Trinchero, R. (1998) Sistemi di voto e preferenze individuali: un’analisi com-parata con l’uso della simulazione al calcolatore, mimeo.


53 Spatial Competition Between Two Candidates of Different Qual-ity

Enriqueta Aragones and Thomas PalfreyInstitut d’Analisi Economica, CSIC

[email protected]

This paper examines competition in the standard one-dimensional Downsianmodel of two-candidate elections, but where one candidate (A) enjoys an ad-vantage over the other candidate (D). Voters’ preferences are Euclidean, butany voter will vote for candidate A over candidate D unless D is closer to herideal point by some fixed distance d. The location of the median voter’s idealpoint is uncertain, and its distribution is commonly known by both candidates.The candidates simultaneously choose locations to maximize the probability ofvictory. Pure strategy equilibria often fails to exist in this model, except un-der special conditions about d and the distribution of the median ideal point.We solve for the essentially unique symmetric mixed equilibrium with no-gaps,show that candidate A adopts more moderate policies than candidate D, andobtain some comparative statics results about the probability of victory andthe expected distance between the two candidates’ policies. We find that bothplayers’ equilibrium strategies converge to the expected median voter as A’sadvantage shrinks to zero.

We test the predictions of this model using laboratory experiments, and findstrong support: the better candidate adopts more centrist policies than theworse candidate; the equilibrium is statistical, in the sense that it predicts aprobability distribution of outcomes rather than a single degenerate outcome;and the equilibrium varies systematically with the level of uncertainty aboutthe location of the median voter. We also show that there is a “centrist bias”,in the sense that both candidates adopt centrist positions significantly morethan predicted by the theory. This bias is explained by Quantal ResponseEquilibrium theory.


Cournot Games

54 The Value of Public Information in a Cournot Duopoly

Diego Moreno, Ezra Einy and Benyamin ShitovitzUniversidad Carlos III de Madrid

[email protected]

In a Cournot duopoly where there is uncertainty about the market demandand/or the cost function, the value of public information is determined bythe curvature of an associated real-valued function: if this function is convex(concave) then the value of public information is positive (negative). Theseresults allow us to identify interesting subclasses of industries where the value ofpublic information is positive (negative), and are useful in specific applications.


55 Information Acquisition: A (enhanced) two-stage approach

Eliane CatilinaWashington and Lee University

[email protected]

This paper presents an alternative or enhanced approach to informationacquisition in Cournot markets with stochastic demand in which the cost ofinformation acquisition is endogenously determined by firms’ information pur-chasing strategy. I propose a two-stage model in which in the first stage eachfirm decides whether it will join a coalition to purchase information and there-fore share the cost of information acquisition, to individually purchase infor-mation, or to remain uninformed. In the second-stage firms engage in Cournotcompetition to choose output. The model I propose encompasses the main as-sumptions of the current view on information acquisition, mainly those relatedto the role of information and how it affects firms’ profits. However, I willargue that by adding natural assumptions on oligopolists’ behavior, I can offera model that provides a better description of firms’ actions and trade-offs thanthe standard view. The paper also offers experimental evidences supportingthe new approach to information acquisition.


56 Inventories and Price Volatility in a Dynamic Cournot Duopoly

Henry ThilleUniversity of Guelph, Canada

[email protected]

Inventories can have strategic effects on sales and production in a Cournotduopoly with convex production costs. I examine the feedback equilibrium ofan infinite horizon game in which firms face uncertain demand and supply.They can use inventories to buffer the effects of the random variables. Thestrategic use of inventories can lead to more or less volatile prices depending onthe source of the uncertainty. In particular, prices are more volatile when theuncertainty is primarily due to randomness in demand and less volatile whenthe uncertainty is primarily due to randomness in costs.

Inventories have strategic value because they can represent a relatively in-expensive source of supply for a firm that faces increasing marginal costs ofproduction. Inventories are not costless, they must have been produced atsome point in the past, and the firm incurs a cost to store them. As these costsare sunk at the beginning of a period, a firm that holds inventories lowers itscost of sales in that period by selling from inventories. In a Cournot game, thislower cost of sales results in increased sales and profit for the firm using theinventories and reduced sales and profit for the other firm.

The game is a Linear Quadratic one in which firms observe the current stateof demand and cost at the beginning of a period. Once the current demand andcost is known, firms choose that period’s production and sales quantities withthe residual production added to the firms stock of inventories to carry to thenext period. I show that the average level of sales and output price are the sameas occur in the static Cournot game. This occurs because the strategic effectsmanifest themselves through changes in inventories, which are zero on average.However, the inventories do not respond as strongly to the random demandand cost realizations as they do when strategic considerations are absent (suchas with perfectly competitive firms or a two-plant monopolist). This reducessmoothing of random fluctuations leads to increased price response to demandshocks and decreased price respond to cost shocks.


Matching Markets

57 A Generalized Assignment Game

Ester Camiña Centeno ([email protected])Universidad Autónoma de Barcelona

We study the interaction of a finite number of sellers and buyers in a market.The fact that agents in these markets belong, from the outset, to one of twodisjoint sets, sellers and buyers, and the bilateral nature of exchange that exists,allows us to treat them as “two-sided matching markets”. The proposed gameis a natural extension of the Assignmnet Game (Shapley and Shubik, 1972) tothe case where each seller owns a set of possibly different objects instead ofonly one indivisible object, and each buyer wants to buy at most one object.Sotomayor (1999) studies a special case of our model where all the objectsa seller owns are equal. The game is a many-to-one matching model withmoney. An outcome of the game specifies a matching between buyers andsellers and a vector of prices. The worth of a potential transaction is given bya nonnegative real number associated with each possible pair of a buyer and anobject. The set of sellers in our model can also be seen as multiproduct firmswith one vacancy per division and the set of buyers as workers, where salariesare determined as part of the outcome of the game. Two natural questions arewhat kinds of partnerships we can expect to observe and how agents will dividetheir gain. The answers to these questions involve the choice of an appropriateconcept of equilibrium that in these kinds of games is called stability. Thereare two different ways of defining stability in a many-to-one matching model:considering deviations only of pairs of agents (pairwise stability) or deviationsof groups of agents (group stability). We define a new concept of pairwisestability that takes into account the fact that the objects that one seller ownsare possibly different, which implies that the gain that a buyer and a sellercan share is no longer the same, but depends on the object bought. We alsopropose a definiton of group stability adapted to our framework. This is theconcept of stability we will concentrate our results on, since it is the moreadequate concept for a many-to-one matching model. In contrast with theprevious models, we prove that group stability is sufficient but not necessaryfor pairwise stability. We prove existence of both pairwise and group stableoutcomes and study the structure of the group stable set. We show that thegroup stable set is the Cartesian Product of the set of group stable payoffsand the set of optimal matchings, and that it forms a complete lattice with anoptimal group stable payoff for each side of the market. We also show that thecore of a given market equals the set of group stable outcomes. We relate thegroup stable set with the set of competitive equilibria and compare our resultswith the Assignment Game.


58 Many-to-one matching when the colleagues do matter

Pablo RevillaUniversidad Pablo de Olavide de Sevilla

[email protected]

This paper studies many-to-one matching market in which each agent’s pref-erences not only depend on the institution that hires her, but also on hercolleagues, that are matched to the same institution. With an unrestricteddomain of preferences the non-emptyness of the core is not guaranteed. Wepresent some conditions on agents’ preferences, that determines two possiblesituations in which, at least, one stable allocation exists. The first one reflectsreal-life situations in which the agents are more worried about an acceptable setof colleagues that the frim hiring them. The second one is related to marketsin which a workers’ ranking is accepted by everyone (workers and firms) in themarket.


59 Matching, Search, and Intermediation with Two-Sided Hetero-geneity

Nadia BuraniDepartamento de Economía, Universidad Carlos III de Madrid

c/ Madrid 126, 28903 Getafe, [email protected]

This paper analyzes the role played by intermediation in the context of adecentralized market, where trade is carried out through bilateral bargaining,and where the bargaining outcome depends on the process of search for suitabletrading partners. To this purpose, first a model of random-matching and two-sided search with heterogeneous agents is developed. The characterization ofpure search equilibria is provided and it is shown that sorting externalitiesmight cause inefficiencies.

Secondly, the incentive for intermediation to arise endogenously within thisenvironment is considered. Intermediation might induce separation of theagents’ types, thus serving as an efficiency-enhancing mechanism.


Production Games

60 LP Games with Committee Control: behavior of the core underdirect market replication

Elisenda Molina, and Juan TejadaUniversidad Complutense de Madrid & Universidad Miguel Hernández

[email protected]

Some optimization situations, in which several independent decision makersare involved, can be modeled as cooperative games with side payments. This isthe case of those economic production situations in which economic agents poolresources to produce finished goods which can be sold at a given market price.Since Owen (1975) defined linear production games arising from situations inwhich the production process is linear, several generalizations of the originalmodel have been proposed. Generalizations of Owen’s model are given in Dubeyand Shapley (1984), Granot (1986) and Curiel, Derks and Tijs (1989).

Owen (1975) makes use of the duality theory of linear programming to obtainequilibrium price vectors and to prove the non-emptiness of the core. He showsthat in linear production games, a part of the core elements can be foundby using shadow prices for the resources. When each player receives a payoffthat equals the value of his/her resource bundle under the shadow price, thisforms a core element in the production game. Moreover he shows that, underreplication, in the non-degenerated case, the core converges finitely to the set ofcompetitive equilibria. Afterwards, Samet and Zemel (1984) obtain a necessaryand sufficient condition for finite convergence, under the equivalent approachof refinement.

Here, we consider the generalization proposed by Curiel et al. (1989): linearproduction games with committee control. In this model, it is assumed thatresources are controlled by committees of players. Now, each coalition hasaccess to certain parts of the total resource bundle. The access to each ofthese parts is governed by a simple game. When each of these simple gameshave a non-empty core, there is a possibility to assign a part of the resourcebundle to each of the players by using core allocations of each of the controlgames and to assign to each player the shadow value of the part of the resourcebundle he/she obtained. This payoff vector is once again a core allocation ofthe associated production game. Following Owen’s previous work, our goal isto study the limiting behaviour of the economy when the number of players ofeach type increases. To be specific, the central problem we tackle in this paperis the convergence of the set sequence composed by the core of the LP-gamesobtained when the player set is successively refined. We concentrate on the


relation between the limit of the cores of the r-refinements and the set of core-competitive-equilibria or shadow-price-based core allocations, which is definedas the set of all payoff vectors obtained by the process described before.

First, we face the problem of extending the Curiel et al. model, i.e., wedescribe how to define the LP-game with committee control which arises whenthe player set is refined (or equivalently replicated). The refinement we haveconsidered is based on the definition of a “direct market generated by a game”introduced by Shapley and Shubik (1969) and it is suitable when players areactivity owners. Then, we proceed on the analysis of the limiting behaviourof the core. We give necessary conditions for convergence, as well as for finiteconvergence, to the set of core-competitive-equilibria.


61 Competition and cooperation in non-centralized linear produc-tion games

M. G. Fiestras-Janeiro, F. R. Fernández, I. García-Jurado and J. PuertoUniversidade de Vigo and Universidad de Sevilla

[email protected]

In this paper we consider non-centralized linear production situations. Inone of those situations, each producer i of a set N has an optimal productionplan xi0 for a linear production problem given by

maxcixi : Aixi ≤ bi, xi ≥ 0.

In order to maximize the benefits, the producers decide to take profit of theirsurpluses

(b1 −A1 − x10, . . . , bn −Anxn0 ).

We study the games which describe this interactive situation when playersdo not cooperate, when players cooperate and side payments are possible, andwhen players cooperate and side payments are not possible.


62 Profit Sharing in Enterprises

Vito Fragnelli and Maria Erminia MarinaDipartimento di Scienze e Tecnologie Avanzate


We consider a problem in which a set of agents have to carry out an enter-prise; if they succeed in performing it they have to pay a cost, but they receivea benefit. In our situation the cost depends on the agents involved, while thebenefit is fixed.

We may refer to connection situations where the objective and the benefitis the connection of two different nodes and the cost is the minimum costconnection. Another possibility is represented by several firms that have toproduce a machine that is sold at a fixed price; each firm may produce allthe components of the machine, but they have different technologies, so thatthe components have different costs. If a subset of firms agree on a jointproduction, they can save on the costs, while the selling price remains fixed.Also bankruptcy situations match our hypotheses with the estate in the roleof the fixed benefit, while the costs are the claims of the agents not in thecoalition. Finally the model fits insurance situations where a set of companiesmay reduce the costs if they share the risk, but the amount of the premium isfixed and independent from the number of companies.

Formally we consider a set of agents N = 1, 2, . . . , n, a cost functionP : 2N → R ∪ +∞, a fixed benefit φ ∈ R and define the characteristicfunction v of a TU-game as v(S) = max0, φ− P (S), for all S ⊆ N .

We want to investigate how the agents may share the benefit, taking intoaccount their charges of the costs. For this class of games we have the followingresults.

• There exists φ such that the game is balanced if and only if φ ≤ φ or thegame is balanced for every φ ∈ R.

• Under suitable hypotheses on the cost function P it is possible to give asimple expression for φ.

• In the particular case of φ = φ we show that the core, the nucleolusand the τ -value coincide with the marginal solution, that assigns to eachplayer exactly his marginal contribution.


Simple Games

63 Dimension of complete simple games with minimum

María Albina Puente and Josep FreixasUniversitat Politècnica de Catalunya

[email protected]

The main contribution of this paper is to calculate the dimension of com-plete simple games with minimum using their characteristic invariants. As aconsequence of this result we can deduce that there are complete simple gamesof every dimension. In a weighted majority game we can line up the voters inthe order determined by noncreasing weight. Speaking informally, it certainlyseems as if this same line-up should be one of creasing or decreasing influence(the exact definition of player i has more influence than player j is given by thedesirability relation introduced by Isbell (1956)) over outcomes in the votingsystem modeled by the game. Indeed, the phrase “carries more weight” is usedcolloquially to mean “has more influence”. Complete games can be describedas those for which it is possible to line up the players in some linear orderingfor which influence is decreasing. This informally justify that every weightedmajority game is complete. Then, a natural extension of weighted majoritygames are complete games, for which the desirability order is total.

Carreras and Freixas (1996) associate a system of quantities (characteristicinvariants) with every complete simple game and state their basic properties.Then they show these quantities determine the game (uniqueness) and that ev-ery system (admissible system) is associated with some complete simple game.A partial motivation to introduce characteristic invariants is the issue of char-acterizations of weightedness and we notice that the assumability condition ofElgot (1960) and the trade robustness of Taylor and Zwicker (1992) are adaptedto the representation given by characteristic invariants.

A particular case of complete simple games are those with minimum thathave been studied by Freixas (1997) and Freixas and Puente (1998). The firstwork provides necessary and sufficient conditions to determine when a game ofthis type is weighted. In the second one the characteristic invariants are used tofacilitate the calculus of different kind of solutions like the nucleolus, the kerneland semivalues. Different situations described through rules of decision canbe modeling using complete simple games with minimum; among them, theUnited Nations Security Council and the procedure to amend the CanadianConstitution.

It is well known that every simple game can be represented as intersectionof weighted majority games. It nevertheless becomes of interest to ask how


efficiently this can be done for a given simple game. The question of efficiencyleads to the definition of dimension. A simple game is said to be of dimensionk if and only if it can be represented as the intersection of exactly k weightedmajority games, but not as the intersection of k − 1 weighted majority games.Most naturally occurring voting systems in use are modeled by simple gamesof dimension one or two. Interesting examples of dimension 2 are the UnitedStates Federal System and the procedure to amend the Canadian Constitution.A central question to suggest itself is whether or not there are simple gamesof every dimension. This was answered by Taylor and Zwicker (1999). Theyproved that for all positive integer n, there is a simple game of dimension n.However, the Taylor and Zwicker’s proof is based in a class of games highly notcomplete. In this situation we raise the next question: is it possible to definefamilies of complete simple games such that their dimension increases with thenumber of players?, i.e., for all positive integer n, there is a complete simplegame of dimension n? The answer to this question is given in this paper by thedetermination of the dimension of complete simple games with minimum. Thisresult shows that the complexity of the game is not closely connected with thefact that the desirability relation is total.


64 On the convenience to form coalitions or partnerships in simplegames

M. Dolors Llongueras and Francesc CarrerasUniversitat Politècnica de Catalunya

[email protected]

Kalai and Samet (1987) introduced the concept of partnership in coopera-tive games. A partnership is a coalition that has an internal structure and,simultaneously, behaves as an individual member.

Furthermore, when one o more players decide to form a coalition a coalitionstructure (i.e. a partition) arises and the quotient game is defined.

In this work the Shapley and Banzhaf values are used to describe the conve-nience to form either coalitions or partnerships in a simple game. In fact, thedifference between the Shapley (and Banzhaf) value for the partnership playersand for their representative in the quotient game is analyzed. Maximum andminimum values for this difference are established.

Finally, partnership and coalition structure imposition are studied in weightedmajority games, a fraction of simple games.


65 Structure of voting simple games with several levels of approval

Josep Freixas and William S. ZwickerUniversitat Politècnica de Catalunya

[email protected]

In modeling the type of voting intended to register collective approval ordisapproval of a proposal, the literature on voting systems has confined it-self almost exclusively to the mathematical structures known as simple votinggames. By their nature, simple games allow for exactly two possible levels ofapproval in the “input” (voters must choose between “yes” and “no”) and twolevels in the “output” (the proposal either passes or does not pass). Yet inmany of the real voting systems that have been modeled by these games, suchas that of the United Nations Security Council, abstention plays a key role asa middle level of voter approval.

One factor that may have retarded the study of voting games with abstentionis the absence of a completely satisfactory definition of weighed voting in thiscontext. In a weighted simple game with no abstentions allowed, each voter isassigned a numerical weight, and a proposal is approved if the total weight castin favor meets or exceeds some preset threshold. There are two tempting waysto modify the definition in order to account for abstention: collective approvalcan require that the ratio of total weight cast in favor to total weight castagainst meet or exceed a preset threshold, or that the difference between thesetotals meet or exceed such a threshold. These two approaches are distinct,though each reduces to the standard definition when no voters abstain. Withabstentions allowed, the UN Security Council voting system fails to be weightedunder either definition.

We argue in favor of a third definition of weighted voting with abstention,which is strictly more general than either of the two mentioned above. TheUN Security Council system is weighted under this new definition. The newdefinition, generalizes to the structures we call (j, k) games, in which thereare j ordered levels of input approval and k ordered levels of output approval.While it is the (3, 2) games and (3, 3) games that are probably most relevant tolegislative voting, these are not empty generalizations. Our examples of gradingsystems suggest that level structures with larger j and k values arise naturallyas models of certain types of decision making.

The study of the weightedness of (j, k) simple voting games is solved bymeans of our “grade trade robust” notion. However two possible extensionsof (j, k) weighted games arise in a natural form. We also focus our interest indeveloping them.

The first extension involves preference relations among voters intending tomeasure, what it means for one voter to have more influence than another.


Those games for which the associated “desirability” relationship is total willbe called “linear”. There are several subclasses of (j, k) weighted voting gamesthat deserve be considered, and we will show that each one of these subclassesmay be extended by means of an appropriate linear desirability order.

A second natural extension is that of considering (j, k) simple games definedas intersections of (j, k) weighted games. It is of interest to ask how efficientlythis can be done for a given (j, k) simple game and whether each (j, k) simplegame admits a finite intersection. We check the affirmative answer to the secondquestion, providing a notion of “dimension” for (j, k) simple games.


TU Games I

66 On coalitional semivalues

M. Josune Albizuri and José Manuel ZarzueloUniversidad del País Vasco — Euskal Herriko Unibertsitatea

[email protected]

In 1977 Owen defined and axiomatized the coalitional value for games withtransferable utility, providing in this way a generalization of the Shapley valueto the coalitional framework. Later, Hart and Kurz (1983) gave an alternativeaxiomatic characterization of the coalitional value by considering games withan infinite universe of players. In this paper we obtain two results. First, wepropose another axiomatic characterization of the coalitional value by employ-ing the same games as Hart and Kurz (1983). In fact, in our characterizationwe employ the same axioms in Hart and Kurz’ characterization (1983) exceptone which is withdrawn. Moreover, we change one of their axioms, namelyanonymity, becoming it stronger so that it turns also into a stability axiom.Second, we define the coalitional semivalues providing a generalization to thecoalitional context of the semivalues, defined by Dubey et al. (1981). Theseauthors obtained the semivalues by removing efficiency from the classical ax-iomatization of the Shapley value (1953) and by adding other axioms verifiedby the Shapley value. In our paper, to define the coalitional semivalues, we willemploy our characterization of the coalitional value. We will remove also effi-ciency, we will add the translations to the coalitional framework of the axiomsadded by Dubey et al. (1981) and we will add a new axiom which is specific forthe coalitional context and is verified by the coalitional value. The coalitionalsemivalues could also be got in a similar way as Owen obtained the coalitionalvalue. Owen (1977) defined the coalitional value of a game by applying twicethe Shapley value. First, the Shapley value is employed at the level of the coali-tions of the coalitional structure formed by the players, and a new game foreach coalition of the coalitional structure is obtained. After that, the Shapleyvalue is applied to these new games. These results yield precisely the coalitionalvalue of the original game. So, we can say that the coalitional value is obtainedby means of a “composition” of the Shapley value with itself. In this paperwe will show that the coalitional semivalues we have defined can be obtainedalso by means of a “composition” of semivalues. Furthermore, if one additionalaxiom is considered, the coalitional semivalues that result are “compositions”of a semivalue with itself. It can be pointed that if we consider the charac-terization provided by Hart and Kurz (1983) and we define the correspondingcoalitional semivalues, we do not obtain all the “compositions” of semivalues,but only those in which a semivalue is “composed” with the Shapley value.


67 On the serial cost sharing

José Manuel Zarzuelo, M. Josune Albizuri and Juan Carlos SantosUniversidad del País Vasco — Euskal Herriko Unibertsitatea

[email protected]

In this paper we study a solution for discrete cost allocation problems,namely, the serial cost sharing method. We show that this solution can becomputed by applying the Shapley value to an appropriate TU game and wepresent a probabilistic formula. We also define for cost allocation problems amultilinear function in order to obtain the serial cost sharing method as Owen(1972) did for the Shapley value in the cooperative TU context. Moreover weshow that the pseudo average cost method is equivalent to an extended Shapleyvalue.


Decision Theory

68 Dynamic Consistency in Extensive Form Decision Problems

Nicola DimitriUniversità di Siena, Italy

[email protected]

In a stimulating paper Piccione and Rubinstein (1997) argued how a decisionmaker could undertake dynamically inconsistent choices, when in an extensiveform decision problem she exhibits a particular type of imperfect recall named“absentmindedness”. Such imperfection obtains whenever an information setincludes histories along the same decision path. Starting from work focusingon the Absentminded Driver example, and independently developed by Segal(2000) and Dimitri (1999), the main theorem of this paper provides a generalresult of dynamically consistent choices, valid for the entire class of finite ex-tensive form decision problems (with any kind of imperfect information) with-out nature. The result obtains under two main behavioral assumptions, thatfully take into account the decision maker’s awareness of her possibly beingabsentminded. The first assumption says that upon reaching an informationset, beliefs to be at a certain decision node should be consistent with the be-havioral strategy chosen at that set, and not with the one adopted prior toreaching the set itself. The second assumption fully incorporates into the deci-sion maker’s reasoning the idea that, for the decision maker herself, the gamehas the same value at all nodes within an information set, most notably thosewhere absentmindedness is exhibited. The paper Dimitri (1999), only on theabsentimindeddriver, was presented at the Bologna 1999, Meeting organised bythe Italian Game Theory Association.

References

N. Dimitri (1999) A Note on Time Inconsistency in a Game with Absentmind-edness, Quaderni del Dipartimento di Economia Politica 273, Universityof Siena.

M. Piccione and A. Rubinstein (1997) On The Interpretation of Decision Prob-lems with Imperfect Recall, Games and Economic Behavior 20, 3—24.

U. Segal (2000) Don’t Fool Yourself to Believe that You Won’t Fool YourselfAgain, Economics Letters 67, 1—3.


69 Systemic Game Theory: Werterian Approach

Alexandru W. Popp4650 Clanranald 16, Montreal, Quebec, H3X 2R9 CANADA

[email protected]

The equilibrium in a game changes once additional information is beingintroduced. The equilibrium as we know it in game theory becomes absolutewhen we are looking at the system in which the game is played and who theplayers are, not only at the game. I make the distinction between the currentstand of game theory and the systemic game theory. I critique the currentstand of game theory. It stagnated and became an inert system that does notrepresent actual situation in the world. I am looking for answers to questionslike: taken into consideration not only the players and the game, but also,the experiences that the players have and the system in which the game isplayed, how does the solution of the specific game change (or, how does thegame change)? Under what assumptions is a game transformed? How is ittransformed?

Game theory, presently, is a photo camera: it takes a snapshot at the playersand the game. Because of this, it creates a two dimensional (2D) image. It hasbeen argued, and rightfully so, that the theory is an idealistic one in which thegames as well as the players, are ’ideal’. If one looks at the world, one realizesthat it is not two dimensional, but a three dimensional system that is not ideal.

Systemic game theory takes into consideration the system in which a spe-cific game is played. The types of players are analyzed and categorized. Ingeneral, people want the maximal benefits (payoffs) with the minimal involve-ment. Systemic game theory takes into account almost all factors that influencethe players and the game in any way.

Regarding the players, we have two main types: rational and irrational. Ican state that the rational player is using an egoistic judgment: the more hegets the better it is (I call this the 1st level of rationality). For this player,any other method of judgment is irrational. Regarding the irrational player,we can characterize him in two ways: idiot; or altruistic (I call the altruisticapproach the 2nd level of rationality). The players can have three possiblestrategies characterized by their psychological aspect: maximizers, deniers, andcooperators.

A new methodology of analysis is introduced. There is a return to the scien-tific method. After describing the overarching system and provide some usefultools: multiplicative multiattribute utility models; Klee’s algorithm; Church-man and Ackoff approximate measure of value; global convergence; method ofdimensional analysis. Using these operational devices, we can distinguish what


are the important characteristics of the players and the game and we can com-bine them. The less important elements are discarded, and the end analysisbecomes more representative of real life.

2×2 matrix examples are given in support of this new methodology. Analo-gous as well as polar strategies are analyzed in specific games. Moreover, I amlooking at systemic game theory through an economic mirror. I analyze realworld circumstances; that is why I am looking also at games without equilib-rium.

Even in games in which game theory cannot find an equilibrium becauseit does not exist, systemic game theory has methods of finding the higherpower point using higher power strategies. Moreover, my approach takes inconsideration what the players want: to achieve the higher power point, or notto achieve it.

At the end of my paper, I compare current game theory and systemic gametheory. One can see the theoretical differences. Some of the axioms of the firsttheory do not hold in the second one.

In order to be more specific and accurate regarding situations of the realworld, one must not analyze them with the current approach of game theory.Systemic game theory gives the best approach to life and it is a better andprecise tool. Looking at the broader picture, one can understand and predictspecific situations, outcomes in the real world, not only in a laboratory.


Games in Extensive Form

70 Modeling Deception in Extensive Form Games

Oliver J. BoardDepartment of Economics

University of [email protected]

The standard model of an extensive form game rules out two phenomenaof importance in situations of dynamic strategic interaction: deception andunreliability. We show how the model can be generalized to incorporate thesephenomena, and give some examples of the richer model in use.

We say that deception takes place when one player tricks another into be-lieving that she has done something other than what she actually did. Thestandard model does allow moves to be uninformative (whenever informationsets are non-singleton), in that it is not revealed which of several moves hasbeen made. But they cannot be deceptive: the actual move made is neverruled out. Our extension of extensive form games relaxes the assumption thatthe information sets partition the set of nodes. In particular, the set of nodesbelieved possible after a certain move is made might not include the actualnode.

Consider the story of the wooden horse in Virgil’s Aeneid Book II. TheGreeks have two choices, to go home and give up the war or to stay and attemptto sack Troy. The latter seems hopeless until Odysseus suggests the followingplan: they should sail their ships out of sight behind the island of Tenedosand leave a gigantic wooden horse in front of the city. Believing the Greekshave really gone home, the Trojans accept the horse as a gift and break downtheir walls to wheel it into Troy. The Greeks then leap out of the horse andsuccessfully sack the city. What we have here is a genuine case of deception:one move is made, but another is observed. Note that while standard gametheory does not rule out an agent having false beliefs (for example the beliefsthat justify a rationalizable strategy may well be mistaken), nothing in thestructure of the game itself forces her to have these beliefs. When deceptiontakes place, on the other hand, the agent is forced to have false beliefs. Sodeception is not simply a case of an agent coming to a false belief on the basisof a mistaken prior. Rather, the agent receives false information about whatmove has been made, and updates her beliefs according to that information.

Formally, we replace the information sets of the standard model with aninformation correspondence Ii for each player. Ii is a function from the setX of nodes of the game tree to 2X , the set of subsets of X. Ii(x) is to be


interpreted as the set of nodes which player i considers possible when actuallyat node x. A player is deceived at node x if x is in Ii(x). To model the deceptionof the Greeks described above, we allow them three choices at the beginning ofthe game: to go home (reaching node h); to sail behind the island of Tenedos(t); and to stay put (s). In each case, the Trojans can choose to accept or rejectthe wooden horse. The Trojans’ information at these three nodes is defined asfollows: Ii(h) = h; Ii(t) = h; Ii(s) = s. It is clear this cannot be representedby a partition.

If deception is allowed, an agent may receive contradictory information aboutwhere she is in the game. Further on in Aeneid Book II, we find that Sinon andLaocoon give the King of Troy two opposing accounts of the true purpose ofthe wooden horse. In such cases, some of the information must be unreliable.We further enrich the standard model by attaching reliability weights to eachnode. These weights have quantitative as well as qualitative significance, andthey tell us how an agent will evaluate various conflicting pieces of information.Her beliefs are updated using a modified version of the belief revision systemproposed by Spohn (1988) (used in a game-theoretic context by Board 1998(TARK VII). The new system implies a relaxation of the AGM axiom of success,so that the latest piece of information is not always accepted. In the exampleof Sinon vs. Laocoon, although Laocoon spoke second and warned King Priamthat the wooden horse was a trick, the king decided that Sinon was a morereliable witness and accepted it as a genuine gift. We model this situation byattaching a greater reliability weight to the nodes following Sinon’s move thanto those following Laocoon’s move.

In order to analyze rational behavior, we show how to construct epistemicmodels for these games. These models tell us what the players believe aboutthe game and about each other, and also how these beliefs will be revised asthe game progresses. This allows us to give a formal definition of rationalityas expected utility maximization. We then consider various solution concepts.It can be shown that a player cannot be deceived along the path of a (perfectBayesian) equilibrium, since the requirement that beliefs be consistent withstrategies dictates that those beliefs be correct. In certain games, deception isinconsistent with common belief in rationality as well. The intuition behindthis result is as follows: for a player to believe that her opponent is rationalrequires her to have beliefs about the structure of the game, and in particularabout the moves that are available to him. If these beliefs are correct, and ifonly one of these moves is rational, then she can infer what he will do. Shecannot therefore be deceived at the resulting node.


71 Games in Code Form

Cristina Peixoto and Manuel Alberto M. FerreiraDep de Matemática (ESTV) Viseu

Dep Métodos Quantitativos (ISCTE) [email protected]

The aim of this paper is to present and discuss a new representation forgames with more than 2 players. This formalization centres the consequencesof different strategies without suppressing the meticulousness of the extensiveform. In other words this representation simultaneously provides all informa-tion, which the extensive and normal forms representation establish separately.Several games are presented to illustrate this point.


Bankruptcy Games II

72 The constrained equal award rule for bankruptcy problems witha priori unions

Luisa Carpente, Peter Borm, Balbina Casas, and Ruud HendrickxUniversidad de la Coruña

[email protected]

We consider bankruptcy problems where the creditors are grouped in a struc-ture of a priori unions. Our main objective is to define a rule for dividing theavailable estate among all the creditors, which takes the structure of unionsinto account.

In a first step we introduce disjoint issue allocation situations. These are aspecial case of multi-issue allocation situations as introduced in Calleja, Bormand Hendrickx (2001); those in which every player is involved in exactly oneissue. Every bankruptcy situation with a priori unions gives rise to a disjointissue allocation situation, where the issues correspond to the unions. Thisallows us to define a class of cooperative games with transferable utility inorder to analyse bankruptcy problems with a priori unions. These games turnout to be exact.

We provide different ways to extend a rule on the class of the standardbankruptcy problems to the class of bankruptcy problems with a priori unions.We concentrate in a rule that is an extension of the constrained equal awardrule for standard bankruptcy problems. We study properties and provide twoaxiomatic characterizations of this rule. We also see that for a bankruptcyproblem with a priori unions, this rule coincides with a certain cooperativesolution concept for games corresponding to bankruptcy problems with unions.We also define consistency properties for this kind of problems, based on O’Neill(1982). Finally, we give some applications.


73 Bankruptcy situations with references

Joaquín Sánchez-Soriano, Manuel Pulido, Peter Borm, Ruud Hendrickx andNatividad Llorca

Universidad Miguel Hernández and Tilburg [email protected]

In a bankruptcy situation, one has to divide a given amount of money (es-tate) among a set of agents, each of whom has a claim on it. The total amountclaimed typically exceeds the estate available, so not all the claims of the agentscan be fully satisfied. In Pulido et al. (2001) a real bankruptcy situation, aris-ing from the university management, is analyzed. In this analysis two types ofinformation were used. On the one hand, the ‘claims’ of the different agentsand, on the other hand, the ‘references’ obtained by applying objective crite-ria. Hence, this additional information was exogenous to the claimants butclosely related to the problem. Pulido et al. (2001) focused the problem as a‘bankruptcy situation with references’ and used the additional information as astarting point to begin to distribute the estate. In their study the total amountprovided by the reference point was less than the estate.

This paper deals with bankruptcy situations with references in the generalcase. We introduce to new games associated with any bankruptcy problem withreferences and a new monotonicity property for standard bankruptcy rules,which will play an important role in the paper. We analyze compromise solu-tions in this context.


Learning Models, Contracts

74 Equilibrium Properties of Reinforcement Learning by Imitation

Antonio J. MoralesDepartment of Economics, University of Malaga

[email protected]

We investigate the ability of pure strategy imitation rules to lead the decisionmaker towards optimal actions when he faces repeatedly a decision problem.We study the perfomance of a given imitation rule in two different frameworks:in an arbitrary environment, i.e. in an environment in which the populationbehaviour is given and fixed, and in an environment in which all other agentsuse this rule. In the first framework, we show that there are no approximatelymaximising rules, i.e. rules which lead the decision maker to play in the longrun the expected payoff maximising action with probability arbitrarily closedto one, independent of what the true payoff distribution is and regardless ofthe given and fixed population behaviour. In the second framework, we findthat there are no maximising rules, i.e. rules that lead the entire population toplay the expected payoff maximising strategy.

In the absence of a maximising rule, we investigate individual agents’ incen-tives to adopt any particular rule. We define an appropriate payoff functionand for given population and fixed imitation rule, we define the set of imitationrules which are best responses to the fixed imitation rule, An imitation rule iscalled an equilibrium rule if it belongs to the set of best responses to itself. Wecharacterise the set of equilibrium rules in some restrictive settings.


75 Bidding in a Pool with Contract Market

Beatriz de Otto LópezUniversidad de [email protected]

We study the rational for buying electricity by means of forward agreementseven in absence of risk aversion. Contracting is profitable for a buyer of electric-ity because it reduces the seller’s incentive to manipulate up the prices in thespot market. In a symmetric equilibrium with contracting all the generatorscontract and ask prices which are less than in a symmetric equilibrium withoutcontract market. A generator is willing to contract only if the contract priceis greater than the expected spot market price given the equilibrium biddingfunctions. We prove that there is a set of contracts satisfying this conditionwhich, however, are such that the cost of electricity for the buyer is less thanwithout contracting.

More interesting, we show that when contracts are for a quantity whichis large enough, then the price that any generator asks at equilibrium is lessthan the cost of production for this generator. Indeed, beyond reducing theincentive to push the spot prices up, contracts to supply means that generatorsneed becaming buyers in the spot market unless they manage to be dispatched.


76 Mixed Equilibria in Games of Strategic Complements are Un-stable

Federico Echenique and Aaron EdlinUniversidad Torcuato Di Tella, Argentina

[email protected]

In Games of strict strategic complementarities, properly mixed Nash equi-libria —equilibria that are not in pure strategies— are unstable for a broad classof learning dynamics.


http://www.utdt.edu/∼fechenique/instab3.pdf

http://www.utdt.edu/~fechenique/instab3.pdf


Repeated Games

77 The Repentance Strategy

Miguel AramendiaUniversidad del País Vasco

[email protected]

It is well known that repeating a game may lead to the emergence of newnoncooperative equilibriums. In his seminal paper on trigger strategies, Fried-man (1971) showed that cooperative outcomes could be sustained by thesestrategies as subgame perfect equilibrium provided that the players are suffi-ciently patient. Friedman’s strategy prescribes cooperating to begin with andas long as the others cooperate; if any of the others deviates it recommendsswitching to the one-shot equilibrium strategy and playing it forever. Its maindrawbacks are that any deviation precludes cooperation forever in the futureand all the players suffer equally.

In a large class of infinitely repeated Cournot duopoly games with discount-ing and perfect monitoring, Segerstrom (1988) defined the repentance strategiesand showed that the symmetric monopoly payoff (cartel benefits) could be sus-tained by these strategies as subgame perfect equilibrium for the same rangeof discount factors as the trigger strategy does. In these repentance strategies,the nondeviating player forgives the deviating player and both revert back tocollusive behavior after the cheater repents paying a punishment in several pe-riods. During the repentance phase, the punisher receives an entire payoff thatis higher than the payoff of the punished player.

The context of our work is the repeated Cournot oligopoly model with dis-counting and perfect monitoring. We define repentance strategies that keepsall the properties mentioned above and also has a special feature: during therepentance phase, each of the nondeviating players is playing his best responseto the output specified to the other players. In other words, each of the pun-ishers does not have any incentive to deviate during the repentance phase. Webelieve that this repentance strategy could have some applications in the fieldof industrial organization. With the Segerstrom’s assumptions, we show thatthe symmetric monopoly payoff could be sustained by these strategies as sub-game perfect equilibrium for at least the same range of discount factors as thetrigger strategy does.


78 The economics of information transmission

Penelope Hernandez, Olivier Gossner and Abraham NeymanCORE, Université Catholique de Louvain

[email protected]

It is a well known phenomenon that communication between a group ofplayers affects the set of equilibria of a repeated game. This communicationcan be costly. Its cost may be measured in terms of complexity, or payoffs.In particular, a payoff cost of communication occurs when players communi-cate through their actions, and one is interested in characterizing the optimalcommunication.

We study a class of games that models the communication cost of onlinecoordination. For example: A sequence X = (X(t)) of zeroes and ones is chosenat random, and announced to a player, the seer, but not to his teammate, thefollower. A repeated game then proceeds in which the follower and the seerchoose actions Y (t) and Z(t) in 0, 1 at stage t, and their (common) stagepayoff is 1 whenever X(t) = Y (t) = Z(t).

We address the question of the optimal communication level from the seerto the follower in order to maximize payoffs. We characterize this level usingthe notion of entropy, and provide construction of optimal strategies for theteam: the seer and the follower. In particular, we prove that against a familyof i.i.d. random variables (1/2, 1/2), the maximum expected payoff v to theteam is the solution of the equation

H(x) + log (3) (1− x) = 1;

where H is the entropy function and log is taken in basis 2. Moreover, we provethe existence of pure strategies of the prophet and the seer that guarantee vagainst all sequences X = X(t). Hence, the zero-sum game in which player Iis the Team (no correlation is allowed between its members), player II is theopponent, and the payoff to I is as above, has a value, and this equals v. Ourresult has direct implications for repeated games played by finite automata, orby players with bounded recall.


79 Repeated Games, Automata, and Growing Strategy Space

Abraham Neyman and Daijiro OkadaThe Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, ISRAELRutgers University, 75 Hamilton Street, New Brunswick, NJ 08901-1248, USA

[email protected] & [email protected]

A (pure) strategy in a repeated game is a mapping from histories, or, moregenerally, signals, to actions. We view the implementation of such a strategy asa computational procedure and attempt to capture in a formal model the fol-lowing intuition: as the game proceeds, the amount of information (history) tobe taken into account becomes large and the “computational burden” becomesincreasingly heavy.

The number of strategies in repeated games grows double-exponentially withthe number of repetitions. This is due to the fact that the number of historiesgrows exponentially with the number of repetitions and also that we countstrategies that map histories into actions in all possible ways. Any model thatcaptures the intuition mentioned above would impose some restriction on theway the set of strategies available at each stage expands.

We point out that existing measures of complexity of a strategy, such asthe number of states of an automaton that represents the strategy needs tobe refined in order to capture the notion of growing strategy space. Thus wepropose a general model of repeated game strategies which are implementableby automata with growing number of states with restrictions on the rate ofgrowth. With such model, we revisit some of the past results concerning therepeated games with finite automata whose number of states are bounded by aconstant, e.g., Ben-Porath (1993) in the case of two-person infinitely repeatedgames.

In addition, we study an undiscounted infinitely repeated two-person zero-sum game in which the strategy set of player 1, the maximizer, expands “slowly”while there is no restriction on player 2’s strategy space. Our main result isthat, if the number of strategies available to player 1 at stage n grows subexpo-nentially with n, then player 2 has a pure optimal strategy and the value of thegame is the maxmin value of the stage game, the lowest payoff that player 1 canguarantee in one-shot game. This result is independent of whether strategiescan be implemented by automaton or not. This is a strong result in that anoptimal strategy in an infinitely repeated game has, by definition, a propertythat, for every c, it holds player 1’s payoff to at most the value plus c aftersome stage.


Voting Procedures, Game Models in Ethics

80 Candidate Stability and Probabilistic Voting Procedures

Carmelo Rodríguez-ÁlvarezUniversitat Autònoma de Barcelona

[email protected]

We extend the study by Dutta, Jackson and Le Breton (Econometrica, 2001)of the strategic incentives of the candidates to withdraw their candidacy toprobabilistic environments. A probabilistic voting procedure is a rule that foreach set of candidates at stake and each profile of voters’ preferences selects alottery on the set of candidates. It is exit stable if a candidate never benefitsby leaving the fray unilaterally. It is candidate stable at a given agenda ifno running candidate has incentives to quit. We assume that the candidatesare Expected Utility maximizers and show that any exit stable and unanimousprobabilistic voting procedure is a probabilistic combination of dictatorial rules.The results for candidate stability depend on the size of the agenda. If thereare at least four initial candidates, candidate stability implies the selection tobe consistent with a random dictatorship whenever a candidate quits, but thisis not the case when all the candidates are at stake. However, when thereare only three candidates, sub-additive distributions of the veto power are alsoadmitted.


81 The Impossibility of Ethics

Juan Perote-PeñaUniversidad Pablo de Olavide de Sevilla

[email protected]

In this paper I analyze the possibility of making ethical judgements in termsof “good” or “bad” individual actions in social situations modelled as well-defined extensive form games. After requiring some “minimal” social desir-ability properties of ethical codes in this context, we find that impossibilitytheorems can be easily generated and trade-offs between ethical properties areproved to exist. We argue that this normative approach to Game theory cangive useful insights into the study and design of ethical rules, moral maximaand religious codes that are now considered as allocation problems of “moral la-bels” to individual actions that societies can use to help enforcing the desirableequilibria in different social institutions.

Game theory is concerned with self-interested individuals and the likely equi-libria that can be predicted when individuals try to maximize their own payoffin a given social situation. Therefore, although it has traditionally been consid-ered as a positive theory of human behavior, moral considerations can also beanalyzed within the rich description of extensive form games. Notice, however,that we are not trying to develop a kind of Game theory approach to Ethics,since moral issues clearly demand a normative approach that we consider is faraway from the positive perspective of Game theory. We are basically borrowingthe tools of describing interactive decision-making situations from Game theoryto apply a normative axiomatic analysis to moral issues. Moreover, we followthe Game theory approach on the implicit assumption that every relevant as-pect of the situation is embodied in the complete definition of an extensiveform game, with a sole important and obvious difference: we no longer con-sider the moral status of actions included into the final individual payoffs, butwe restore the moral considerations as being endogenous to the model. There-fore, the moral status of the individual actions is not considered fixed in thesituation, but we will find answers to the questions of the allocation of moralvalue to actions that respect minimal moral principles.


Utility and Subjective Probability

82 A Theory of Choice over Time

Marco Mariotti, and Paola ManziniDepartment of Economics, University of Exeter

[email protected]

Human beings are affected by a fundamental cognitive shortcoming: theperception of future events becomes increasingly ‘blurred’ as the events arepushed further in time. Deciding between alternatives to be obtained in thedistant future is intrinsically more complex than deciding between the samealternatives available earlier. For instance, an individual may find 1000 eurosnow distinctly preferable to 1500 euros in one year time, but may neverthelesshesitate when asked to decide between 1000 euros in five years or 1500 eurosin six years. That is, the distance in time makes comparisons more difficult; itconfounds what would be easy choices if made between the same alternativesavailable earlier.

In this paper we propose a novel approach to modeling time preferenceswhich has this type of cognitive feature at its core. This approach allows usto explain some types of behaviour which are incompatible with the standardexponential discounting model (EDM), according to which an outcome x avail-able at time t is evaluated as δtu (x), with δ a constant discount factor and theinstantaneous utility of x. In the last decade evidence has accumulated whichhas contradicted the predictions of the EDM. In particular, it is clear that in-dividuals often exhibit the so-called preference reversal phenomenon, wherebyan initial preference between an earlier outcome and a later one is reversedonce both outcomes are delayed by the same amount of time. Recently, analternative model, which is consistent with this type of evidence, has enjoyedconsiderable prominence and success in the literature.

This is the model of hyperbolic discounting (HDM), in which the discountfactor is a hyperbolic function of time. The central point of this approachis that the rate of time preference between alternatives is not constant butvaries, and in particular decreases, with the passage of time. The simplestand most popular form of hyperbolic discounting occurs in a two-parametermodel which may be termed the β − δ model. In this model, the rate of timepreference between a present alternative and a future one is βδ, whereas therate of time preference between two future alternatives is δ. Though the HDMapproach is extremely fruitful in resolving some of the issues raised by theexperimantal evidence, we believe that a more fundamental assessment of thecognitive aspects of time preferences is needed. Our view is in line with an


observation formulated by Rubinstein (1998) who shows experimentally thatprecisely the same type of decision situations that may create a difficulty forthe EDM may also be problematic for the HDM. He argues that, in this sense,the change from a constant to hyperbolic functional form to discount futureutility is not radical enough.

We view the evidence against the EDM as indicating that human agents useheuristic procedures to make choices over time which differ fundamentally fromthe stark maximisation of a complete preference ordering which is commonto both EDM and HDM. In our theory we propose that behaviour can berationalised by a combination of three factors: (a) impatience; (b) ‘vagueness’in the perception of future alternatives; and (c) a suitable heuristics to make achoice when vagueness is ‘high’. We provide a parsimonious representation ofindividual preferences in which a measure of an individual’s vagueness is addedto the standard individual’s discount factor (which embodies the ‘pure timepreference’ component of utility) as a factor influencing individual choice. Inthis way we provide an explanation for the phenomenon of preference reversalwhose interpretation seems to us more adherent to the psychological basis for it.The model can also account for cyclical patterns of choice, which are obviouslyincompatible with the maximisation of a preference ordering. Moreover, ourframework can explain a phenomenon of time inconsistency which we dub the‘slippery slope’: an agent avoids an activity altogether for fear that engagingin it once will generate the temptation to indulge in it to an undesirable extent(e.g gambling, pregnacies, and so on). The slippery slope phenomenon hingesdirectly on a cyclical pattern of preference (as well as time-inconsistency).

Finally, we move from individual decisions to strategic behaviour. In partic-ular, we study the implications of our theory to bargaining behaviour. In analternating offer setting, we find support for equal division. Our theory is closein spirit to Rubinstein’s (2001) suggestion to extend his own similarity-basedapproach from risk to time preferences. He argues that, when evaluating al-ternatives, agents may perceive these alternatives to be ’similar’ in either thetime or the outcome dimension. This notion of similarity is closely related toour notion of vagueness. The crucial difference is that in our model vaguenessapplies to alternatives (outcome-dates pairs), not only to each date or outcomedimension separately. In our opinion, vagueness (or similarity) in one dimen-sion may not be defined independently of the other dimension. For example, 1Milion today may be clearly distinct from 1.01 Milion today; but from today’sperspective the two amounts in in thirty years time may well be consideredsimilar. This difference is important. Indeed, Rubinstein (1988) shows that hissimilarity approach generates a transitive preference relation on alternatives,and takes it as evidence that utility theory is hardly reconcilable with exper-imental data. On the contrary, as observed above, our theory, though utilitybased, can easiliy explain cyclical preferences.


83 Forward Induction and the Minimum Revision Principle

Andrés PereaDepartment of Quantitative Economics

Maastricht [email protected]

In this paper, we present a model of rationality in extensive form games inwhich players, during the course of the game, may revise their conjectures aboutthe opponents’ preferences, while imposing common belief of rationality at eachinformation set. Since players are assumed to be expected utility maximizers,their preferences, at each of their information sets, are given by a utility functionover the reachable terminal nodes and some subjective probability distributionover the opponents’ strategy choices compatible with this information set. Aplayer may therefore revise his conjecture about an opponent’s utility function,or his subjective probability distribution, or both, whenever this opponent haschosen a move which would be suboptimal given the previously conjecturedpreferences.

We refer to these two types of revisions as utility revisions and probabilityrevisions, respectively. A profile of conjectures about the opponents’ prefer-ences, as described above, is required to be self-enforcing in the sense that, ifthese conjectures were to be made public, no player, at any of his informationsets, would have an incentive to change his conjecture about the opponents’preferences. More precisely, if player i believes that player j, at informationset h, assigns positive probability to some player k strategy, then this strategyshould be optimal for player k, given player j’s conjecture at h about playerk’s preferences. We refer to this property as sequential rationality.

In addition, we impose a restriction on the revision of conjectures aboutthe opponents’ utility functions, called utility consistency, which states thata player, who at information set h decides to revise his conjecture about anopponent’s utility function, should do this in such a way that the opponent’sutilities following h remain unaltered. A last property we require is that, atthe beginning of the game, the players hold common conjectures about theother players’ utility functions and, for every player i, there exists a commoninitial conjecture about the initial subjective probability distribution attachedto player i’s strategy choice. A profile of conjectures satisfying the latter prop-erty, together with sequential rationality and utility consistency, shall be calleda preference conjecture equilibrium.

It is shown that, for every game tree and every possible profile u of utilityfunctions at the terminal nodes, there exists a preference conjecture equilib-rium in which the players’ initial common conjectures about the players’ utility


functions coincide with u. This result thus establishes that, within this frame-work, common belief of rationality at every information set is always possiblefor every given profile of initial common conjectures about the players’ util-ity functions. However, there may exist preference conjecture equilibria thatcould be rejected on the ground that they contain “too many” utility and/orprobability revisions. Intuitively, one would wish for a theory of rationality inwhich the number of preference conjecture revisions, needed to maintain com-mon belief of rationality throughout the game, is minimal. We will refer to thisproperty as the minimum revision principle.

As to the question how to measure preference conjecture revisions, we adoptthe view that utility revisions should have a higher weight than probability re-visions. The reason is that a player’s utility function is, in some sense, a more“robust” characteristic for this player than his subjective probability distribu-tion about the opponents’ strategies since the latter depends crucially uponhis perception of the other players’ preferences, whereas the former does not.Formalizing this point of view leads to the concept of minimum revision equi-librium, selecting those preference conjecture equilibria for which the set ofutility revisions is minimal, and for which there is no other equilibrium withthe same set of utility revisions but fewer probability revisions. The forwardinduction flavour of minimum revision equilibrium is being illustrated by itsperformance in several examples to be discussed in this paper. In particular,we prove that the concept of minimum revision equilibrium uniquely selectsthe forward induction outcome in every outside option game, as defined in vanDamme (1989). We show that in perfect information games with generic utilityfunctions u, every preference conjecture equilibrium for which the initial com-mon conjectures about the players’ utility functions coincide with u, alwayssupports the backward induction procedure in this game.


Strategic Equilibrium

84 Cooperation with Strategy-Dependent Uncertainty Attitude

Nicola DimitriUniversità di Siena, Italy

[email protected]

The paper shows that in a one shot Prisoner’s Dilemma Knightian uncer-tainty, as formalised by multiple priors, may entail cooperation at a GeneralisedNash Equlibrium. The main idea is that players may have an attitude towardsuncertainty that depends upon their available strategies. In particular, if play-ers anticipate to be sufficiently more optimistic when choosing to cooperate,than when defecting, then they may indeed cooperate. Though uncommon ineconomic modelling, choice (action, in general state) -dependent uncertaintyattitude formalises a behaviour which could instead be well understood andwidely accepted by cognitive psychologists, within the theory of Cognitive Dis-sonance. An example may clarify this notion. Suppose a person thinks a jobto be particulalry uninteresting (this is her prior). Then, imagine that cir-cumstances are such that she has to take that job. In taking it she would,ex-post, face a “dissonance” with her prior beliefs and this would produce anunpleasant state of internal tension. Such tension may be resolved by cognitivereadjustment, namely ex-post changes in beliefs. More explicitly, the personinterviewed after having been engaged for some time on that job, may now saythat the job is not so bad after all. In the Prisoner’s Dilemma this could offera possible explanation to justify a strategy-dependent uncertainty attitude.

References

N. Dimitri (1995) Randomization and Cooperation with Knightian Uncer-tainty, Quaderni del Dipartimento di Economia Politica 191, Universitàdi Siena.

N. Dimitri (1999) Independence Between Events with Non-Additive Probabil-ities, in Uncertain Decisions, Bridging Theory and Experiments, L. Luini(Ed.), Kluwer Academic Publishers.

J. Dow and G. Werlang (1994) Nash Equilibrium under Knightian Uncer-tainty; Breaking Down Backward Induction, Journal of Economic Theory64, 197—204.

J. Eichberger and D. Kelsey (1999) Non-Additive Beliefs and Strategic Equi-libria, Games and Economic Behavior 30, 183—215.


P. Feitinger (1957) A Theory of Cognitive Dissonance, Stanford UniversityPress, Stanford.

I. Gilboa and D. Schmeidler (1989) Maxmin Expected Utility with Non-UniquePrior, Journal of Mathematical Economics 18, 261—291.


85 Nash equilibria in normal form games

Hannu SalonenUniversity of Turku, 20014 Turku, Finland

[email protected]

We show that there exists a Nash equilibrium in mixed strategies in allnormal form games such that pure strategy sets are nonempty compact metricspaces and utility functions are continuous. The player set can be arbitrarynonempty set. The utility functions of the players are defined on the set ofpure strategy profiles, the product space of individul players’ pure strategy sets.This product space is equipped with the product topology and the productsigma-algebra, so the product space is a compact Hausdorff space, and thecontinuous utility functions are also measurable. Continuity and compactnessensure that each utility function depends only on countably many coordinates.This suggests that the existence of a Nash equilibrium in the general casefollows actually from the existence of equilibrium in games with countably manyplayers. This indeed is the case. In the proof of the main result the player setis equipped with a well-ordering, so the Axiom of Choice is assumed. However,both the Tychonoff theorem and the product measure existence theorem (or theKolmogorov’s existence theorem) already depend on the Axiom of Choice. Theformer ensures that the set of strategy profiles is a nonempty compact space,and the latter ensures that players can “calculate” expected utilities associatedwith mixed strategy profiles. Therefore we need the Axiom of Choice (at leastimplicitly) already if we want to talk about non-trivial normal form games witharbitrary player sets.


Values of Cooperative Games

86 Potential, Value and Probability

Federico Valenciano and Annick LaruelleUniversidad del País Vasco y Universidad de Alicante

[email protected]

From Shapley’s (1953) seminal paper a copious family of “solutions” hasgrown in different directions with many ramifications1 : nonsymmetric values,semivalues, weak semivalues, weighted weak semivalues, probabilistic values,least square values, coalitional values, coalitional semivalues, etc., as well asNTU or non atomic extensions of some of these notions, or their restriction tospecial subdomains as, e.g., simple games. In fact, a lot of energy in cooperativegame theory has been and is still devoted to the study of these ramifications,and the search of new “axiomatic” characterizations of these objects, be itthe very seminal concept or any of its extensions around. A weak point in theliterature concerned with these extensions is the lack of a clear motivating goal,apart from the purely mathematical motivation (dropping axioms and seeingwhat happens, finding weaker or more appealing ones, etc.). And as a rule the“stories” behind these notions are not very compelling to tell the truth.

One of the stories that are told in connection with most of these notions isone in probabilistic terms. Some of these notions are covered by Weber’s notionof “group value”. Nevertheless this covering requires a different probabilisticmodel for every case: the formation of the grand coalition in a certain order,all orders being or not equally probable; a vector of probabilistic assessmentsof the players of the coalition every player will join with different “consistency”requirements, etc.

In this paper we focus on the probabilistic point of view and propose anextremely simple probabilistic model that provides a single and simple storyto account for some of these “solutions”. Namely, it gives a clear probabilis-tic meaning to certain extensions of the Shapley value, as semivalues, weaksemivalues, weighted weak semivalues and random-order values, which appearas particular cases of this single general model, as of course the Shapley valueitself. Moreover, some of the most interesting conditions or notions that havebeen introduced in the search of alternatives to Shapley’s seminal characteri-zation, as “strong monotonicity”, “transfer”, “balanced contributions” and the

1 In order to avoid an unreadable abstract crowded with references, we omit the too many

that we should add in connection with all these extensions. We do the same when referring

later to some of the best-known alternative characterizations of the Shapley value. We

apologize for so many obvious omissions.


“potential”, can be reinterpreted from this same point of view. In this newlight some of these notions acquire a clear, simple and compelling meaning.In particular, the notions of potential and balanced contributions lose their“mystery” and admit a transparent interpretation. Surprisingly enough, ac-cording to this interpretation the “weighted” variants of both notions result tobe the “natural” ones in a precise sense (in fact “weights” evaporate as such,becoming an “optical effect”), while the non-weighted ones turn out somehowad hoc adaptations of these more general and natural conditions. Similarly,some of the characterizing results in the literature involving the above men-tioned extensions become straightforward from the point of view provided byour probabilistic model.


87 The Hamiache value for union stable cooperation structures

E. Algaba, J. M. Bilbao, N. Jiménez and J. J. LópezUniversidad de Sevilla

[email protected]

The most extensively studied solution concepts in communication situationshave been the Myerson value (1977) and the position value (see Borm, Owen,and Tijs, 1992). These concepts have been generalized to union stable cooper-ation structures by Algaba et al. (2000, 2001). Hamiache (1999) presents anew solution concept for the case in which the communication relations amongplayers are modelled by means of an undirected graph. In this work, we studythe generalization of this value in other models of partial cooperation where thecommunication links may be represented by hypergraphs.

A union stable system is a pair (N,F) with F ⊆ 2N verifying that for allS, T ∈ F , with S ∩ T = ∅, it is satisfied that S ∪ T ∈ F . If (N,F) is a unionstable system, the set B (F) = F \ D (F) where

D (F) = G ∈ F : G = A ∪B, A,B ∈ F , A = G, B = G, A ∩B = ∅.

is called basis of F and the elements of B (F) are called supports of F .Throughout this work, we assume that the union stable systems (N,F)

satisfy the following properties: a) i ∈ F for all i ∈ N, b) if S ∈ F and thereexists B ∈ B (F) such that S ∩ B = ∅ and S ∪ B = N, then there exists noF ∈ F so that S ⊂ F ⊂ N. Let (N,F) be a set system and let S ⊆ N. Thecoalition T ⊆ S is a F-component of S if T ∈ F and there exists no T ′ ∈ Fsuch that T ⊂ T ′ ⊆ S. For all S ⊆ N we denote by CF (S) the set of the F-components of S. Also, we denote by FS = F ∈ F : F ⊆ S . A union stablecooperation structure is a triple (N, v,F) where (N, v) is a cooperative gameand (N,F) is a union stable system. The set of all union stable cooperationstructures with players set N will be denoted by USN .

Definition 1. Let (N, v,F) be a union stable cooperation structure. The F-restricted game is the triple

(N, vF ,F

)where vF : 2N → R is given, for all

S ⊆ N, by

vF(S) =

∑T∈CF(S) v(T ) if CF (S) = ∅,

0 otherwise.

We denote by RUSN =(

N, vF ,F): (N, v,F) ∈ USN

. López (1996) showed

that the collection of unanimity games uR : R = ∅, R ∈ F is a basis of thevector space

vF : (N, v,F) ∈ USN

.As consequence, vF =

∑R∈F:R =∅ cRuR

and the coefficients cR can be calculated using the Möbius function. For every


nonempty coalition S ∈ F , we have v (S) =∑

R∈F:R⊆S cR and taking intoaccount that (F ,⊆) is a finite partially ordered set, we can apply the Möbiusinversion formula and we obtain cR =

∑S∈F:S⊆R µ (S,R) v (S) .

Definition 2. A solution on USN is a function ψ which associates to eachunion stable cooperation structure (N, v,F) a vector ψ (N, v,F) ∈ R

N wherethe coordinate ψi (N, v,F) represents the payoff to player i.

Let (N, v,F) be a union stable cooperation structure and S ∈ F . We con-sider the set

S∗ = i ∈ N : there exists B ∈ B (F) with i ∈ B and B ∩ S = ∅ .

Note that S∗ =⋃B ∈ B (F) : B ∩ S = ∅ , and hence S ⊆ S∗ ∈ F . Given

a solution φ on USN and (N, v,F) ∈ USN , we define its associated game(N, v∗

φ,F

)by

v∗φ (S) =

v (S) +∑

j∈S∗\S

∑B∈B(j,S) φj (S ∪B, vS∪B,FS∪B)− v (j)

|B (j, S)|if S ∈ F ,

∑T∈CF (S) v

∗φ(T ) if S /∈ F ,

where B (j, S) = B ∈ B (F) : j ∈ B and B ∩ S = ∅ . In the following we de-note the associated game by (N, v∗,F) and the subgames by (S, v,Fs) . Wenow consider a reasonable set of axioms.

Axiom 1. (Component-independence) For all (N, v,F) ∈ USN and i ∈ N ,φi (N, v,F) = φi (M,v,FM) , with M ∈ CF (N) such that i ∈ M.

Axiom 2. (Component-efficiency) For all (N, v,F) ∈ USN and M ∈ CF (N) ,∑k∈M φk (N, v,F) = v (M) .

Axiom 3. (Linearity with respect to the game) For all α, β ∈ R and (N, v,F) ,(N,w,F) ∈ USN , φ (N,αv + βw,F) = αφ (N, v,F) + βφ (N,w,F) .

Axiom 4. (Independence of irrelevant players) If (N,F) is a union stable sys-tem and R, T ∈ F with R ⊆ T, then φi (N,uR,F) = φi (T, uR,FT ) for alli ∈ T.

Axiom 5. (Positivity) For all R, T ∈ F with R ⊆ T, we have φi (T, uR,FT ) ≥0 for i ∈ T and φi (R, uR,FR) > 0 for i ∈ R.

Axiom 6. (Consistency) For (N, v,F) ∈ USN , φ (N, v,F) = φ(N, v∗

φ,F

).


Lemma 1. If φ is a solution on USN which satisfies Axioms A1− A6, then,for all feasible coalitions R ∈ F ,

φi (N,uR,F) =

φi (R,uR,FR) if i ∈ R,0 if i /∈ R.

By Axioms 1,2,3 and Lemma 1 we can assume, whithout loss of generality,that N ∈ F .

Corollary 2. If φ is a solution on USN which satisfies Axioms A1−A6, then

φi(N, vF ,F

)=

∑S∈F

∑R∈F:R⊇S,i∈R

µ (S,R)φi (R,uR,FR)

v (S) ,

for all i ∈ N .

Theorem 3. There is only one solution φ on RUSN which satisfies AxiomsA1−A6.

Theorem 4. If φ is a solution on USN satisfying Axioms A1−A6, then

φ (N, v,F) = φ(N, vF ,F

).

References

Algaba, E., Bilbao, J. M., Borm, P., López, J. J. (2000) The position valuefor union stable systems, Mathematical Methods of Operations Research52, 221—236.

Algaba, E., Bilbao, J. M., Borm, P., López, J. J. (2001) The Myerson valuefor union stable systems, Mathematical Methods of Operations Research54, 359—371.

Borm, P., Owen, G., Tijs, S. H. (1992) On the position value for communica-tion situations, SIAM J. Discrete Math. 5, 305—320.

Hamiache, G. (1999) A Value with Incomplete Communication, Games andEconomic Behavior 26, 565—78.

López, J. J. (1996) Cooperación parcial en juegos de n-personas, Tesis Doc-toral. Universidad de Sevilla.

Myerson, R. B. (1977) Graphs and cooperation in games, Mathematical ofOperations Researchs 2, 225—229.


88 Axiomatizations of the Shapley value for cooperative games onantimatroids

R. van den Brink, E. Algaba, J. M. Bilbao, and A. Jiménez-LosadaFree University, De Boelelaan 1105, 5000 LE Amsterdam, The Netherlands

[email protected]

A cooperative game describes a situation in which a finite set of playersN can generate certain payoffs by cooperation. In a cooperative game theplayers are assumed to be socially identical in the sense that all subsets ofN are feasible coalitions. However, in practice there exist social asymmetriesamong the players and there are restrictions on coalition formation. For thisreason, the game theoretic analysis of decision processes in which one imposesasymmetric constraints on the behavior of the players has been and continues tobe an important subject to study. Important consequences have been obtainedof adopting this type of restrictions on economic behavior. As examples wemention the games with a priori coalition structures, and games with a limitedcommunication graph.

In this paper we study cooperative games which restricted set of feasiblecoalitions is an antimatroid. A set of feasible coalitions A is an antimatroid onN if it is a set of subsets of N satisfying the following three properties:

1. the empty set is feasible,

2. A is accessible, meaning that for every nonempty S ∈ A there is at leastone player i ∈ S such that S \ i ∈ A,

3. A is closed under union.

In an earlier paper we showed that cooperative games on antimatroids gener-alize both the conjunctive and disjunctive approach to cooperative games witha permission structure. In these games coalition formation is restricted becausethere are players that need permission from so-called predecessors before theyare allowed to cooperate. A coalition is conjunctive feasible if for each playerin the coalition all its predecessors belong to the coalition. A coalition is dis-junctive feasible if for each player in the coalition (that has predecessors in thepermission structure) at least one of its predecessors belongs to the coalition.

In this paper we provide an axiomatic characterization of the restricted Shap-ley value for games on antimatroids. This characterization generalizes the ax-iomatizations of both the conjunctive and disjunctive Shapley permission valuesfor games with a permission structure. In particular, we unify both disjunctiveand conjunctive fairness (meaning that deleting a permission relation betweena predecessor and successor in a permission structure has the same effect on


the payoffs of these two players, respectively, on the payoffs of the successorand each of his other predecessors.) After this we give special attention tothe so-called poset antimatroids. These are antimatroids that are closed underintersection. These are exactly those antimatroids that can be the set of con-junctive feasible coalitions in a permission structure. We show that deletingthe fairness axiom from the set of axioms mentioned before characterizes therestricted Shapley value for the class of cooperative games on poset antima-troids. Moreover, it turns out that the class of games on poset antimatroids ischaracterized as that class of games on which the restricted Shapley value isthe unique solution satisfying these axioms.


Sequencing Games

89 On the Balancedness of Semi-infinite Sequencing Games

Vito FragnelliDipartimento di Scienze e Tecnologie Avanzate


In these last years several classes of TU games were extended to the semi-infinite case. Preserving a finite number of players and bounded values forthe characteristic function other elements were supposed to have a countablecardinality. The aim of this paper is to study the balancedness in the semi-infinite case of the class of sequencing games, introduced by Curiel, Pederzoli,Tijs (1989), considering a countable number of jobs and a finite number ofplayers. Curiel, Pederzoli and Tijs show that sequencing games are convex and,consequently, balanced; moreover it is possible to determine a core allocationwithout computing the characteristic function of the game. They propose toshare equally between two players the gain produced by their switch and callthis rule the Equal Gain Splitting Rule.

We redefine some elements of a sequencing situation in order to include thecase in which each agent may own more jobs. We suppose that if an agentowns several jobs, he pays the cost of each one until it is completed. Given amultiple-job sequencing situation it is easy to define a multiple-job sequencinggame, that results to be convex. Finally we add the condition that the setof jobs has a countable cardinality. When this happens it is possible that thecost of an order and consequently the values of the characteristic function, isunbounded. In order to avoid this situation we introduce suitable hypotheseson the total service time and on the costs of the jobs. We prove that in thesehypotheses the corresponding semi-infinite sequencing game is balanced.

Finally we propose some possible future developments modifying the charac-teristic function for the multiple-job case and considering an infinite sequenceof jobs in which only a finite number of them can be exchanged.

Main References

Curiel, I., Pederzoli, G., Tijs, S. (1989) Sequencing Games, European Journalof Operational Research 40, 344—351.

Fragnelli, V., Patrone, F., Sideri, E., Tijs, S. (1999) Balanced Games Arisingfrom Infinite Linear Models, Mathematical Methods of Operations Re-search 50, 385—397.


90 On Games corresponding to Sequencing Situations with Prece-dence Relations

Flip Klijn, Herbert Hamers and Bas van VelzenUniversitat Autònoma de Barcelona and Tilburg University

[email protected]

Sequencing situations are characterized by a finite number of jobs lined up infront of one (or more) machine(s) that have to be processed on the machine(s).Because every time unit a job is in the system will incur costs, a single deci-sion maker wants to determine a processing order of the jobs that minimizestotal costs (or maximizes cost savings). This single decision maker problemcan be transformed into a multiple decision makers problem by taking agentsinto account that own (at least) one job. In such a model a group of agents(coalition) can save costs by cooperation. For the determination of the maximalcost savings of a coalition one can use techniques from linear and combinatorialoptimization theory.

A subsequent question that arises is how to allocate the revenues among theparticipants in the project. This second problem is an issue that is addressedin cooperative game theory. One first constructs a sequencing game, which isa cooperative game that associates to every coalition the cost savings of thatcoalition if its members (players) decide to cooperate. Next, one can studythe properties of the sequencing game to determine ‘fair’ allocation(s) of therevenues.

Two important properties that a sequencing game may satisfy are balanced-ness and convexity. A game is balanced if there is a core allocation, i.e., a wayto distribute the revenues of the grand coalition among the players in such waythat no subset of players can be better off by seceding from the rest of theplayers and act on their own behalf. The convexity condition expresses thatthe incentives of an arbitrary player for joining a certain coalition increase asthe coalition grows. Convex games are known to have nice properties, in thesense that some solutions concepts for these games coincide and other haveintuitive descriptions. For example, for convex games the core (i.e., the setof all core allocations) is equal to the convex hull of all marginal vectors (cf.Shapley (1971) and Ichiishi (1981)), and as a consequence, the Shapley (1953)value is the barycenter of the core (cf. Shapley (1971)). Moreover, the bargain-ing set coincides with the core and the kernel coincides with the nucleolus (cf.Maschler et al. (1972)).

Sequencing games were introduced in Curiel et al. (1989). They consideredthe class of one-machine sequencing situations in which no restrictions like duedates and ready times are imposed on the jobs and the weighted completion


time criterion was chosen as the cost criterion. It was shown that the corre-sponding sequencing games are convex and hence balanced.

Hamers et al. (1999) and Calleja et al. (2002) extended the number of ma-chines. Hamers et al. (1999) considered sequencing situations with parallel andidentical machines and the weighted completion time criterion. They provedbalancedness for the case of two machines. They also showed balancednessfor special classes with more than two machines. Calleja et al. (2002) estab-lished balancedness for a special class of sequencing games that arise from twomachine sequencing situations with a maximal weighted cost criterion.

Instead of considering multiple machines, Hamers et al. (1995) and Bormet al. (2002) extended the class of the one-machine sequencing situations con-sidered by Curiel et al. (1989) by imposing ready times and due dates on thejobs. In this case the corresponding sequencing games are balanced, but are notnecessarily convex. For special subclasses of these sequencing games, however,convexity could be established.

In this paper we continue the latter line of research by imposing precedencerelations on the jobs. Precedence relations arise when certain job(s) have topreceed some other job(s). In this regard, one can think for example of theproduction, assembly, and testing of different parts of a consumer product.

Our main result is that a precedence game is convex if precedence relationsconsist of chains. This means that the set of jobs can be partitioned into groupssuch that: 1) there are no precedence relations between two jobs from differentgroups, and 2) the precedence relation within each group can be described bya complete and transitive ordering.

We also provide examples in which we show that the convexity result cannotbe extended to several larger classes of precedence games, e.g., the class inducedby tree precedence relations.


91 Sequencing games without initial order

Estela Sánchez and Flip KlijnUniversidad de [email protected]

In this paper a sequencing situation consists on a finite number of agents,who have one job to be processed on a single machine. Before the processingof the machine starts either agents know their positions in the queue, i.e. theinitial order, or the order can be partially known, or completely unspecified.

If an initial order has been established, the initial rights of the jobs are de-termined. Under this assumption, Curiel, Pederzoli, and Tijs (1989) associatedto each sequencing situation a cooperative TU game that reflects the savingsof each coalition of jobs if they reorder their positions in an optimal way. Thevalue of a coalition is defined as the maximal cost savings that a coalition canmake by admissible rearrangements. Furthermore, Curiel et al. (1989) presenta single-valued solution, called the equal gain splitting rule that always is inthe core of the sequencing game. Besides, it was also shown that these gamesare convex and, hence, balanced.

What we study in this paper are sequencing games associated to sequencingsituations in which there is uncertainty on the order of arrival to the queue, orsimply, that information is not relevant, and therefore, it will not be taken intoaccount in the model. There are several ways to deal with uncertain sequencingsituations: one can either have information on the probability distribution ofthe initial orders or have not information at all. A solution for the first casewas pointed out in Hamers and Slikker (1995).

We present different classes of games attending to the information thatagents have about the initial order: the probabilistic cost game, the pessimisticcost game and the tail cost game, providing relations among them and describ-ing their main properties, with special attention to the core.


Multicriteria Problems

92 Multicriteria Linear Bargaining Games

Luisa Monroy, Miguel A. Hinojosa and Amparo M. MármolUniversidad de Sevilla

[email protected]

Bargaining games are inherently multicriteria problems. This multicriterianature is even more evident if each agent has to take into account several criteriawhen deciding which of the feasible solutions it is convenient to agree on. In thispaper we consider the class of multicriteria linear bargaining games. In thesegames the payoff set is defined by linear constraints, and the coordinates ofthe joint payoff space are split in different subsets corresponding to the criteriaof the agents. These linear bargaining models are important because they canrepresent directly a wide range of applications.

In general, it is difficult to deal with multicriteria bargaining games dueto the double multidimensionality involved. Nevertheless, in this paper wewill exploit the special properties of the problem derived from the linearity inorder to establish procedures to obtain, both the disagreement points and thesolutions for this class of bargaining problems. We illustrate the ideas withthe analysis of a finite n-person game with multiple payoffs as a multicriteriabargaining game.


93 Solutions for Multiple Scenario Cooperative Games

Miguel A. Hinojosa and Amparo M. MármolUniversidad Pablo de Olavide (Sevilla)

[email protected]

Multiple scenario cooperative games is a particular class of cooperative n-person vector valued transferable utility games, where a single resource is allo-cated but the strength of the different coalitions is valued in different scenariossimultaneously or under different states of nature.

These games differ from conventional cooperative games in the dimension ofthe coalition values, and they also differ from multicommodity games, in thata single good has to be allocated among the players.

The objective of this paper is to propose and analyze different solution con-cepts for the class of multiple scenario cooperative games from a multicriteriaperspective. We deal with the vector valued characteristic function keeping itsmultidimensional nature and propose solution concepts for these games thatextend the existing notions for conventional cooperative games. Moreover andmost important, these solutions can be calculated solving multicriteria linearproblems for which the appropriate software is available.


94 Pareto Optimality in Multicriteria Bargaining

Amparo M. Mármol, Luisa Monroy and Victoriana RubialesUniversidad de Sevilla

[email protected]

A multicriteria bargaining problem is a generalization of the classical bar-gaining problem where each player has a set of criteria to value any decision.To analyze these situations we have to jointly consider two decision problems:one related to the preferences of the players with respect to their own criteria,and the other problem of selecting a solution that could be accepted by all therational players.

In the paper we assume that the utility function of each of the players isnot known explicitly. As a first step to find a solution, we have to establishthe minimal requirements for the outcomes that the rational players will bewilling to accept. In other words, we have to extend the Pareto-optimalityor nondominance condition inherent in the axiomatic bargaining solutions, tothe case of multicriteria bargaining. Depending on the concept of dominanceconsidered in both, the set of players and the sets of criteria of each player, wepresent a range of possibilities for the generalized Pareto-optimality axiom andexplore the relationship among them.

In this sense, we provide a general framework where we locate the existingsolution concepts for multicriteria bargaining problems and that permits us todefine other concepts based on different dominance structures.


Environmental and Experimental Games

95 Application of Game Theory in a Groundwater Conflict inMexico

Raquel Salazar and Abraham RojanoUniversidad Autónoma Chapingo, Mexico

[email protected]

Game theory is applied to groundwater management conflict in the Alto RioLerma Irrigation District located in the state of Guanajuato in Mexico. Themajor agricultural constraint in this irrigation district is water. From Octoberto June there is a deficit of rainfall with respect to evapotranspiration. Theshort period of rainfall is reflected in high groundwater withdrawals from wells,generating both aquifer overexploitation (overexploitation coefficients between1.2− 1.4) and the consequent subsidence (the most affected places have 8 cmper year in the average). Besides, the current management practices in theirrigation district, like high fertilizers and pesticides application together withthe irrigation management, produces environmental problems: nitrates andpesticides percolation that may cause aquifer contamination. On the otherhand, farmers need to maximize their net benefit increasing yields. In summary,the problems in the ARLID may be classified as economic and environmental.

The goal is to find the best compromise or satisfying solution, the increasein economical benefits reducing the negative environmental effects and at thesame time using the water in the most efficient way. Because these are con-flicting objectives, improvement in one objective could imply the worseningin the other the application of game theory was necessary. The players arefarmers in the irrigation district and the community. The economic attributesare obtained by using linear programming to identify optimal cropping pat-terns. The Groundwater Loading Effects of Agricultural Management Systems(GLEAMS) was used to simulate the amounts of nitrates, water and pesticidesin runoff and percolation which account for some environmental attributes. Theaquifer overexploitation coefficient and pumping cost were considered for eachgroundwater withdrawal scenario. The objective is to maximize economic ben-efits but at the same time minimize the negative environmental consequences inthe region. The decision variable is the groundwater withdrawals. The paretofrontier was generated representing the trade off between economic benefits andnegative environmental effects.

The game theory methods applied were: The Non-symetric Nash solution,Non-symetric Kalai- Smorodinsky solution, Non-symetric area monotonic solu-tion and Non-symetric equal loss solution. The application of different methods


based on different notions of fairness and compromise allow the decision makerto decide between different options depending on the importance weights as-signed to environment. When environment and economics have equal impor-tance, the decision is to withdraw between 360 to 370 million of cubic meterswhich correspond to the seven groundwater extraction scenario.


96 Professionals Play Minimax

Ignacio Palacios-HuertaDepartment of Economics, Brown University

64 Waterman Street, Providence, RI 02912, [email protected]

During the last few decades game theory has contributed to a reshaping ofimportant aspects of the methodology of Economics and other social sciences.In large part this is because the language, concepts, and techniques of non-cooperative game theory have become central to many areas of the discipline.Undoubtedly, studying the interaction of ideally rational players greatly aidsour understanding of the behavior of real individuals, firms and other agents.Moreover, as Kreps (1991) remarks, "studying the former should aim at under-standing the latter. The point of game theory is to help economists understandand predict what will happen in economic, social and political contexts."

Theoretical contributions should thus feed back to empirical analysis. How-ever, testing the implications of the theory has proven extremely difficult inthe literature. The primary reason is that many predictions often hinge onproperties of the utility functions and the values of the rewards used. Evenwhen predictions are invariant over classes of preferences, data on rewards areseldom available in natural settings. Moreover, there is often great difficulty indetermining strategy sets and in measuring individuals’ choices, effort levels,and the incentive structures they face. As a result, even the most fundamentalpredictions of game-theoretical models have not been supported empirically inreal situations yet.

In view of the substantial problems with testing theoretical predictions usingnatural data, many authors have been compelled to test them in experimentalsettings. Interestingly enough, despite the controlled structure of experiments,the results of many experiments during the last few decades typically reject theassumption that subjects are playing according to the theoretical implicationsof equilibrium play. For instance, a number of experiments have evaluatedthe empirical validity of von Neumann’s Minimax Theory for two-person zero-sum games, a theory that occupies a central position in our understanding ofstrategic situations. The results have been mixed and often rather negative(see, for instance, Brown and Rosenthal (1990), Rapoport and Boebel (1992),Camerer (2001), and the references therein).

It is rightly argued, however, that in experimental settings individuals areoften exposed to games and situations that they have not faced previouslyand that, despite their usual simplicity, it may not be possible for them to


become very proficient in the limited time frame of an experiment. This inturn has contributed to generating an important literature dealing with out-of-equilibrium play, or "learning," in experimental games (See Camerer andHo (1999), Erev and Roth (1998), Stahl (2000), and many references therein.)Consequently, the state of affairs is such that even though the ability of gametheoretical models to explain the diversity of individual behavior in socioeco-nomic environments has significant theoretical and practical implications, themajor equilibrium predictions of the theory have received little or no empiricalsupport.

This paper offers a test of some of the main predictions of the theory ofstrategic interactions. A fundamental concept in many strategic situations isthat one’s actions not be predictable by one’s opponent. The theory of mixedstrategy play, including von Neumann’s Minimax Theorem and the general no-tion of a Nash equilibrium in mixed strategies, remains the essential basis of ourtheoretical understanding of strategic situations that require unpredictability(Osborne and Rubinstein (1994) discuss a number of interpretations of mixedstrategy equilibrium). In this paper we use data from a natural strategic playin professional sports to provide an empirical test of the theory of mixed strat-egy equilibrium, an approach that has been recently followed by Walker andWooders (2000). Moreover, the specific features of the play and its environ-ment allow the analysis to overcome the usual difficulties that have plaguedmost empirical attempts in the literature, both in natural and experimentalsettings.

In professional sports participants are experts at their games. This clearlyposes a notable advantage over many experiments, as the individuals are alreadythe most proficient agents of the situations of the game. However, despite thefact that situations that require unpredictability play and mutual outguessingare pervasive in sports, it is normally not feasible to conduct accurate empiricaltests of the Minimax Hypothesis in these scenarios. The reason is that play-ers generally have many strategies available and outcomes are numerous. Inaddition, outcomes are generally not decided immediately after players choosetheir initial actions. There are subsequent strategic choices that usually play acrucial role in determining final outcomes. As a result, it is essentially impos-sible to obtain detailed data on all relevant variables to conduct informativestatistical tests.

All of these usual difficulties and drawbacks are overcome in the naturalsetting we examine in this paper. We focus on a specific one-shot two-personzero-sum game in professional sports that requires unpredictability and mutualoutguessing. As in the typical experimental setting, the game has a preciselydefined set of rules, few strategies are available, outcomes are decided immedi-ately after strategies are chosen, and all relevant information is observable. Incontrast with most experiments, however, professional players are highly mo-


tivated and experts at the game. The analysis is concerned with a play of theworld’s most popular game: penalty kicks in soccer. To test the implications ofthe Minimax Theorem, we exploit a unique data set of more than 1,400 penaltykicks in professional soccer games that includes very detailed information onall relevant aspects of the play, especially strategies and outcomes.

As a brief summary of the main results, we find that, as predicted by thetheory of mixed strategy play: (i) winning probabilities are statistically iden-tical across strategies, and (ii) players’ strategy choices are independent drawsfrom a random process.

The first result has been notably difficult to obtain in the literature. As tothe second, essentially all previous tests of randomness in experimental researchin psychology and economics have found that individuals "switch strategies"too often to be consistent with random play. When individuals are asked togenerate or identify random sequences these sequences often have negative au-tocorrelations (see Bar-Hillel and Wagenaar (1991) and Camerer (1995) forreviews of the literature). In our tests of the Minimax hypothesis, professionalplayers are found to be capable of behaving perfectly randomly. Their se-quences neither exhibit negative or positive autocorrelation, and choices do notdepend on one’s own previous play, on the opponent’s previous plays or on pastoutcomes.

We take these two results as consistent with the implications of the MinimaxTheorem. In this sense, they represent the first time that the fundamentalnotion of Nash equilibrium in mixed strategies is supported with real data.

Paper can be downloaded from:

http://www.econ.brown.edu/∼iph/

http://www.econ.brown.edu/~iph/


97 Game Theory Applied to Water Problems

Abraham Rojano and Raquel SalazarUniversidad Autónoma Chapingo, Mexico

[email protected]

Water use in current time brings about the following problem. Urban waterdemand is growing up while rainfall keeps somehow constant. Agriculture areadoes not increase and water savings are feasible with better irrigation tech-niques. Recent experiences in California Valley and Chinese regions give someidea how to deal with these scenarios before the problem exploits by itself. Theproblem requires to construct rational solutions. From a set of solutions weshould choose the best one in order to satisfy given assumptions. Here, it isthe point, because the answers depend on the relevant information possessedor collected either. Utilization of total information is not given to anyone andpolitical decisions should be supported on logical basis. Thus, the goal of thispaper is to establish the optimal solution working on the mathematical form inwater problems searching the demands and supplies.

The demand comes from the urban areas and the supply from the agriculture.Demand from surface water and supply from groundwater. The problem couldgo beyond these simple scenarios, for instance when water quality is includedor groundwater depends on the irrigation techniques.


Semivalues and Power Indices

98 Commutation Games and Multilinear Extension

José Miguel Giménez and Rafael AmerUniversitat Politècnica de Catalunya

[email protected]

The semivalues, introduced by Dubey, Neyman and Weber in 1981, form anample family of solutions for cooperative games with transferable utility andinclude, among others, the solutions of Shapley and Banzhaf.

For one of these solutions, it can happen that different games obtain a samepayoff vector. We say that two games are inseparable by semivalues if bothgames obtain the same payoff vector for any semivalue that is considered.

The linearity of the semivalues allows to reduce the problem of the separationto the null game and to consider the games simpler than they are inseparablefrom the null game: the commutation games. It has been proven that allinseparable game from the null game is a linear combination of commutationgames, which allows to know the dimension of this vector subspace for eachspace of cooperative games.

For each semivalue, a modification for games with structure of coalition canbe considered using at level of the modified quotient and within each coalitionblock the induced semivalue, in analogous way to as Owen in 1977 introducesthe coalitional value from the Shapley value or in 1981 the modified Banzhafvalue from the Banzhaf solution.

The multilinear extension of the commutation games has a specially simpleexpression; as much it is so it can be verified that, for games with four players,the concepts of separable by semivalues and separable by semivalues modifiedfor games with coalition structure are coincident. It does not happen the samefor games with five or more players, where the commutation games are nowseparable from the null game by modified semivalues.

But, the definition of commutation game can be adapted and thus we obtainthe definition of expanded games for commutation games with four players.This expanded game fulfils the following property: its multilinear extensionagrees with the multilinear extension of the commutation game with four play-ers in a space of only four players. This way is able to obtain a basis of thesubspace of inseparable games from the null game by modified semivalues.

The incorporation of the coalition structures is able to reduce of significantway the dimension of the subspace of inseparable games from the null gamewhen the cardinality of the set of players is equal or greater than five.


99 Several ways of taking decisions by voting

Antonio Magaña and Francesc CarrerasUniversitat Politècnica de Catalunya

[email protected]

Usually, organisms that take decisions by voting are exclusively modeledin terms of simple games. Then, to measure the “power" of each player inthe game/organism, some index of power (as the Shapley—Shubik or Banzhafindices) is used.

The problem is that the same organism does not always use the same proce-dure to take decisions and, so, simple games (and the power indices related tothem) do not describe accurately enough the position of voters in the organism.Take, for instance, a Parliament: daily affairs are commonly decided by relativemajority.

In this work we study and compare different rules used in practice whenvoting. To make an appropriate description of relative majority rules, we usesimple games with alternatives and the extension of the Shapley—Shubik andBanzhaf indices to this context. We apply these rules to an example: the PaísVasco Parliament during the 2001—2005 Legislature.

The different voting procedures that we analyze are:

(a) Voting on a single motion when approval requires absolute majority.

(b) Voting on a single motion when approval requires a qualified majority(greater than 1/2).

(c) Voting on a single motion when approval requires relative majority.

(d) Voting on two simultaneous motions when approval of any of them re-quires relative majority.

To end the work, we discuss the situation that arises when a player announcesits preferences before voting. In fact, the proposal that voters vote is presentedfor one (or more) of them and, consequently, this voter will vote for it.


100 Characterization of Binomial Semivalues and their Modifica-tions

José Miguel Giménez and Rafael AmerUniversitat Politècnica de Catalunya

[email protected]

Each semivalue defined on a space of cooperative games with transferableutility comes determined by a vector of weights. These coefficients offer thepayment to the different players weighting the marginal contributions fromeach one of them according to the cardinality of the coalitions to which theybelong.

Those semivalues whose weighting coefficients are arranged in geometric pro-gression form the family that we designate as binomial semivalues. The Banzhafvalue is the particular case in which the reason is the unit. For this solutiondifferent systems of axiomatic characterization exist, like the proposed by Owenin 1978, Feltkamp in 1988 or Lehrer in 1995.

In this work we propose a characterization for each one of the membersof the binomial family. We obtain these systems of characterization from thehand of a family of games that are introduced previously under the name ofdelegation games, which allow to establish an exclusive property for each one ofthe members of the family who is wanted to characterize. It deserves a specialconsideration the extreme cases corresponding to the marginal index and thedictatorial index.

On the other hand, each binomial semivalue has a natural extension to aconcept of solution modified for games with coalition structure, like the valueof Shapley leads to the coalitional value of Owen (1977). For each member ofthis family of solutions also an axiomatic characterization sets out, so that thisprocedure includes to obtain an axiomatic system for the solution of Banzhafmodified for games with coalition structure.


Cooperative Games III

101 Buyer-seller exactness in the assignment game

Marina Núñez and Carles RafelsUniversitat de Barcelona

[email protected]

Assignment games associated to different assignment matrices may have thesame core (no examples will be found in 2× 2 matrices). Nevertheless, amongall matrices leading to the same core, we are interested in those which havemaximal entries. We say an assignment game is buyer-seller exact if eachmixed-pair coalition attains the corresponding matrix entry in the core of thegame. When an assignment game is buyer-seller exact, no matrix entry can beincreased without modifying the core.

A representation result is obtained: for a given assignment game (N,w)defined by matrix A, a unique buyer-seller exact assignment game Ar with thesame core is proved to exist. The proof makes use of the characterization ofthe extreme core allocations of the assignment game (Núñez and Rafels, 2001).

Once existence is proved, we would like to characterize those assignmentgames which are buyer-seller exact in terms of the matrix entries, and, at thesame time, provide a method to compute Ar for a given matrix A. In Solymosiand Raghavan (2001) exact assignment games are characterized in terms of ma-trix properties: an assignment game is exact if its matrix is dominant diagonaland doubly dominant diagonal. But for a given assignment game, the exactgame with the same core which does exist, might not be an assignment game,this is why buyer-seller exactness seems more suitable for assignment games.

In order to identify the matrix of the buyer-seller exact game related to agiven assignment game, expressions for attainable upper and lower core boundsfor mixed pair coalitions are found: the maximum joint core payoff of buyer iand seller j is w(N)− w(Ni, j).

A new property called strongly dominant diagonal is introduced. All stronglydominant diagonal matrices are doubly dominant diagonal. Finally, an assign-ment game is proved to be buyer-seller exact if and only if its matrix is stronglydominant diagonal.

Some consequences will be deduced. First an open question posed by Quint(1991) searching for a canonical representation of a 45 degrees lattice by meansof the core of an assignment game can now be answered. The second conse-quence is a characterization of those assignment matrices such that there is noother one leading to the same core. Finally we analyze when a buyer-sellerexact game is exact: this happens when the correspondent matrix, in additionto being strongly dominant diagonal, is dominant diagonal.


References

Núñez, M. and Rafels, C. (2001) The extreme core allocations of the assign-ment game, Documents de Treball, Collecció d’Economia E01/65, Uni-versitat de Barcelona.

Quint, T. (1991) Characterization of cores of assignment games, InternationalJournal of Game Theory 19, 413—420.

Shapley, L. S. and Shubik, M.(1971) The assignment game I: The core, Inter-national Journal of Game Theory 1, 111—130.

Solymosi, T. and Raghavan, T.E.S. (2001) Assignment games with stable core,International Journal of Game Theory 30, 177—185.


102 Inventory Games with Discrimination

Ana Meca and Andrés ToledoUniversidad Miguel Hernández de Elche

[email protected]

Several classes of Inventory Games, which arise from basic inventory models,are introduced in Meca et al. (1999), Tijs et al. (2000), Meca et al. (2001),Meca and Toledo (2001). The first one, which is the starting point, studiesseveral situations where a collective of firms minimizes its joint inventory cost bymeans of cooperation. Depending on the information revealed by the individualfirms, related cooperative TU games are analyzed and focus on proportionaldivision mechanishms to share the joint cost.

Tijs et al. (2000), following a slightly different line, study a situation whereone agent has available an amount of storage facility and the other agents havesome goods, part of which can be stored generating benefits. The problem ofsharing the benefits produced by full cooperation between agents is tackled byintroducing a related cooperative game, which turns out to be a Big Boss Gamewith interesting properties.

Meca et al. (2001) and Meca and Toledo (2001), are two different extensionsof the starting point. In the first one is showed that the more general basicinventory model, EPQ with shortages, leads to exactly the same class of costgames studies in Meca et al. (1999). Moreover, the above inventory situationis studied from a non cooperative point of view, providing necessary and suf-ficient conditions for the existence(and uniqueness)of a so-called constructiveequilibrium in which all players make joint orders.

On the other hand, in Meca and Toledo (2001), Inventory Games with Dis-counts are introduced. It is a new class of games which arises from thoseinventory situations in which several firms, demanding a same product, coop-erate acording to different criteria, and the purchasing cost established by thesupplier is modelled by a certain decreasing function depending on the ordersize, which satisfies several smothness properties. According to the classes ofgames introduced in Meca et al. (1999), the price policy fixed by the supplierdoes not affect the decisions made by agents about cooperation and cost shar-ing. However, this policy influence on both of the former, for Inventory Gameswith Discount. This will also happen with the class of games presented in thispaper. It is another extension of the initial Inventory Games, in which now wesuppose that the supplier apply different and constant (not depending on theorder size) prices to each one of his customers.

In an economical context, there are several reasons according to a suppliercould apply this price policy: fidelity or affinity customer-supplier, privilegegranted by the supplier to a customer because of his economical power, etc.


Games arising from this situation, called Inventory Games with Discrimination,turn out to be totally balanced ones. For each one, the core can be obtain asthe intersection of, at least |N |, cores of simpler games. A core allocation ruleover the above class is presented and characterized.

References

Meca A., Timmer J., García-Jurado I. and Borm P. (1999) Inventory Games.Discussion Paper 9953, CenTER, Tilburg University, The Netherlands.

Tijs S., Meca A. and López M. A. (2000) Benefit sharing in holding situa-tions. Technical Report I-2000-01. Operation Research Center, MiguelHernández University, Spain.

Meca A. and Toledo A. (2001) Inventory Games with Discounts. TechnicalReport I-2001-22. Operation Research Center, Miguel Hernández Uni-versity, Spain.

Meca A. and Toledo A. (2001). A note on Inventory Games with Discounts.Technical Report I-2001-26. Operation Research Center, Miguel Hernán-dez University, Spain.

Meca A., García-Jurado I. and Borm P. (2001). Cooperation and Competitionin Inventory Games. Technical Report I-200-28. Operation ResearchCenter, Miguel Hernández University, Spain.

Meca A. and Toledo A. (2002). Inventory Games with Discimination. Tech-nical Report Preprint. Operation Research Center, Miguel HernándezUniversity, Spain.


103 A model of competition in inventory games

Ignacio García-Jurado, Ana Meca and Peter BormUniversidad de Santiago de Compostela

[email protected]

Inventory cost games are introduced in Meca et al (1999). These games arisewhen considering the possibility of joint ordering in n-person EOQ inventorysituations. Moreover, the SOC-rule is introduced and analysed as a cost al-location rule for this type of situations. In the current paper it is seen thatn-person EPQ situations with shortages lead to exactly the same class of costgames. Furthermore, an alternative characterization of the SOC-rule is offered,primarily based on a transfer property which constitutes a special form of addi-tivity. Necessary input variables for the SOC-rule are the (optimal) individualaverage number of orders per time unit in case there is no cooperation. As-suming that these average numbers are observable but not verifiable, we allowthe players to select them strategically, while knowing that the SOC-rule willbe (consecutively) applied as the cost allocation principle. Necessary and suf-ficient conditions are provided for the existence (and uniqueness) of a so-calledconstructive equilibrium in which all players make joint orders.

References

Meca A., Timmer J., García-Jurado I. and Borm P. (1999) Inventory Games.Discussion Paper 9953, CenTER, Tilburg University, The Netherlands.


Price theory and market structure

104 On Auctions and Posted prices

Pekka Aarnisalo and Klaus KulttiHelsinki School of Economics and Business Administration

[email protected]

Kultti (1999 GEB) studies auctions and posted prices in a general equilib-rium model of an urn-ball type or many-to-one matching. In the model sellershave unit supplies and buyers unit demands. All the sellers are alike and allthe buyers are alike. The former value the good for sale at zero and the lat-ter at unity. Consummating trades using auctions turns out to be utilitywiseequivalent to a model where sellers post prices, and buyers decide which sellerto visit based on the observed prices.

It is known that the equivalence does not hold if the agents’ utilities arenot linear. However, it is not known whether the equivalence holds when theagents’ valuations come from a non-degenerate distribution.

In the present paper the model set up is much the same as in Kultti (1999)except that buyers are now assumed to draw their valuations from a commoncontinuous probability distribution with a bounded support and cumulativeprobability function F . Buyer’s valuation is distributed over the interval [0, z],where z is the highest valuation that buyers have. Sellers however, still value thegood at zero. This set up differs from previous studies in a way that formerlybuyers were assumed to have a fixed valuation of the good.

We investigate the equivalence of two transaction mechanisms, auctions andposted prices. Mechanisms are said to be equivalent when sellers and buyers areequally well-off regardless of which mechanism is used. The crucial parameter isthe buyer/seller ratio, and we are interested in whether the possible equivalencedepends on this ratio. As this turns out to be the case we try to track thosebuyer/seller-ratios that give equal amount of utility for both sellers and buyersin either mechanism.

In order to compare the utilities we derive recursive formulae for agents ex-pected life time utilities in both mechanisms. Formulae turn out to be muchmore complex than in the case of a degenerate distribution. Utilities and equi-librium price for the posted price mechanism form a system of equations thatonly in very simple cases can be solved directly. This complicates the analysisof the system greatly.

In the posted price mechanism a formula for a Nash-equilibrium price isderived. The existence and uniqueness as well as some other properties of thatprice are shown.


Finally we calculate some numerical examples showing that in general equiv-alence result does not hold. As an example we present a case where buyers’valuations are evenly distributed with support [0, z]. It turns out that thereexists no buyer/seller ratio that equates the utilities in both mechanisms. Areplicator dynamics kind of analysis is used to derive the equilibrium marketstructure when both mechanisms are used simultaneously.

Non-equivalence result is shown to hold also when valuations are drawn froma two-peaked (binary valuations) distribution in case where both buyer groupshave valuations that are above their acceptance price. If, instead the otherpeak is located below the acceptance price and one is above it the model isreduced to one-peak valuation problem that Kultti (1999) has shown to havethe equivalence property.


105 Who searches?

Juha Virrankoski, Klaus Kultti, Antti Miettunen and Tuomas TakaloHelsinki School of Economics and Business Administration

[email protected]

In a general equilibrium framework where buyers and sellers do not meetautomatically but search each other, it becomes a strategic choice whether tosearch for a partner or wait for a partner. We study the question whetherto search or wait in random matching models where trades are consummatedeither in auctions or in bargaining, and we determine their Nash equilibria.

Most search models assume that either buyers or sellers search. Burdett,Coles, Kiyotaki and Wright (1995 AER) endogenise the search decision in amodel of fiat money and explicit search costs. Herreiner (1999 discussion paperB-462, University of Bonn) studies a market similar to ours, but she ignoresprice formation that is central in our model.

The matching process in our model is of urn-ball type or many-to-one, withno explicit search costs. We ignore the co-ordination aspect where only onetype searches, since everyone expects this type to search. Instead, we considera mixed strategy equilibrium where both buyers and sellers can search and wait,and we find that it exists for a large interval of parameter values. The mixedstrategy tells the likelihood of searching versus waiting for buyers and sellers,and it provides insight into the relevance of standard practice of postulatingthe identity of searchers and waiters. We investigate different mechanisms ofprice determination, including auction and bargaining. The main result is thatthe more numerous party searches.

We also determine the equilibrium of, to our knowledge, a new setup wherebuyers and sellers are in locations. When a, say, seller decides to search, his lo-cation becomes empty and a searching, say, buyer may contact this empty loca-tion. This can be thought to depict an environment of very little co-ordinationwhere agents on both sides of the market may end up without a partner, unlikein the basic model. We compare the efficiency of the above models to thismodel. This is done numerically.


106 About trading mechanisms in a model of endogenous money

Klaus Kultti and Toni RiipinenHelsinki School of Economics and Business Administration

[email protected]

Three commonly applied trading mechanisms or methods to determine theterms of trade in general equilibrium search or matching models are bargain-ing, auctions and price posting. It is known that, for instance, in search modelsof labour markets pairwise bargaining typically result in inefficient outcomes,while auctions result in efficient outcomes (eg. Jansen M. 1999 manuscript,Shimer R. 1999 manuscript). Similarly it is known that price posting resultsin efficient outcomes (eg. Acemoglu and Shimer 1999 IER, Peters M. 1984Ecma). Further, Kultti (1999 GEB) shows that auctions and price postingare utilitywise equivalent mechanisms. In none of the above models there arebasically any budget constraints, and the amount of production is fixed. Torelax these assumptions I turn to the standard model of endogenous money(eg. Kiyotaki and Wright 1993 AER) where some agents are endowed with aunit of indivisible and intrinsically useless object, i.e. fiat money. The onlyway to trade in a monetary equilibrium is to exchange this unit of money to aconsumption good. To get meaningful prices we assume that the amount of thegood to be exchanged is a variable as it is assumed to be produced on the spot.To motivate anonymous monetary exchange, or the lack of credit relationships,it is assumed that the agents meet in a random manner. To make the study ofdifferent trading mechanisms meaningful it is assumed that the meetings areof many-to-one or urn-ball-type. The economy proceeds infinitely in discretetime. The agents are either in a role of buyers aka money holders, or sellersaka producers. Upon meeting the terms of trade, i.e. the amount produced isdetermined, and it depends on the trading mechanism. Bargaining is modellednon-strategically as a division of surplus using the Nash-bargaining solution.Price posting is a strategic game for sellers. The trade-off is that by lower-ing the posted price a seller attracts more buyers who choose mixed strategiesdetermining which sellers to visit. We determine the Nash-equilibrium price.Auctions are of the second price sealed bid variety where the seller is assumedto be one of the participants. In the model the reservation values arise endoge-nously since the parties can always leave the current partner and wait for nextperiod to meet new ones.

We use specific functional forms to demonstrate that the there is basically nohope for any kind of equivalence results between any of the trading mechanisms.Further, it turns out that the trading mechanisms cannot be ordered in a welfaresense. Depending on the number of money holders any of them may be themost efficient. We also produce some illustrative numerical examples.


Noncooperative Games

107 On Nash Equivalence Classes of Normal Form Games

Fabrizio GermanoUniversitat Pompeu Fabra

[email protected]

The paper introduces a procedure that uses basic topological characteristics ofequilibrium correspondences of standard equilibrium concepts like correlated andNash equilibrium to define broad equivalence classes of (generic) normal formgames. The proposed procedure is a potentially useful way of both organizingthe underlying spaces of games as well as of comparing different equilibriumconcepts with each other. The focus of the paper is mainly on equivalenceclasses induced by the Nash equilibrium concept. However, equivalence classesinduced by the concepts of rationalizability and correlated equilibrium are alsoconsidered and some relations between them are derived.

The representation of a strategic situation by means of a normal form gameis one of the most fundamental constructions of game theory. But, while muchwork has been done devising and studying different equilibrium concepts fornormal form or extensive form games, not much has been done to systematicallydifferentiate and understand the different “types” of games that can arise. Someexceptions are Rapaport, Guyer and Gordon (1976), Conway (1976), Barany,Lee and Shubik (1993), which systematically study equivalence classes of spe-cific classes of games.

The present paper takes standard equilbrium concepts like rationalizabil-ity, correlated or Nash equilibrium, to define equivalence classes of normalform games by means of the corresponding equilibrium correspondences. Morespecifically games are distinguished on the basis of very basic topological orgeometric characteristics of the graphs of the equilibrium correspondences. ForNash equilibria, the geometric characteristics are defined using notions of sin-gularity or transversality; for correlated equilibria, the dimension of the set ofcorrelated equilibria is used. Distinguishing games on the basis of such criteriadivides the space of games into a finite number of connected components ofgames, which constitute the equivalence classes. This leads to classifications ofnormal form games that, through the geometries of the equilibrium correspon-dences, are naturally related to the corresponding equilibrium concepts.

For example, applying the procedures for the Nash and correlated equilib-rium concepts to the space of 2× 2 normal form games leads, in both cases, tojust three equivalence classes of games, namely (i) games with a unique pureNash equilibrium (e.g., prisoners’ dilemma) (ii) games with a unique strictly


mixed equilibrium (e.g., matching pennies), and (iii) games with two pure andone mixed equilibria (e.g., battle of the sexes or chicken).

More generally, the classifications obtained yield a way of highlighting as-pects of normal form games distinguished by different equilibrium concepts. Atthe same time, they provide natural organizations of the underlying spaces ofgames into well-defined equivalence classes, whereby, in order to keep the anal-ysis as useful and essential as possible, the topological characteristics inducingthe equivalence classes are chosen so as to obtain as broad classes as possible.

Besides providing a better understanding of the underlying equilibrium con-cepts, and especially of the geometries of the corresponding equilibrium corre-spondences, the procedures also provide a transparent way of comparing equi-librium concepts with each other. In fact, the proposed procedures can beextend to other equilibrium concepts beyond the three studied in this paper.Furthermore, the present approach also forces one to think about what thefundamental characteristics describing a strategic situation may be and why.


108 A Communication Protocol for Random Consensus in the Co-ordinated Attack Problem

Amparo Urbano and José VilaUniversity of [email protected]

Consensus problems arise in many situations. A particular consensus prob-lem is the “coordinated attack problem”: a fundamental of reaching consensuson a setting where messages may be lost. The impossibility result for determin-istic systems is quite well known. This suggests that there is little that can bedone to solve basic consensus problems in the face of unreliable communication.One approach to strength the model is to make some probabilistic assumptionswhile keeping the system deterministic. This is Rubinstein’s Electronic MailGame (1989) consisting of a two-players game with incomplete information,which is the extension of an underlying incomplete information game G. In G,nature has two possible states a and b and informs player 1. Each player hastwo actions to chose, denoted by A and B. Players are interested in choosingaction A in state a and action B in sate b. In Rubinstein’s game there is onlyone Nash equilibrium where player 1 plays A in state a. In other words, bothplayers play A independently of the number of messages received.

Our paper also deals with the coordination paradox in the coordinated at-tack problem. Our concern is with the practical coordinated-action problem.We consider an adaptation of the Electronic Mail Game. As in Rubinstein’smodel, players are located at two different sites and they communicate only byelectronic mail signals. There is also a “small” probability that a message doesnot arrive at its destination. Under our approach players use a randomized com-munication protocol with some probability of miscoordination. The protocolconsists of a communication network, which transmits the level of informationof players and a decision rule to choose actions. However, the electronic mailnetwork is set up differently from Rubinstein’s model. Communication lasts afinite number of periods r. The decision rule to take actions depends on aninteger K, chosen uniformly at random from the interval [1, r−1]. The commu-nication network adds to Rubinstein’s design an additional movement in periodr − 1, where with some probability the players exchange the last message. Weshow that to follow the communication protocol is an interim Bayesian-Nashequilibrium of such a game, supporting coordinated attack with probability

r − 2

r − 1.


109 The Coordination Paradox in the Electronic Mail Game

José Vila and Amparo UrbanoUniversity of [email protected]

This paper analyzes the coordination paradox in the coordinated attackproblem via Rubinstein’s Electronic Mail Game. In his 1989 paper, Rubinsteinconsiders a two-players game with incomplete information, which he called theElectronic Mail Game, EMG, hereafter. This game stylizes the key features ofthe coordinated attack problem, substituting message-doves by an automaticsystem of e-mails delivery, with a positive probability of failure. The EMG, isthe extension of an underlying incomplete information game G. In G, naturehas two possible states a and b. a is the most likely state and it appears withprobability (1 − p), where p < 1/2. Each player has two actions to chose,denoted by A and B. Players are interested in choosing action A in state a andaction B in sate b, but only Player 1 gets informed of the actual state of nature.Rubinstein shows that there is only one Nash equilibrium of EMG where player1 plays A in state a. In this equilibrium, both players play, A independently ofthe number of messages received. Moreover, the ex-ante equilibrium payoff isthe same as if no e-mail communication system had existed.

We propose an alternative scheme of cheap talk. This scheme is not ableto generate common knowledge, but it induces common p-beliefs on the eventsrelated to the attack, in such a way that a non-deterministic coordination cantake place in every state of nature. The key feature of our communicationscheme is that player 1, the informed agent, is the only agent who sends e-mails. The role of our communication scheme is to guarantee that players’probabilistic information on the actual state of nature is accurate enough tosupport coordination. The main difference with Rubinstein’s construction isthat such an accuracy increases with the length of the communication phase.This fact allows the players to communicate by sending a number large enoughof messages.


Communication Networks

110 On Economies of Scale and Cooperation in Some Communi-cation Networks

Darko Skorin-Kapov and Jadranka Skorin-KapovAdelphi University, Garden City, New York

[email protected]

The objective for designers of communication networks is often to minimizethe cost, while satisfying various constraints on reliability, congestion, speed,capacity and coverage. Economies of scale play an important role in theirconsiderations. Namely, the creation of high capacity links and concentration offlows reduces the number of needed links and the unit flow cost. Applications ofthis approach are seen, for example, in high capacity lines of backbone networksin telecommunications, and in high volume traffic between major airports inair transportation networks.

Most of the related work in the literature was performed in the context ofthe so-called hub networks. In those networks a certain subset of focal nodes(i.e. hubs) is fully interconnected while other nodes are connected to thosehubs and the economies of scale are achieved by discounting the cost of trafficamong hubs. The hub networks were extensively studied over the last coupleof decades and numerous computational studies show that hub networks arequite attractive and practical. Nevertheless, the restrictions imposed with thehub network model are sometimes to prohibitive. Recently, authors togetherwith some collaborators introduced the network model in which each pair ofnodes can communicate via any path, and the cost of sending flow through eachlink is discounted if and only if the amount of flow exceeds a certain threshold.This approach also gives incentive to concentrate flows. It seems however, thatthe above threshold based discounting model is even more ‘efficient’ than hubnetworks in its use of a relatively small number of links and in the exploitationof economies of scale. We will refer to the threshold based discounting networkmodel as to the hub-like network model.

In this paper, we study the cost allocation problem in hub-like networks.Namely, the cost of services delivered through the hub-like network is dis-tributed among its users who may be individuals or organizations with possiblyconflicting interests. The cooperation of these users is essential for the exploita-tion of economies of scale. Consequently, there is a need to find a fair distribu-tion of the cost of providing the service among network users. It is well knownthat in general, the game theoretic network cost allocation solution conceptsare computationally prohibitive even for relatively small problems. Moreover,


there are no general practical algorithms for the computation of these solutions.Consequently, researchers have concentrated on individual classes of networkgames to demonstrate that computation of network cost allocation solutionconcepts is sometimes feasible in the context of a particular problem.

In our study we take similar approach. Namely, we develop some cooperativegame theory based mechanism to efficiently characterize cost allocation solu-tions for networks that allow threshold based discounting. Specifically, whilepaying special attention to users’ contribution to economies of scale, we formu-late the associated hub-like network cost allocation games in the characteris-tic function form. Since the optimization problem is NP-hard, we depend onour heuristic algorithms to obtain optimal or best-known optimization results.These results are used to determine the value of the associated characteristicfunction. We use our four and three-dimensional optimization formulationsfor small and large size problems respectively. In the four dimensional casewe define players to be pairs of users, while in the three dimensional case weset players to be individual users. We demonstrate that hub-like games aredecomposable into link games. We then show that the cores of those gamesare not empty and that they can be characterized with polynomial number ofconstraints. Finally, we show that the nucleolus of the above hub-like networkgames can be efficiently computed.

Key words: network design, economies of scale, heuristic algorithms, combina-torial optimization, cost allocation, cooperative games.


111 Semivalues and Centrality in Social Networks

Daniel Gómez, Enrique González, Manuel Conrado, Guillermo Owen, Mónicadel Pozo and Juan Tejada

Universidad Complutense de Madrid & Naval Postgraduate [email protected]

A social network can be modelized as a graph, which shows the possibledirect communications between individuals. A cooperative game in charac-teristic function form can be considered to reflect the interest that motivatethe interactions. Then, the graph-restricted game (the Myerson-game) is usedto represent the economic possibilities of coalitions taking the available com-munications into account. Every semivalue in a game can be considered asactor’s power index. A centrality measure is defined as the semivalue of a givensymmetric communication game. Conditions like symmetry, supperaditivity orconvexity are neccessary to reach desirable properties for these measures.

In Gómez et al. (2001) is proved that when the chosen semivalue is theShapley value, the centrality measure satisfies the following properties:

1. Is symmetric.

2. The centrality of a node in a disconnected subgraph coincides with thecentrality of that node in the connected subgraph to which it belongs.

3. Isolated nodes have minimal centrality.

4. In a chain centrality increases from the end node to the medain node.

5. Of all connected graphs with n nodes, the minimal centrality is attainedby the end nodes of a chain.

6. Of all connected graphs with n nodes the maximal centrality is attainedby the hub of a star.

In this communication we explore the extent to which previous propertiesare satisfied when we consider other semivalues.


112 Stochastic Minimum Spanning Tree Games

C. Fernández, F. R. Fernández, and M. J. ZafraUniversidad de Sevilla

[email protected]

Stochastic cooperative games are a new area of interest within Game The-ory, in which the worth given to any coalition through the characteristic func-tion is a random variable. In this paper, we introduce a stochastic version ofthe minimum-cost-spanning-tree game (MCST-game). For this special class ofgames we give a family of core solutions that are based on proportional al-locations of the worth allocated to the grand coalition. In addition, we givesufficient conditions ensuring that such allocations exist.


113 Multicriteria Minimum Spanning Tree Games

M. A. Hinojosa C. Fernández, F. R. Fernández, and J. PuertoUniversidad de Sevilla

[email protected]

The minimum cost spanning tree game (MCST-game) is a well-known modelwithin Operations Research Games that has been widely studied in the litera-ture. In this paper we introduce the multicriteria version of the MCST-gameas a set-valued TU-game. We prove that the extension of Bird’s rule providesdominance core elements in this game. We also give a family of core solutionsthat are different from the previous one; these solutions are based on propor-tional allocations obtained by supported scalar solutions of the multicriteriaspanning tree problem. Besides, we prove sufficient conditions ensuring thatBird’s rule belongs to the preference core of this game.


Market Models

114 Matching, Heterogeneity, and Noisy Observable Types

Hector ChadeArizona State University, U.S.A.

[email protected]

This paper studies a two-sided search with heterogeneous agents and non-transferable utility in which each individual observes only a noisy signal of thetype of a potential partner, as in marriage and entry-level labor markets.

It is shown that the optimal strategy for an agent takes into account notonly the signal observed, but also the information contained in the event thatthe agent is accepted. The latter depends on the characteristics of the strategyfollowed by the agents on the other side of the market, which complicates theanalysis of the model substantially.

An equilibrium in increasing strategies is characterized using some recentresults on monotone methods and Bayesian games. In that equilibrium, positivesorting of agents by types holds in a stochastic sense: on average, agents withhigher types mate together.


115 The Inner Core of Concave Exchange Economies

Geoffroy de Clippel and Enrico MinelliCORE, Université Catholique de Louvain

[email protected]

Shapley (1969) proposed a general procedure that allows to extend TU solu-tion concepts to NTU games by considering fictitious transfers of utility. Whenapplied to the TU core, this procedure leads to the inner core.

In the context of well-behaved exchange economies with concave utility func-tions, the inner core appears to be a relevant refinement of the core. First, itcontains the set of competitive equilibria. As a consequence, convergence andequivalence results for the core can immediately be extended to the inner core.Second, the inner core coincides with the set of feasible allocations such that nocoalition can improve on it, even if coalitions are allowed to use some randomplans: Given a proposed allocation x of commodities, we consider that individ-uals in a coalition S are able to write deviating contracts between themselveswhich specify a probability p of refusal, a feasible (for S) allocation y to berealized in this case, and a reallocation z among themselves of the commoditiesthey would receive in case of acceptance of x. The status quo allocation x canbe L-improved if

(∀i ∈ S) : ui(xi) < pui(yi) + (1− p)ui(zi),

for some such S, p, z, y. The standard notion of improvement is the specialcase in which p = 1. The notion of L-improvement involves the computationof expected utilities, and exploits the cardinal content of the concave utilityfunctions.


116 Competitive equilibrium as the outcome of a capacity-pricegame

Massimo A. De Francesco,Department of Economics, University of Siena

Piazza S. Francesco 7, 53100 Siena, [email protected]

The paper analyses Bertrand competition under free entry for a single marketof a homogeneous commodity. It is shown that,similarly as in the Cournotsetting (see Novshek, 1980, Novshek and Sonnenschein, 1978), in a “large”market the equilibrium of the capacity-price game exhibits long-run competitivefeatures. Production is assumed to take place in the short run at constantvariable unit cost c up to capacity; in the long run, average cost at full capacityis assumed to be constant at c + k, k being the cost of a unit of capacity.One peculiar feature of the model is the recognition of indivisibilities in capitalequipment; as a result, capacity can be chosen within a grid and the minimumfeasible capacity is bounded away from zero. Under these assumptions, in thecompetitive equilibrium with free entry total capacity is at the largest levelsuch that active firms make nonnegative profits at the market-clearing price (i.e, at the largest level such that the demand price of total capacity is not lessthan c + k); any industry configuration –number and size of active firms–consistent with this condition can occur at the competitive equilibrium.

Next the capacity-price game is analysed. Entry, capacity and price decisionsare structured as follows. In the first stage, a large number of potential entrantsmake simultaneously entry and capacity decisions, which become known tomarket participants at the beginning of the second stage; then active firmschoose prices, whereupon the buyers choose where and how much to buy, whichin turn affects the firms’ output decisions. What is crucial is how many firmsare required to produce the long-run competitive output when each of them isof the minimum feasible size. If this number (n∗) is sufficiently large –i.e., ifthe economy is sufficiently “large”– then the capacity-price game yields thelong-run competitive outcome, with active firms operating at the minimumsize. This result is obtained through the sequence of propositions summarizedbelow.

Proposition 1. Suppose there are n∗ active firms, each with the minimum fea-sible capacity so that total capacity is the one obtaining at the long-run competi-tive equilibrium. With n∗ sufficiently large, all firms charge the market-clearingprice at an equilibrium of the price subgame.

The proof of this proposition is standard (see Vives, 1986). It rests on thefact that, when the size of the firm is small relative to the total capacity of


rivals, the elasticity of its residual demand is large so that it does not pay thefirm to charge a price higher than rivals’. (Incidentally, it is worth remindingthat, in such a case, the equilibrium of the price game reproduces the Cournotoutcome.)

As a further, intermediate step, we derive a few properties of the mixedstrategy equilibria of the price subgame, which in fact extend similar propertiesof mixed strategy equilibria in a two-firm setting (see Kreps and Scheinkman,1983). Taking advantage of these properties, it is established that the industryconfiguration discussed by Proposition 1 is an equilibrium of the capacity-pricegame.

Proposition 2. An industry configuration with n∗ active firms, each with theminimum feasible capacity, is part of an equilibrium of the capacity-price game.

We are also able to exclude other industry configurations –whether com-petitive or not– from being equilibria of the capacity-price game.

Proposition 3. (a) An industry configuration with total capacity at the com-petitive level and with less than n∗ active firms cannot obtain at an equilibriumof the capacity-price game; (b) An industry configuration with total capacity dif-ferent from the competitive level cannot occur at an equilibrium of the capacity-price game.

References

Kreps, D. M and Scheinkman, J. A. (1983) Cournot pre-commitment andBertrand competition yield Cournot outcomes, Bell Journal of Economics14, 326—338.

Novshek, W. (1980) Cournot equilibrium with free entry, Review of EconomicStudies XLVII, 473—486.

Novshek, W., and Sonnenschein, H. (1978) Cournot and Walras equilibrium,Journal of Economic Theory 19, 223—266.

Vives, X (1986) Rationing rules and Bertrand-Edgeworth equilibria in largemarkets, Economics Letters 113—116.


117 Market Equilibria via Integer Programming

Ivan Arribas and Amparo UrbanoUniversity of [email protected]

We study a model of price competition in a multiproduct oligopoly market.The products are of general nature. Sellers fix prices to maximize their profitsand the representative buyer, after observing prices, selects a bundle of productsaccording to his/her reservation value. Consumers are characterized by a setfunction which determines their willingness to pay for every subset of products.The existence and characterization of Nash equilibria is reached via integerprogramming.

We start with the simplest model where firms produce only one product andextend the equilibrium characterization of Tauman, Urbano and Watanabe(1997) to non-monotonic consumer value functions. Then, we generalize themodel to multiproduct firms, and show that Nash equilibria with non-linearprices always exists. We also offer the sufficient conditions which guarantee theexistence of linear-price equilibria.


Conflicts, Collective Decision

118 An Amendment to Final-Offer Arbitration

Dao-Zhi ZengFaculty of Economics, Kagawa University, Japan

[email protected]

Compulsory arbitration has become an increasingly frequent mean of resolv-ing labor disputes in the private and public sectors and a variety of arbitrationprocedures have been proposed and used.

Conventional arbitration (CA) is the most traditional procedure to settledisputes that arise in the negotiation of labor agreements and also in commer-cial disputes. CA requires that a third party arbitrator listen to the offers fromdisputants, based on which the arbitrator imposes a final settlement which isfair in his/her own view. In CA, although the arbitrator is free to fashion asettlement of his/her own choosing, and the arbitration decision needs not tobe a compromise between the disputants’ offers, it is observed that arbitrators’decisions are influenced more or less by disputants’ offers. Therefore, CA pro-vides an incentive for the disputants to avoid pre-arbitration concessions, orCA “chills” the negotiation process which precedes arbitration.

Final-offer arbitration (FOA) was proposed by Stevens (1966). It requiresthat the arbitrator choose one of the disputants’ offers without compromise. Inthis way, “winning” becomes important so disputants have to give compromis-ing offers. Although this procedure is actually used in the determination of thewage in disputed contracts for major league baseball players and other manypublic sector labor disputes in US, theoretically and practically we know thatthe offers still diverge (see Farber (1980) and Brams and Merrill (1983)).

Some procedures are proposed to improve FOA. Theoretically, it is knownthat combined arbitration (CBA) of Brams and Merrill (1986) and double-offerarbitration (DOA) of Zeng et al. (1996) can induce offers’ convergence withsome additional conditions to ensure the existence of a Nash equilibrium. Thepurpose of this paper is to provide a simple amendment of FOA so that it’soffers converge in quite general situation. Specifically, in the original FOA, thefinal arbitration result is determined by the winner’s offer. However, in our newarbitration procedure, the final arbitration result is determined by the loser’soffer.

We furthermore illustrate two variations of the amended FOA. One of themgeneralizes the CBA of Brams and Merrill (1986).

Finally, we consider a practical issue of the amended FOA. If two disputantsfail to give convergent offers due to some casual reasons, the final arbitrationresult may be quite extreme. We propose two methods to solve this problem.


119 Bargaining with Commitment under an Uncertain Deadline

Vicente Calabuig, Gonzalo Olcina and Antoni CunyatUniversity of Valencia

[email protected]

The horizon of a bargaining game is an important factor in determining theoutcome of a negotiation process. There exist situations in which a last periodis not perceived, i.e, infinite horizon games. On the contrary, there are nego-tiations in which the final period determines, from the beginning of the game,the strategic behavior of the players. However, in some situations both types ofhorizon may emerge. For example, many negotiations on environmental issuesor labor conditions are conducted under the pressure of a possible interventionof the government by imposing a deadline if the parties do not reach an agree-ment. Therefore, when players engage in such negotiations, even though theymay perceive that there is room for a counterproposal, they can also perceivethat at any time the government can impose a deadline after which there willnot be scope for an agreement.

On the other hand, commitment is an essential component in many real-life negotiations because it both clearly affects the bargaining power of thenegotiator and it may lead to inefficiencies. An (endogenous) commitment,in which players cannot accept a less generous offer than an offer previouslyrejected (combined with a deadline), can lead to delay in negotiations as itwas first shown by Fershtman and Seidmann (1993). This assumption can bejustified by the existence of representatives. In this sense, a representativewould have problems in “explaining” to his principal that he had accepted aworse offer than an offer previously rejected.

Another justification for an uncertain deadline appears in this context. Ifa principal has a fixed deadline, which is private information, he may prefernot to communicate it to his representative until the very last moment sinceby knowing a priori the exact date where the cake vanishes, he will be in aweaker bargaining position. A result in this direction has been obtained bySandholm and Vulkan (1999) in a context of automated negotiations wheresoftware agents bargain on behalf of their users. These authors even suggestthat “a user is better off by giving her agent a time discount function insteadof a deadline since a deadline puts her agent in a weak bargaining position”.All these arguments imply that from the point of view of a representative thedeadline can be an uncertain event.

In the present work, we analyse how the presence of both features, an uncer-tain deadline and bargainers with endogenous commitments, affect the outcomeof the bargaining process. More precisely, we analyse an infinite horizon bar-


gaining game in which there is a positive probability that a deadline appearsafter any rejection, and where players possess an endogenous commitment.

Although apparently our model resembles a negotiation with probabilisticbreakdown, this is not the case. There are two basic differences. First, whereasin the model with a breakdown there is a chance that, after a rejection, thebargaining terminates, in our model after the appearance of the deadline, play-ers remain in the same bargaining relationship, but they face, unexpectedly,the last chance of reaching an agreement in the relationship. Secondly, thepayoffs in the model with breakdown are exogenous to the bargaining process,whereas in our work the players’s payoffs depend on the history of offers andcounterproposals of the game.

We show that, for any truncation of the infinite horizon game, when theplayers are sufficiently patient and possess an endogenous commitment, theywill fail to reach an agreement before this truncation point, unless a deadlineappears. However, since the deadline is probabilistic, this will result in delaysin equilibrium and, therefore, in inefficiencies.

Finally, note that in Fershtman and Seidmann’s framework with an infinitehorizon there exists a subgame perfect equilibrium which yields an immediateagreement to the Rubinstein division of the surplus. However, our model showsthat this result is not robust to a slight perturbation in the horizon of the game,namely, a small exogenous probability of the appearance of a deadline in eachperiod.


120 Political Competition as a War of Attrition

Rene LevinskyFaculty of Economics, University of Freiburg, Germany

[email protected]

The essence of the situation to be studied in this paper is the following: fourplayers are involved in a competition that is resolved by playoff elimination.The heterogenous agents characterized by their “capacities” play sequential“concession” game to win the prize. Thus, the strategic variable of the playersis the level of “effort” they are willing to spend in each round. It is assumedthat the sum of effort spent in the two rounds cannot exceed the “capacity”of the player. The player investing higher effort in a partial conflict wins theparticular conflict, the first round winners can participate in the final and theoverall winner get the prize.

The brief description of the game suggests that the analyzed situation isa variant of the (sequential) game of attrition introduced by Maynard Smithin 1982 to study the evolutionary stability of behavioral patterns in animalconflicts. However, our approach differs from classical war of attrition in tworespects. First, we generalize the payoff function of the winner of particularconflict as a function of effort of both winner and loser, while the payoff of thefollower in the war of attrition depends just on the time, in that the leadergive up the game. We suppose that the effort that the player plan to spend inpartial conflict is not only of the matter of “time (and money) that she is wilingto spend during the game,” but also “time (and money) that she sacrifices forthe training and preparation for the game.” While in the first case can thewinner (i.e., follower) stop her expenditures directly at the moment in that heropponent concedes, the later expenses are payed “a priori” and there are lost.

The other generalizations of the war of attrition employed in our paperfollows from our primary intention to consider the game in the framework ofthe political competition. We motivate our approach in the sense that thequalification round of the game stays for the intra-party competition and thefinal represents the general elections. Therefore, we try to model strategicinteractions and phenomenons that are observable in the political context.

First, we examine the above described modification of the standard warof attrition. Having this benchmark in hands we allow for side-payment andrepeating. The first case we model as a “bribing” the first-round opponentto save the own capacity for the final round. In this scenario the players ex-ante commit themselves to ex-post re-distribution of the prize to strategicallychange the incentives of the opponents to fight. Then, in the repeated gameframework, we describe various “political contracts” regarding the restriction tothe number of period in that can the (overall) winner subsequently participate


in the political competition. Here, we are mainly interested in the impact ofthe constitutional rules on intensity of intra party competition.

Technically, we assume that the capacities of players are public knowledge.The mixed strategy Nash equilibria of particular games are described, discussedand compared.


121 On Efficiency and Sustainability in Collective Decision Prob-lems with Heterogeneous Agents

Ori Haimanko Michel Le Breton and Shlomo WeberBen-Gurion University of the Negev, Israel

[email protected]

In this paper we examine a collective decision problem, where the set ofheterogeneous individuals is partitioned into several groups, each choosing itsown policy (e.g., location of a public project) from the given policy space.The model is that of “horizontal product differentiation” where individualsdisplay distinct preferences over the policy space. We first consider the notion of“efficient” partition that minimizes the total policy-related costs and aggregatepersonalized costs. (The latter are incurred when an individual belongs to agroup that does not choose her most preferred, ideal, policy.) We then examine“sustainable” partitions, in which the policy-related costs can be distributed ina way that no subgroup (belonging to the partition or not) has an incentive tobreak away from the rest and to set its own policy. Our main result is that,with a unidimensional policy space and single-peaked personalized costs, everyefficient partition is sustainable.

We further describe some important features of efficiency by characterizingthe efficient distribution (and number) of policies chosen from the policy spacewhen their cost is small. It turns out that efficiency is achieved when thedistribution of policies follows the square root of the density of individuals’ideal choices.


Solution Concepts

122 The Not-quite Non-atomic game: the Role of Differentiabilityfor the Core of Production Games

Martha Saboyá, and Guillermo OwenUniversidad Autónoma de Madrid & Naval Postgraduate School

[email protected]

In Aumman and Shapley (1974), hereafter referred to as A/S, is presenteda remarkable study of non-atomic games. In that paper the question of therelation between the core and the Shapley value of very large games is studiedin detail. In particular, productive economies are analyzed. In that regard,one assumes a continuum of individuals, each of them is regarded as a profitmaximizing producer endowed with his technology and resource bundle, poolinghis private endowments to achieve better outcomes. They proved under certainregularity conditions, that the core of the corresponding production game isnon-empty and it consists of a single point, which coincides with the non-atomic Shapley value. This value is based on a vector of equilibrium prices forthe goods in an endowment.

However, the continuum assumption is justified by the claim that a contin-uum of agents is a good approximation to a large but finite number of players.It is important to test this assumption by proving a limit theorem, showing thatthe behavior of a finite game is indeed a good approximation to a continuumgame, as the number of players becomes unbounded large.

In a previous paper by Flam and Owen (2002), the non-emptiness of the coreis discussed. Specifically, they showed that, under similar conditions (to thoseof A/S), the core of the finite games might well be empty. However, point basedon a Lagrangian price vector will be “almost” in the core, in the sense that foreach coalition S the “excess” will not be greater than some (relatively) smallpositive number (“absolute deficit”). The “relative deficit”, i.e. the additionalamount of payoff needed to reach a core point (measured as a fraction of totalavailable utility), converges to zero, so that, “in the limit”, a non-empty coreis obtained. We prove in our work that differentiability gives us a strongerresult: not merely the “relative”, but the “absolute” deficit, goes to zero underreasonable regularity conditions.

Specifically, we observed that, for one allocation based on a Lagrangian pricevector, there are at most so many agents as number of resources. For theseagents, the value of concavified production function at their optimal allocationis strictly greater than the value of original ones. We construct an allocationthat will be “almost” in the core, by redistributing the resources of these players


among all agents. Under certain conditions of the production functions, a goodestimation of the deficit is obtained.

To prove that the absolute deficit goes to zero, we have to allow the numberof agents to grow unboundedly. One of the several ways to do it is by “marketreplication”. We show, by using our estimate of deficit, that in this case thedeficit goes to zero as the number of players increases. But market replicationimposes a very regular increase in the number of players. We analyze a weakerpossibility of increase in players, which depends strongly of two parameters ofour above estimate and we get the desired convergence to zero.

References

Aubin, J. P. and Ekeland, I. (1976) Estimates of the duality gap in nonconvexoptimization, Mathematics of Operations Research 1, 225—245.

Aumann, R. J. and Shapley, L. S. (1974) Values of Non-Atomic Games,Princeton University Press, New Jersey.

Edgeworth, F. Y. (1881) Mathematical Psychics: An essay on the Applicationof Mathematics to the Moral Sciences, London.

Flam, S. D. and Owen, G. (2001) Large Production and Market Games.

Flam, S. D. and Owen, G. (2002) The Not-quite Non-atomic game: Non-emptiness of the core in production games.

Hildebrand, W. (1974) Core and Equilibria in a Large Economy, PrincetonUniversity Press, New Jersey.

Mas-Colell, A. (1977) Competitive and value allocations of large exchangeeconomies, J. Economic Theory 14, 419—438.

Owen, G. (1975) On the core of linear production games, Mathematical Pro-gramming 9, 358—370.

Owen, G. (1993) The Not-Quite Non-Atomic Game: Normal Approximation,Int. J. Game Theory 405—418.

Shapley, L. S. and Shubik, M. (1966) Quasi-cores in a monetary economy withnon-convex preferences, Econometrica 34, 805—827.

Shapley, L. S. and Shubik, M. (1969) On Market Games, J. Economic Theory1, 9—25.

Wooders, M. H. (1994) Equivalence of games and markets, Econometrica 62,1141—1160.


123 Serial cost sharing: Discrete and continuous case

Emilio Calvo, Esther Gutiérrez and Juan Carlos SantosUniversitat de Valè[email protected]

In this paper we study cost allocation problems and the asymptotic relationbetween the serial cost sharing method defined in the discrete case (Moulin,1995), and two different definitions of the serial cost sharin rule in continuouscost allocation problems.

Discrete cost allocation problems arise when goods are produced in indivis-ible units, and continuum cost allocation problems arise when all the outputsare perfect divisible goods and the level of production can be measured by a realnumber. We will consider a continuum cost allocation problems as the “limit”of discrete cost allocation problems. These problems are a discrete version ofthe original one, allowing only a finite number of levels, instead of a continuum.Following this asymptotic approach, we show the relation between the discreteand the continuum serial cost sharing methods.

We construct two different sequences. In the first sequence it its assumendperfect comparability between the units of different goods. Then we show theconvergence of the discrete serial cost rule to the continuum extension definedby Friedman and Moulin (1999). In the second sequence, we compare the unitsof different goods by their stand alone production cost. In this case we showthe convergence with a new serial extension, that we call individual cost pathsolution.

Moreover, we characterized axiomatically these serial cost sharing methods(on discrete and continuum CAP) by a system of similar axioms.

References

Friedman, E., and Moulin, H. (1999) Three Methods to Share Joint Cost orSurplus, Journal of Economic Theory 87, 275—312.

Moulin, H. (1995) On Additive Methods to Share Joint Costs, The JapaneseEconomic Review 46, 303—332.


124 The Nucleolus as a Power Index

María MonteroUniversity of Dortmund, Germany

[email protected]

This paper studies a situation in which n players vote on the division of abudget. Very general voting rules are considered, namely all the rules such thatno two disjoint groups of agents have a majority (i.e., the associated character-istic function game is proper). A player’s expected share of the budget can beinterpreted as a measure of his power. This type of power is known as P -power,as opposed to the probability of being pivotal, or I-power (see Felsenthal andMachover, 1998). This paper addresses the question of what is a good measureof P -power. From the axiomatic point of view, the answer seems to be theShapley value. We approach this question from a strategic point of view, usingan extension of Baron and Ferejohn’s (1989) noncooperative bargaining model.

Bargaining proceeds as follows. An agent is selected at random to propose adivision of the budget. If a winning coalition votes in favor of the proposal, theproposal is implemented; otherwise, a new proposer is selected at random, al-ways using the same probability vector. Baron and Ferejohn limited themselvesto symmetric majority games in which all players are selected to be proposerswith equal probability; we consider arbitrary proper simple games and proba-bility vectors. Like Baron and Ferejohn, we look at stationary subgame perfectequilibria.

The are two measures of a player’s power in this game: the probabilityof being a proposer (an input for the game) and the expected share of thebudget in equilibrium (an output of the game). A power index is called self-confirming if it can be obtained as the equilibrium expected payoff of the gameusing the index itself as probability vector. The nucleolus is the only well-known value allocation that has this property. This is surprising, since thenucleolus was introduced as a fair division rule given that the grand coalitionforms, which does not generally happen in equilibrium (indeed, in constant-sum homogeneous games the grand coalition never forms in equilibrium). Theproof uses the characterization result by Kohlberg (1971) reinterpreting thebalancing weights as mixed strategies.

The self-confirming property means that the nucleolus is a fixed point ofthe noncooperative process we have described. It does not imply that thereis any particular reason to use the nucleolus as a probability vector in thefirst place. In particular, the nucleolus often assigns a value of zero to playersthat are not dummies. However, for constant-sum weighted majority gamesin homogeneous representation the nucleolus is proportional to the number ofvotes of the players. If we identify the players with parties and the number of


votes with the number of seats of each party, a probability vector proportionalto the number of votes seems natural since it treats all representatives equally.

We conclude that the nucleolus can compete with the Shapley value as ameasure of P -power in proper simple games. While the Shapley value looksmore appropriate from the axiomatic point of view, it lacks a compelling non-cooperative interpretation for this type of games. Hart and Mas-Colell (1996)obtain the Shapley value as an equilibrium of a bargaining game with ran-dom proposers for a wider class of games. However, they require unanimousagreement, which does not seem appropriate for majority games.


125 A survey on the potential approach in cooperative game theory

Theodorus S. H. DriessenUniversity of Twente at Enschede, The Netherlands

[email protected]

In 1989, the discrete potential approach to the solution theory for cooperativeTU-games was introduced by Hart and Mas-Colell resulting in the character-ization of the known Shapley value as the unique efficient value that admitsa potential representation (prescribed by the discrete gradient of a so-calledpotential function). In 1997, Calvo and Santos established that the Shapleyvalue is the very best representative of the family of (not necessarily efficient)values that admit the (standard) potential representation. Our main goal isto survey the main recent results in the next two different extensions of thestandard potential approach:(1) In the framework of the so-called weighted pseudo-potential approach toefficient values, the revised representation of the value may incorporate, besidesa fraction of the discrete gradient, a fraction of the underlying pseudo-potentialfunction itself, as well as a fraction of the average of all the components of thegradient. Concerning the class of efficient values that admit a pseudo-potentialrepresentation, a number of appealing examples are put forward and moreover,a characterization of the full class of such values is stated (e.g., by linearity).(2) The standard efficiency principle refers to the addition (sum) of payoffs tothe single players, whereas the (standard) potential representation of a player’svalue in a game refers to the subtraction (difference) of two evaluations by thepotential function (at the initial game and an induced subgame). It is shownthat the equivalence theorem by Calvo and Santos can be revised whenever theefficiency principle and potential representation are revised in the sense thatthe addition (and subtraction) are replaced by multiplication (and quotient).In even stronger words, no change of the several equivalence theorems happensin case an arbitrary binary group operator on the set of real numbers (whichguarantees the existence of the inverse of any real number) replaces the addi-tive or multiplicative operator. Hence, in the setting of a group structure, acounterpart of the standard Shapley value arises as the very best representativeof the family of (not necessarily group-efficient) values that admit the potentialrepresentation (with reference to the underlying group structure).


Preferences, Rationality

126 Hold up and Intergenerational transmission of Preferences

Gonzalo Olcina and Concepción PeñarrubiaUniversidad de [email protected]

This paper focuses on the formation, evolution and stability of the distribu-tion of preferences in the population and its relationship with the investmentand bargaining strategies in a simplified hold up problem. More precisely, inour model a population of infinitely-lived players (say, for example, firms) withhomogeneous selfish or self-regarding preferences is pair-wise matched at eachperiod with a population of an equal size of short-lived players (say, for ex-ample, workers) with heterogeneous preferences. Both types of player play atwo-stage game. In the first stage, they decide separately but simultaneouslywhether to make a general or a relation-specific investment. The latter type ofinvestment is more efficient, that is, yields a higher surplus, but entails a higherindividual cost. Moreover, both of the current investments also determine thebargaining power of the partners in the second stage of the game, when theynegotiate the division of the surplus. If both players have made the same kindof investment they will have the same bargaining power, but if one has made ageneral investment while the other has made a specific investment, the formerhas all the bargaining power, although the surplus to be divided is smaller.Using this simplified game, we capture a stylized hold up situation. Playersare afraid of making costly specific investment because if their partners makea general investment, they run the risk of being exploited.

Preferences in the population of short-lived players are heterogeneous. Ineach period, there is a fraction of selfish players, but there is also a fractionof players motivated by reciprocal altruism. In particular, we use the con-cept of inequity aversion of Fehr and Schmidt (1999). More precisely, stronglyinequity averse players behave very differently from selfish players in a nego-tiation. Namely, they reject very unfair offers when they are responders andthey are generous when they are proposers. Any short-lived player lives fortwo periods in an overlapping generation situation. In the second period, as anadult, each one plays the investment game already described, but she also has adescendant and makes a costly decision on education effort, trying to transmither own preferences.

The reason for this behaviour is that they are altruistic towards their off-spring but in a particular form of altruism called imperfect empathy (see Bisinand Verdier (1998)). Namely, parents evaluate their child’s well-being throughthe filter of their own preferences. If this vertical transmission of preferences


does not succeed, then children acquire preferences from the social environ-ment (oblique transmission). The distribution of preferences in the populationof workers will, therefore, evolve over time, depending on the education effortof both types of parents which is determined itself by the actual distribution ofpreferences (since oblique transmission is a substitute for vertical transmission),and by their expectations about the firms’ investment and negotiation policy.But in turn, the firms’ policy will depend on the distribution of preferencesin the population of short-lived players, which is all the information the firmshave.

We find that in any stable steady state of this economy there is a mixedstationary distribution of preferences where both types are present in the pop-ulation of short-lived players. In general, the presence within the populationof a fraction of strongly inequity averse players alleviates the post-contractualopportunism and improves the efficiency in the optimal long-run investmentdecisions. The driving force for this improvement in efficiency is not any par-ticular “compulsion” or tendency of the inequity averse players to make spe-cific investments as compared to selfish players. In fact, it is the opposite insome cases: for instance, when there is a conflict between efficiency and risk-dominance in the investment game, inequity averse short-lived players makegeneral investments while selfish players ( long-lived and short-lived) makespecific investments. The driving force of the model is that the presence ofa significant fraction of inequity averse players in the population plays the roleof a kind of “social capital” in the economy. In other words, it works as a goodsubstitute for complete contracting, and this occurs basically because of theiraversion to advantageous inequality. Strongly inequity averse players are rathergenerous and fair when they are at the top (i.e., proposers in the negotiation)and this works as a “credible promise” that they will not abuse their partnersafter investments have been made.

Our framework allows us to introduce many important qualifications on thisgeneral result depending on the particular strategic structure of the investmentgame (coordination game, prisoner’s dilemma).


127 Rationalizable Foresight Dynamics: Evolution and Rationaliz-ability

Daisuke Oyama and Akihiko MatsuiUniversity of Tokyo

[email protected]

This paper considers a dynamic situation where rationality is common knowl-edge among infinitesimal agents, but beliefs may not be coordinated with eachother. A rationalizable foresight path is a feasible path of behavior patternalong which every agent takes a strategy that maximizes his expected dis-counted payoff against another path which is in turn a rationalizable foresightpath. A strategy distribution is accessible from another distribution underrationalizable foresight if there exists a rationalizable foresight path from thelatter to the former. A strategy distribution is said to be a stable state un-der rationalizable foresight if no rationalizable foresight path departs from thedistribution. A set of strategy distributions is said to be a stable set underrationalizable foresight if it is closed under accessibility. Stable sets underrationalizable foresight always exist. These concepts are compared with thecorresponding concepts under perfect foresight. Every stable state under ra-tionalizable foresight is shown to be stable under perfect foresight. But theconverse is not true. An example is provided to illustrate that stability un-der rationalizable foresight gives a sharper prediction than that under perfectforesight.Key words: Nash equilibrium; evolution; rationalizability; rationalizable fore-sight dynamics; perfect foresight dynamics; stability.


http://www.grad.e.u-tokyo.ac.jp/∼doyama/papers/rfd.html


128 Robust models for Nash equilibrium

Takashi MatsuhisaIbaraki National College of Technology, Japan

[email protected]

This paper investigates the class of the robust models for a mixed strategyNash equilibrium of a finite strategic form game G.

The concept of Nash equilibrium has become central in game theory, eco-nomics and its related fields. R. J. Aumann and A. Brandenburger (1995,Econometrica 63, 1161—1180) gives epistemic conditions for Nash equilibriumin the model for the modal logic S5. However it is still not yet clear just whatclasses of models leading to a Nash equilibrium in the epistemic point of view.

The purposes in this paper are two points: First to introduce the notionof robustness of the S4 logic models for a mixed strategy Nash equilibrium inthe game, and secondly to characterize the class R(G) of the robust models asthe class E(G) of all the S4 logic models satisfying with common-knowledge ofconjectures about the other players’ actions. Then we show:

Theorem 1. The class E(G) is a non-empty subclass of R(G). FurthermoreE(G) almost coincides with R(G).


129 Rational expectation can preclude trades

Takashi Matsuhisa, and Ryuichiro IshikawaIbaraki National College of Technology, Japan

[email protected]

In a pure exchange economy under uncertainty the traders are willing totrade of the amounts of state-contingent commodities and they know their ex-pectations. Common-knowledge about these conditions among all traders canpreclude trade if the initial endowments allocation is a rational expectationsequilibrium, even when the traders have the non-partition structure of informa-tion without the common prior assumption. In the proof it plays essential roleto extend the notion of a rational expectations equilibrium and to characterizeex-ante Pareto optimal endowments as the equilibrium. From the epistemicpoint of view it is emphasized that the partition structure of information forthe traders plays no roles in the no trade theorem.

In their paper Milgrom and Stokey [1982, Journal of Economic Theory 26,17—27] show the no trade theorem as follows:

Let us consider a pure exchange economy with traders in uncertain envi-ronment. Let Ω = Θ × X and the state of Ω consists of a pair (θ, x) where θranging over the contingencies on which commodities are defined.

The set Θ is interpreted as the set of payoff-relevant events; endowmentsand utility functions may depend on θ. The set X is interpreted as consist-ing of payoff-irrelevant events; these events do not affect endowments or tastedirectly. It is assumed here that the contingent commodities are ex-ante Pareto-optimally allocated, and the traders receive information about the state of Ωrepresentable by information partition, and it is assumed that the traders’ be-liefs are a common prior distribution; we call it the common prior assumption.Now, a trading process takes place where traders try to maximize their ex-pected utilities. We assume that in any equilibrium of this process traders’intended trades are both jointly feasible and common knowledge among them.In this set-up Milgrom and Stokey show that if traders are strictly risk-averse,equilibrium trade is null.

The serious limitations of the analysis in a pure exchange economy under un-certainty such as Milgrom and Stokey’s are its use of the information partitionstructure by which the traders receive information and of the common priorassumption. From the epistemic point of view the information partition struc-ture represents the trades’ knowledge: Precisely, the structure is equivalentto the standard model of knowledge that includes the ‘factivity’ of knowledgeT (what is known is true) and the ‘introspection’ properties Axioms 4 and5 that we know what we do and do not know). The postulate 5 is indeed


so strong that describe the hyper-rationality of traders, and thus it is partic-ularly objectionable. Also is the common-knowledge assumption because thecommon-knowledge operator is defined by an infinite recursion of the knowl-edge operators. The recent idea of ‘bounded rationality’ suggests dropping suchassumptions since real people are not complete reasoners. The common priorassumption also seems to be problematic.

This raises the question to what extent results as the no trade theorem de-pend on both common-knowledge and the information partition structure (orthe equivalent postulates of knowledge.) The answer is that results strengthenthe Milgrom and Stokey ’s theorem can be obtained in two ways: First, Tanaka(2000) investigates the theorem on the information partition by iterated elim-ination reasoning instead of common-knowledge. Secondly, in this paper wedrop the hypothesis that the initial endowments are ex-ante Pareto optimaland we extend the no trade theorem to the reflexive and transitive informationstructure without the traders being risk-aversion and having the common priorassumption. We show the results as follows: In a pure exchange economy underreflexive and transitive information structure, the traders are assumed to havetheir subjective priors not common and to have strictly monotone preferences.Then

Theorem 1. Any price system for which the initial endowments allocation is arational expectations equilibrium allocation can preclude trade if all the traderscommonly know that they are willing to trade of the amounts of state-contingentcommodities and if they know their expectations everywhere with respect to theprice.

To prove it we extend the notion of rational expectations equilibrium foreconomy under uncertainty to that of economy under reflexive and transitiveinformation structure, and we establish the the existence theorem for the equi-librium: The traders are further assumed to be strictly risk-averse.

Theorem 2. There exists a rational expectations equilibrium allocation relativeto a price with respect to which the traders know their expectations everywhere.

Moreover, we show a generalized version of fundamental theorem of welfareeconomics, a part of which plays essential role in proving Theorem 1:

Theorem 3. The initial endowments allocation is ex-ante Pareto optimal ifand only if it is a rational expectations equilibrium allocation relative to a pricewith respect to which the traders are rational everywhere about their expecta-tions.


Effectivity Functions, Partition Functions, . . .

130 Continuous representations of effectivity functions

Hans Keiding and Bezalel PelegUniversity of Copenhagen

[email protected]

An effectivity function specifies for each coalition a collection of subsets ofalternatives such that the coalition can force society’s choice to belong to thissubset. Effectivity functions have found widespread application in implementa-tion theory; much recent research is concerned with the so-called representationproblem of finding a game form having a prescribed effectivity function.

In the general setting where the set of alternatives is compact metric space,the problem of finding a continous representation (a representing game formwith continuous outcome function) is highly non-trivial. In the present paperwe give a new characterization of continuously representable effectivity func-tions using a property of continuous (not necessarily single valued) selectionsof certain operations on the subsets selected by the effectivity function.


131 Bilateral Coalescence and Superadditivity of Partition Func-tions

Ben McQuillinUniversity of East Anglia, U.K.

[email protected]

I consider transformations on the partition function of an underlying cooper-ative game that correspond with coalitional bargaining procedures entailing pe-riodic (sequential) opportunities for bilateral coalescence. I define two notionsof superadditivity (bilateral superadditivity and multilateral superadditivity)on games in partition function form. They equate for games in characteristicfunction form, but mulitilateral superadditivity is stronger for games in parti-tion function form. Both notions have the same intuitive appeal with respectto partition functions as superadditivity (conventionally defined) has for gamesin characteristic function form. But I show that the transformation of the par-tition function associated with any coalitional bargaining procedure entailingperiodic (sequential) opportunities for bilateral coalescence fails to preserve thebilateral superadditivity of all underlying partition functions. Furthermore, itlooks unlikely that we can find a useful sub-class of games in partition functionform such that a property I call “meta-superadditivity” (superadditivity of theunderlying partition function transformed by any t periods of coalitional bar-gaining) holds for some such coalitional bargaining procedure as the numberof players becomes large. This result suggests that there is a conflict betweensuperadditivity as an intuitively appealing characteristic of cooperative games,and sequential bilateral coalescence as an descriptively appealing procedure forcoalition formation. It also means that it is harder than would otherwise havebeen the case to corroborate cooperative game solutions that rest on a linear-ity axiom with non-cooperative game solutions. My paper has a close bearingon Gul’s (1989, 1999) non-cooperative foundation for the Shapley value, whichrests on meta-superadditivity (“value additivity” in Gul’s papers). It transpiresthat procedures, such Gul’s, that satisfy the null-player axiom in their solution,are less likely than others to satisfy the meta-superadditivity condition. Andthe supermodularity (or convexity) condition imposed by Gul (1999) on theunderlying game (so that value additivity equates to efficiency) reduces thelikelihood of meta-superadditivity further.

References

Gul, F. (1989) Bargaining Foundations of Shapley Value, Econometrica 57,81—95.

Gul, F. (1999) Efficiency and Immediate Agreement: A Reply to Hart andLevy, Econometrica 67, 913—917.


132 Aggregation and Disaggregation of European Players in theIMF Compound Games

Carlos BowlesEuropean University Institute, Italy

[email protected]

Should the United States of Europe represent European Nations at the IMF?An aggregation/Disaggregation Approach

We address the question of the desirability of the merger of the representa-tives of European States at the International Monetary Fund using the classicalconcept of value from cooperative game theory. We decompose the whole gameinto a main quotient game (the Executive Directors’ Game) and different sub-games (the European and Constituencies ones). The classical composition ofgames being here inadequate, we axiomatically define a new product opera-tor on the space of simple games. Under this new product, it is shown thatthe value of the product of the games is equal to the product of the values ofthe subgames and the quotient game. Using the probabilistic representation ofvalues, we provide a formal interpretation of this product as an independenceassumption with respect to the probability of coalition formations between thedifferent games that identifies our restriction as an orthogonality condition.This property leads to the nice feature that it becomes possible to aggregateand disaggregate easily the value of the whole game, depending on the kind ofplayer, European Union or European States, we want to consider.

On this ground, we compute the Normalized Banzhaf and the Shapley-Shubik Power Indices for the quotient game and the different subgames toanalyze the impact of a merger of the representatives on both the EuropeanUnion considered as a single player and the European States considered as dif-ferent ones. We also compute an approximation of the indices for the Board ofGovernors large game using Owen’s multilinear extension of the game. A com-parison of the approximates with the exact normalized Banzhaf Index confirmsthat this technique performs quite badly whenever the asymmetry of weightsviolates the conditions under which the Central Limit Theorem holds. We con-jecture a procedure that might easily improve the quality of the estimates inthis case.

Our results indicate that the impact of the merger depends on the importanceof the decision made in the Executive Directors Board. For decisions requiringthe 85% Union and increases symmetrically the power of United States. Forordinary decisions requiring the 50% of European Union is strongly increased.At the state level, the impact of the merger is always negative for the countrieswhose Executive Director is appointed (Germany, France, United Kingdom) as


well as for those who are quite strong in their constituencies (The Netherlands,Belgium and Sweden). On the converse, the merger exerts a leveradge effecton the rights of vote of the countries with little power in their relevant Con-stituencies (Portugal, Ireland, Greece), due to the partial adoption of Europeandecision rules under which they are given more weight.

Both the normalized Banzhaf and the Shapley-Shubik indices support theseresults. The present analysis nevertheless confirms their commonly known con-trasted properties due to the differences in their probabilistic assumptions andasymptotic behaviour. While the normalized Banzhaf Index is more sensitive toany variation in the parameter of the game, the Shapley-Shubik Index exhibitsmore stability.


133 From Cases to Rules

Itzhak Gilboa, Enriqueta Aragones, Andrew Postlewaite and David Schmeidler

Tel-Aviv University and Yale [email protected]

Inductive reasoning aims at finding general rules that hold true in the database.Such rules may state that a certain phenomenon never happens, that certainconditions necessitate another, or that one variable is a function of others. Weshow that several problems associated with these processes are computationalhard. In particular, in the context of linear regression, finding a small set ofvariables that obtain a certain value of R2 is computationally hard.

Computational complexity may explain why a person is not always awareof rules that, if asked, she would find valid. This may explain why one canoften change other people’s minds (opinions, beliefs) without providing themnew information.


Auctions II, Equilibrium

134 Auctions Versus Negotiations Revisited

René KirkegaardUniversity of Aarhus, Denmark

[email protected]

When analyzing the classical monopoly problem, it is natural to start by as-suming that demand is continuous and marginal revenue curves are monotonic.The same assumption, which we will refer to as the regularity assumption, isusually imposed in auction theory. In this paper we will consider departuresfrom this assumption and discuss the robustness of existing results in auctiontheory. We also draw parallels between auction theory and bargaining theorywith incomplete information and a number of buyers. In particular, auctiontheory can be used to explain why certain bargaining procedures are observed.

Our focus is primarily on an issue raised by Bulow and Klemperer (1996).Assuming symmetric buyers with regular demand, they compare standard auc-tions with negotiation. Their main result is that optimally negotiating with nbuyers is revenue inferior to using a standard auction with n+ 1 bidders. For-mally, negotiation is modelled as the optimal mechanism with n buyers, whilethe auction is a standard (suboptimal) auction, such as the English auction,with n + 1 buyers. Hence, “it will often be more worthwhile for a seller todevote resources to expanding the market than to collecting the informationand making the calculations required to figure out the best mechanism”.

We provide two arguments for why we should be careful with the conclusionreached when demand is regular. The first argument is formal, the secondless so. First, the result is not robust to changes in the assumption about thenumber of types. We show that when valuations take one of two values, itmay or may not be optimal to sacrifice negotiation power for more consumers.The second argument is founded on an observation of how to implement theoptimal mechanism. Whether or not demand is regular, revenue maximizationis consistent with a selling procedure that will seem very much like sequentialnegotiation to an outside observer, where the seller is seen to move on to anotherbuyer if the current negotiation stalls. Indeed, this outsider could observe thatthe good is sold before every known buyer have had a chance to bid for it.Based on the result that the seller should try to seek out as many customers aspossible when demand is regular, it would then be tempting to falsely concludethat the seller has failed to maximize his expected revenue.

De Fraja and Muthoo (2000) consider a situation in which a seller bargainswith a finite number of buyers. The bargaining procedure is modelled as a


game with infinite horizon. If the object is unsold at the beginning of a period,the seller selects an agent to approach, and a price to suggest to the chosenagent. Hence, equilibrium can involve switching from one bidder to another.De Fraja and Muthoo observe that this selling procedure is commonly used.In this paper we show why this procedure might be used, i.e. why a sellerwould want to use this procedure or mechanism. We assume as an extremescenario that agents care about the price reached through bargaining but donot care about when an agreement is reached. Then, auction theory tells uswhich conditions a trading mechanism must satisfy in order to be optimal fromthe point of view of the seller. As we have argued, the optimal revenue can beobtained through sequential negotiation similar to the process studied by DeFraja and Muthoo. They find that as the discount factor approaches one, thehighest possible equilibrium revenue of the bargaining game converges. It iseasy to verify using the results in this paper that the point of convergence isthe optimal revenue.


135 An Optimal Auction with Externalities

Hector Chade and Jorge AseffArizona State University, U.S.A.

[email protected]

This paper analyzes the problem of a seller who wants to design an optimalmechanism to sell multiple units of a good to a set of potential buyers whohave single-unit demands. A buyer’s valuation for the good is determined bytwo factors: (i) a taste parameter that is private information; (ii) a positveexternality parameter determined by the identities of the other buyers whoalso obtain a unit of the good.

We characterize the optimal (profit maximizing) selling mechanism in thepresence of private information and identity-dependent externalities, whichclearly illustrates how the existence of external effects alters the allocationand payment rules with respect to the case with just private information. Weprovide several examples that shed light on the main properties of the optimalmechanism; in particular, we show that it leads sometimes to an ex-post inef-ficient allocation, and also that a buyer’s payment for a unit of the good is adecreasing function of the positive externalities he generates.

As an application of the model, we consider the case of the owner of a shop-ping center who wants to allocate retail space to a set of potential firms; thevalue for a firm of locating there depends on the size of the inter-store exter-nalities, which determines the flow of customers. We show that, when inter-store externalities are sufficiently large, then the following sequential mecha-nism which is commonly used in practice is optimal: first, allocate retail spaceto the firms that generate the largest external effects at a deep discount; sec-ond, auction the remaining space among the rest of the firms. However, forintermediate values of inter-store externalities, we show the sequential mecha-nism allocates space to the firms that generate the largest external effects toooften compared to the optimal mechanism.


136 Necessary and sufficient condition for two-player subgame per-fect implementation

Hannu VartiainenJahnsson Foundation, Finland

[email protected]

Implementation theory concerns the problem of designing mechanisms thatdecentralize decision power to the members of a society. Decentralization isnecessary especially if information is dispersed among players. Moore-Repullo(1988) and Abreu-Sen (1990) study implementation of social choice correspon-dences (SCCs) by using subgame perfect Nash equilibrium (SPE) as the solu-tion concept. They find that via an extensive form mechanism one is able toimplement non-monotonic SCCs (see also Herrero and Srivastava 1992). SinceMaskin (1977), monotonicity has been known to be a necessary condition forNash implementation. Moore-Repullo and Abreu-Sen construct a necessarycondition for SPE implementation which, together with no veto power (NVP),forms a sufficient condition for implementation in three or more players caser.The gap between necessary and sufficient conditions in the context of three ormore players is closed by Vartiainen (1999).

Even if the introduced necessary conditions for SPE implementation are validalso in the two-player case, on sufficient side the characterization problem ismuch more complex. In fact, only special results concerning SPE implementablechoice rules are available. E.g. Moore-Repullo introduce domain restrictionswhich together with particular restrictions on SCCs allow SPE implementationin the two player scenario, their treatment is far from complete characteriza-tion: many interesting SCCs not reached by Moore-Repullo characterizationcan be implemented. Herrero and Srivastava (1992) obtain a characterizationfor implementation via backwards induction, a solution which does not permitsimultaneous moves.

In the context of Nash implementation, Moore-Repullo (1990) and Dutta-Sen (1990) provide a complete description of Nash implementable SCCs. Inthis paper, we seek to solve the corresponding problem in the context of two-player SPE implementation, by providing a complete characterization of imple-mentable SCCs.

Our problem is challenging since conditions for SPE implementation arevery weak, even forceless, in the two-player case. In particular, in the two-player case the NVP condition is unacceptably strong, and Condition (a) ofAbreu-Sen (1990) has much less characterization power. To gain some intu-ition on why the two-player case is different, consider a truthful equilibriumin an information revelation game where players announce the state. In the


three-player case, if any player unilaterally deviates from the truthful equilib-rium, it is easy to recognize who that player is and punish him if necessary.However, if there are only two players, then for uninformed planner it is im-possible to see who is deviating and who is playing the truthful equilibrium.Thus, a deviator cannot be punished. In the context of Nash implementa-tion, to characterize implementable SCCs one “only” needs to make sure thatit is never in a single player’s interest to induce the implementation of a dis-agreement outcome. However, in the context of SPE implementation, a similarproblem may manifest in any single stage of the game: a characterization mustcertify that no player gains by inducing the implementation of a disagreementsubgame in any potential stage of a game. To do that, one cannot any longerrestrict the characterization to consist of identification of sequences of outcomesand players having particular properties in each of their steps, as was done inthe three-or-more players context by Moore-Repullo (1988), Abreu-Sen (1990)and Vartiainen (1999). Rather a characterization must consist of identificationof trees assigning an outcome and a player to each of its nodes. The succes-sive nodes are then requiresd to have certain additional properties. Roughly,the complete characterization of SPE implementable SCCs contains three ele-ments: (i) a condition which ensures that a tree with desired properties existsfor all pairs of states, (ii) a restricted veto power condition which ensures thata top-ranked outcome is selected by a choice rule if all above descibed treescontain certain properties, and (iii) a unanimity condition which ensures thata commonly top-ranked outcome is always selected by a choice rule.

In practice, identifiying trees which have the desired properties for each pairsof potential states is of course a very complicated task. Luckily, however, wecan show that certain plausible domain properties (e.g. strict preferences or anexistence of a “bad” outcome) simplify the characterization remarkably.

As an application, we focus on SPE implementation of bargaining solutions.It is well known that no interesting bargaining solutions can be implementedin Nash equilibrium. However, in the context of SPE implementation we showthat any strictly individually rational two-player bargaining solution can beimplemented if and only it satisfies scale invariance. Thus, the result stronglyfavours the use of multi-stage games, such as the alternating offer procedure,in the implementation of bargaining solutions (for results on implementation ofparticular solutions, see e.g. Moulin 1984, Howard 1992). In a sense, this resultanswers to the question analysed in the famous Nash-programme: as such,strategic considerations do not impose any constraint on available bargainingsolutions. Thus, the relevant question is what whether the mechanism, whichimplements the solution, is plausible.


References

Abreu, D. and Sen, A. (1990) Subgame Perfect Implementation: A Necessaryand Almost Sufficient Condition, Journal of Economic Theory 50, 285—299.

Dutta, B. and Sen, A. (1991) A Necessary and Sufficient Condition for Two-Person Nash Implementation, Review of Economic Studies 58, 121—128.

Herrero, M. and Srivastava, S. (1992) Implementation via Backward Induc-tion, Journal of Economic Theory 56, 70-88.

Howard, J. (1992) A Social Choice Rule and its Implementation in PerfectEquilibrium, Journal of Economic Theory 56, 142—159.

Maskin, E. (1999) Nash Equilibrium and Welfare Optimality, Review of Eco-nomic Studies 66, 23—38 (working paper published 1977).

Moore, J. and Repullo, R. (1990) Nash Implementation: A Full Characteriza-tion, Econometrica 58, 1083—1099.

Moore, J. and Repullo, R. (1988) Subgame Perfect Implementation, Econo-metrica 56, 1191-1220.

Moulin, H. (1984) Implementing the Kalai-Smorodinsky Bargaining Solution,Journal of Economic Theory 33, 32—45.

Vartiainen, H. (1999) Subgame Perfect Implementation, the third chapter of aPhD thesis Studies in Mechanism Design and Game Theory, Departmentof Economics Research reports No 83, University of Helsinki.


137 An Anti Folk Theorem? Existence and Evolution of PropertyRights as a Unique Equilibrium

Ricardo NievaUniversity of Minnesota and SUNY Binghamton

[email protected]

The institution of property rights is important for efficiency in market econo-mies. Douglas North and most economists claim that lack of institutions mightbe behind underdevelopment. Hence instead of taking private property rights asexogenous it is necessary to have first a foundational analysis of this institution.In previous literature private property rights can be one of multilple equilibriaoutcomes of infinitely repeated games with conflict (battle of the sexes likegame). In our set up we add to this class of games enforcement technologicalshocks.

After generating naturally a new payoff matrix we use the Folk theoremto conclude that for this “patological” case there is a unique subgame perfectequilibrium outcome of subgames that are reached contingent on this shocks.Loosely speaking the player who gets the technology of enforcement shock willplay the strategy game form “property right” in equilibrium. If shocks happen2 times we get the possibility of players mantaining or changing property rightregimes in equilibrium. We will identify the latter contingence with institutionalchange.


TU Games II

138 An Adaptive Model of Demand Adjustment in CooperativeGames

Alex Possajennikov and María MonteroUniversity of Mannheim, [email protected]

The paper presents a simple best reply adaptive model of demand adjustmentin transferable utility cooperative games. The process is designed to modelcompetition among players. Players set demands for their participation in acoalition. The demands of “scarce” players will be driven up by competition,and the demands of “abundant” players will be driven down. The processoperates in the following way. A player is chosen randomly and looks for acoalition that leaves the highest surplus provided that the demands of otherplayers in the coalition are satisfied. The player then proposes this coalitionand adjusts the demand to coincide with the surplus left. This process hasbeen shown to have aspiration vectors as stationary states, and to converge tothe set of aspirations.

We introduce a particular type of mutations in this process. With small prob-ability, each player changes coalitions only if offered a strictly larger amountthan he is currently getting. This can be interpreted as small transaction costsof changing a coalition, a small friction. If the current demand vector is anaspiration but not a partnered one, then a demand of a strictly larger amountby a “scarce” player leads to a strictly lower demand by another player, whichamounts to a transfer from one player to another. The vector of demands re-sulting from the transfer may not be an aspiration but the base process willthen lead to an aspiration vector.

We show that a sequence of transfers, together with subsequent adjustment,leads to a partnered aspiration. Considering the base process of adjustmenttogether with rare mutations, we show that in a discretized model with suffi-ciently small grid size the stochastically stable states of the process lie in theset of partnered aspirations. For simple majority games it implies that onlyminimal winning coalitions form, and in the apex game the payoff division isproportional to the number of votes. Further selection inside the set of part-nered aspiration can also be considered, ultimately leading to (the interior of)the aspiration core, or the set of balanced aspirations.


139 On Games corresponding to a new class of Parallel SequencingSituations

Pedro Calleja, Peter Borm, Herbert Hamers, Flip Klijn and Marco SlikkerUniversitat de Barcelona

[email protected]

Sequencing or scheduling situations consist of a number of jobs (tasks, op-erations) that have to be processed on a number of machines. The processingtime of a job on a specific machine is the time this machine takes to handlethis job. Sequencing situations can be classified by the number of machines,the specific properties of the machines (e.g. parallel, identical), the chosen re-strictions on the jobs (e.g. ready times, due dates, flow constraints), and thechosen cost criterion (e.g. weighted completion time, makespan). By assumingthat there exists an initial schedule before the machines start processing, onecan establish a relation between cooperative games and sequencing situationsin the following way. Under the assumption that each job is owned by oneagent, a group of agents (a coalition) can save costs by rearranging their jobsin a way that is admissible with respect to this initial schedule. By definingthe value of a coalition as the maximal cost savings a coalition can make inthis way, we obtain a cooperative sequencing game related to a sequencing sit-uation. The above game-theoretic approach was initiated by Curiel, Pederzoliand Tijs (1989) and is used to analyse several classes of sequencing situations.

The specific scheduling problems we consider in this paper are those arisingfrom situations where some complementary jobs, in the sense they can notbe processed on the same machine, need to be finished in order to obtain thefinal output. This paper considers then sequencing situations with 2 differentparallel machines in which no restrictions on the jobs are imposed. Contraryto other papers in this field, it is assumed that each agent owns 2 jobs to beprocessed, one on each machine. The costs of an agent depends linearly onthe final completion time of all his jobs. In other words, it depends on thetime an agent has to wait until both his jobs have been processed. First, wedescribe a rearrangement procedure that provides an optimal processing orderof the jobs of the agents for some particular situations. Secondly, we introducea class of cooperative TU games that arises from these sequencing situations.These games are investigated with respect to the properties of balancednessand convexity. We derive conditions such that the core of the games is non-empty. Finally, some results are extended to parallel sequencing situations inwhich more than two machines are involved.


140 Games with homogeneous groups and convexity

Javier Martínez de Albéniz and Carles RafelsUniversitat de Barcelona and CREB

[email protected]

Standard cooperative game theory deals with players, assuming that eachplayer has its own characteristics. Nevertheless, in most cases, players can andshould be gathered in groups. We present a model in which players belongto groups and the characteristic function captures this reality. Therefore thecharacteristic function is defined over a lattice.

Several concepts, such as the core or the Weber set are presented, and alsothe corresponding Shapley value. Convexity is also studied, wich leads to aspecial class of functions.


141 New rules of amortization

J. Aparicio, T. S. H. Driessen, J. Sánchez-Soriano, M. Pulido, N. Llorca andE. Molina

Universidad Miguel Herná[email protected]

One of the most important problems in accounting is how to determine themethods or rules to carry out the allocation of the depreciation costs of amachine in the annual accounting. At present several empirical rules are used,but they do not cash, in our opinion, the particular features of the problem. Inthis paper, we tackle this problem from a Game Theoretical point of view. Wetry to use the most known solutions concepts in Game Theory to determine newdepreciation rules. For our purpose, we define a game in a suitable way. Thisgame allows us to apply Game Theory. The definition of the characteristicfunction of the game can have different forms, therefore we obtain differentdepreciation rules for each characteristic function. First, by using a particularcharacteristic function for the game, we study under what conditions the usualrules in accounting are stable, in the sense of belonging or not to the core of thegame. Secondly, by using the same characteristic function, we analyze whichkind of rules arise when we use the Shapley value, the nucleolus or the valueof Tijs. The obtained results have simple and reasonable interpretation. Aparticular feature of this kind of game is that these games are a particular caseof airport-games. For this reason it is possible to obtain simple expressionsfor the mentioned values. Lastly, we study alternative models to define thecharacteristic function of the game and analyze the relations with the previousresults.

References

Trigueros, J. A., Sansalvador Sellés, M. E., Reig Mullor, J., González Car-bonell, J. F. and Cavero, J. A. (2001) Fundamentos y práctica de con-tabilidad, Editorial Pirámide.

Littlechild, S. C. (1974) A simple expression for the nucleolus in a special case,International Journal of game Theory 3, 21—29.

Littlechild, S. C. and Owen, G. (1976) A further note on the nucleolus of theairpor game, International Journal of Game Theory 5, 91—95.

Littlechild, S. C. and Thompson, G. F. (1977) Aircraft landing feees: A gametheory approach, Bell Journal of Economics 8, 186—204.

Sprumont, Y. (1990) Population monotonic allocation schemes for cooperativegames with transferable utility, Games and Economic Behaviour 2, 378—394.

AUTHOR INDEX

Aarnisalo, Pekka, 176Albizuri, M. Josune, 123, 124Alcalde, José, 27Algaba, Encarnación, 150, 153Amer, Rafael, 168, 170Aparicio, Juan, 227Aragones, Enriqueta, 107, 216Aramendia, Miguel, 136Arce, Daniel G., 61Arribas, Ivan, 193Aseff, Ben, 219

Barberà, Salvador, 78Berga, Dolors, 85Bergantiños, Gustavo, 45, 72Bertini, Cesarino, 33Bilbao, Jesús Mario, 37, 54, 150,

153Bischi, Gian Italo, 58Board, Oliver, 128Borm, Peter, 131, 132, 175, 225Bossert, Walter, 51Bowles, Carlos, 214Bramoullé, Yann, 46Brams, Steven J., 7Branzei, Rodica, 99Burani, Nadia, 113

Calabuig, Vicente, 195Calleja, Pedro, 225Calvo, Emilio, 202Camiña, Ester, 111Carpente, Luisa, 131Carreras, Francesc, 120, 169Casas, Balbina, 32, 131Catilina, Eliane, 109Chacón, Casi, 54Chade, Hector, 189, 219Chang, Chih, 73Chen, Peng-An, 73

Conrado, Manuel, 186Cunyat, Antoni, 195

Dam, Kaniska, 67de Clippel, Geoffroy, 74, 190De Francesco, Massimo A., 191de Otto, Beatriz, 134Derks, Jean, 51Dimitri, Nicola, 125, 145Driessen, Theodorus S. H., 98, 205,

227

Echenique, Federico, 135Edlin, Aaron, 135Einy, Ezra, 108

Ferdinandova, Ivanna, 21Fernández, Cristina, 187, 188Fernández, Francisco R., 101, 116,

187, 188Fernández, Julio R., 37Ferreira, Manuel Alberto M., 130Fibich, Gadi, 77Fiestras-Janeiro, Gloria, 116Fragnelli, Vito, 105, 117, 155Freixas, Josep, 118, 121Frutos, M. Angeles de, 75

Gómez, Daniel, 186Gambarelli, Gianfranco, 33García-Jurado, Ignacio, 32, 116, 175Gavious, Arieh, 77Germano, Fabrizio, 180Gilboa, Itzhak, 216Giménez, José Miguel, 168, 170González, Enrique, 186González, Paula, 25Gossner, Olivier, 137Gottardi, Piero, 66Goyal, Sanjeev, 46

229


Gutiérrez, Esther, 202

Haimanko, Ori, 199Hamers, Herbert, 156, 225Hendrickx, Ruud, 131, 132Hernandez, Penelope, 137Herrero, Carmen, 25, 44Hinojosa, Miguel A., 159, 160, 188Hochman, Oded, 82Hougaard, Jens L., 80

Iñarra, Elena, 19Ishikawa, Ryuichiro, 210Izquierdo, Josep M., 52

Jarque, Xavier, 75, 96Jiménez, Nieves, 150Jiménez-Losada, Andrés, 54, 153Jones, Michael A., 7

Keiding, Hans, 80, 212Kilgour, D. Marc, 7Kirkegaard, René, 217Kjaer, Christian, 22Klijn, Flip, 156, 158, 225Kopel, M., 58Kuipers, Jeroen, 19Kultti, Klaus, 176, 178, 179

López, Jorge J., 37, 150López-Pintado, Dunia, 46Lahiri, Somdeb, 83Laruelle, Annick, 88, 148Le Breton, Michel, 199Lebrón, Esperanza, 54Levinsky, Rene, 197Llerena, Francesc, 52Llongueras, M. Dolors, 120Llorca, Natividad, 132, 227

Mármol, Amparo M., 159—161Macho-Stadler, Ines, 49Magaña, Antonio, 169

Manzini, Paola, 90, 141Marina, Maria Erminia, 117Mariotti, Marco, 141Martínez de Albéniz, Javier, 226Massó, Jordi, 78Matsuhisa, Takashi, 209, 210Matsui, Akihiko, 208McQuillin, Ben, 213Meca, Ana, 173, 175Meinhardt, Holger, 98Meléndez, Miguel Ángel, 28Miettunen, Antti, 178Minelli, Enrico, 190Moldovanu, Benny, 76Molina, Elisenda, 114, 227Monella, Giovanni, 105Monroy, Luisa, 159, 161Montero, María, 203, 224Morales, Antonio J., 133Moreno, Bernardo, 87Moreno, Diego, 108Moreno-Ternero, Juan D., 42, 44Morgan, Jacqueline, 13Mutuswami, Suresh, 48

Núñez, Marina, 171Napel, Stefan, 40Neme, Alejandro, 78Neyman, Abraham, 137, 138Nieva, Ricardo, 223

Okada, Daijiro, 138Olaizola, Norma, 19Olcina, Gonzalo, 195, 206Ortona, Guido, 105Osterdal, Lars P., 80Owen, Guillermo, 10, 186, 200Oyama, Daisuke, 208

Pérez-Castrillo, David, 48, 49, 67Palacios-Huerta, Ignacio, 164Palfrey, Thomas, 107Peñarrubia, Concepción, 206


Peixoto, Cristina, 130Peleg, Bezalel, 212Perea, Andrés, 143Perote-Peña, Juan, 140Peters, Hans, 51Ponsati, Clara, 90, 96Ponti, Giovanni, 44Popp, Alexandru W., 126Porteiro, Nicolás , 49, 69Possajennikov, Alex, 224Postlewaite, Andrew, 216Pozo, Mónica del, 186Puente, María Albina, 118Puerto, Justo, 101, 116, 188Pulido, Manuel, 132, 227Puy, Socorro, 87

Rafels, Carles, 31, 52, 171, 226Revilla, Pablo, 112Riipinen, Toni, 179Rodríguez-Álvarez, Carmelo, 139Rojano, Abraham, 162, 167Rubiales, Victoriana, 161

Sánchez, Estela, 158Sánchez-Pagés, Santiago, 93Sánchez-Soriano, Joaquín, 132, 227Saboyá, Martha, 200Sakovics, Jozsef, 96Salazar, Raquel, 162, 167Salonen, Hannu, 147Sandler, Todd, 61Santos, Juan Carlos, 124, 202Schmeidler, David, 216Scotti, F., 99Sela, Aner, 76, 77Serizawa, Shigehiro, 85Serrano, Roberto, 66Shitovitz, Benyamin, 108Silva, José A., 27Skorin-Kapov, Darko, 184Skorin-Kapov, Jadranka, 184Slikker, Marco, 225

Stach-Janas, Izabella, 33Szidarovszky, M., 58

Takalo, Tuomas, 178Tarragó, Anaís, 102Tejada, Juan, 114, 186Thille, Henry, 110Tijs, Stef, 3, 99Toledo, Andrés, 173Torre, A., 99

Urbano, Amparo, 182, 183, 193Uriarte, José Ramón, 62

Vázquez, Margarita, 32Valenciano, Federico, 88, 148van den Brink, Rene, 153van den Nouweland, Anne, 32van Velzen, Bas, 156Vartiainen, Hannu, 220Vega-Redondo, Fernando, 46Vidal, Juan José, 45, 72Vilá, Xavier, 21Vila, José, 182, 183Vilella, Cori, 31Villar, Antonio, 42Virrankoski, Juha, 178

Weber, Shlomo, 199Wettstein, David, 48Widgren, Mika, 40Wiese, Harald, 30

Zafra, María José, 187Zarzuelo, José Manuel, 123, 124Zeng, Dao-Zhi, 194Zwicker, William S., 121