Multilateral Bargaining Problems

Ž .GAMES AND ECONOMIC BEHAVIOR 19, 151]179 1997ARTICLE NO. GA970547

Multilateral Bargaining Problems

Elaine BennettU†

Virginia Polytechnic Institute and State Uni ersity, Blacksburg, Virginia 24061; andUni ersity of California, Los Angeles, Los Angeles, California 90024

Received February 15, 1994

In many situations in economics and political science there are gains fromforming coalitions but conflict over which coalition to form and how to distributethe gains. This paper presents an approach to such multilateral bargaining problems.A solution to a multilateral bargaining problem specifies an agreement for eachcoalition that is consistent with the bargaining process in every coalition. Weestablish the existence of such solutions, show that they are determined byreservation prices, and characterize these reservation prices as the payoffs ofsubgame perfect equilibrium outcomes of a non-cooperative bargaining model. Jour-nal of Economic Literature Classification Numbers: C71, C72, C78. Q 1997 Aca-

demic Press

1. INTRODUCTION

In many situations in economics and political science, there are gainsŽfrom forming various coalitions but conflicts about which coalition or

.coalitions should be formed and how the gains should be shared. Exam-ples abound: the formation of a government by political parties in aparliamentary system, trade in an exchange economy, the formation ofjurisdictions, and the production of public goods in a local public goodseconomy, etc. In such situations, the distribution of gains within eachcoalition involves bargaining within the coalition. If each agent can actu-ally participate in at most one coalition, participation in a particularcoalition entails an opportunity cost: the foregone rewards the agent could

U I thank Jan Aaftink, Kenneth Binmore, Randy Calvert, Chew Soo Hong, John Ferejohn,Joseph Greenberg, Bruce Hamilton, John Harsanyi, Martin Hellwig, Michael Maschler, AlvinRoth, Dov Samet, Lloyd Shapley, Richard Small, William Thomson, Myrna Wooders, andespecially William Zame for their comments on various drafts. Financial support from theNational Science Foundation is gratefully acknowledged.

† Elaine Bennett died May 26, 1995, while this paper was under review; final revisions werecompleted by William Zame.

1510899-8256r97 $25.00

Copyright Q 1997 by Academic PressAll rights of reproduction in any form reserved.

ELAINE BENNETT152

have received from participating in another coalition. The bargainingproblem of each coalition is therefore interrelated. We call such aninterrelated set of bargaining problems a multilateral bargaining problem.

The simplest bargaining problem involves only a single coalition. NashŽ .1951 formalized this situation by specifying a set of players, a set of

Žattainable utilities, and a disagreement point the vector of utilities of the.outcome that will result if the agents cannot come to agreement . In the

situation of interest to us, however, if agents in a given coalition cannotcome to agreement in that coalition, they will form other coalitions. Thedisagreement vector therefore must be determined endogenously, in termsof the opportunity costs of not entering into other coalitions.

Our focus here is on the interrelationships between bargaining problemsof the several coalitions, rather than on bargaining within individualcoalitions.1 We shall therefore take as the ingredients of a multilateral

Ž .bargaining problem a set of agents players and, for each coalition ofŽplayers, a set of attainable utility vectors representing a description of the

.potential gains that can be realized by the coalition , and a bargainingŽfunction a mapping from opportunity costs to agreement vectors for the

coalition, representing a summary of the bargaining process within the.coalition . We view the attainable sets and the bargaining functions as

parts of the description of a multilateral bargaining problem, in much thesame way that endowments and utility functions are part of the descriptionof an exchange economy.

To see the way in which the endogenous determination of outsideoptions affect the result, consider four professionals}a biologist,economist, lawyer, and physicist}who each have full-time jobs and alsohave opportunities for outside consulting. The economist and the lawyertogether can represent the electric company at rate hearings for $40,000per year. The economist and physicist together can consult for the waterdistrict for $34,000 a year. The lawyer and the biologist together canconsult on toxic cleanup suits for $20,000 per year. Finally, the biologistand physicist together can write environmental impact statements for$34,000 per year. If these are the only available consulting opportunities,which partner should the economist choose and what should the termsbe?2

How should we analyze problems such as these? One way is to treat thebargaining problem for each pair in isolation, and then integrate theresults. Assume for definitiveness that, gi en an outside option vector,each pair solves its bargaining problem by equal division from the outside

1Determining which bargaining function is appropriate is the domain of traditional bar-Ž . Ž . Ž .gaining theory; see for example, Roth 1979 , Kalai 1985 , and Sutton 1985 .

2 For simplicity we assume utility is linear in money and that only money matters.

MULTILATERAL BARGAINING PROBLEMS 153

option.3 Considering the economistrlawyer partnership in isolation sug-Ž .gests the disagreement point 0, 0 and hence an equal division of

Ž .$20,000, $20,000 }conditional on the economist and lawyer forming apartnership. Similarly, the other coalitions would also divide their incomeequally. In anticipation of these equal divisions, the economist would wishto form a partnership with the lawyer, each of them obtaining $20,000; thiswould leave the biologist and the engineer to form a partnership, eachobtaining $10,000. However, these partnerships are unstable: the economistand the physicist could form a partnership which would yield the economist$22,000 and the physicist $12,000. The source of this instability is plain tosee: the economist and the lawyer each have profitable outside opportuni-

Ž .ties the economist could form a partnership with the physicist, etc. ,and the disagreement point in their partnership should reflect theseopportunities.

It is this endogenous determination of outside options as opportunitycosts that separates a multilateral bargaining problem such as this from adisconnected set of simple bargaining problems.4 In this paper, we takeplayer i’s outside option in coalition S to be the maximum utility he couldobtain by forming an alternative coalition. By constructing outside optionsin the way we do, we leave open the possibility that the outside options ofmembers of a given coalition may not be jointly compatible. For instance,for a coalition consisting of the players 1, 2, the outside options of eachplayer may involve partnership with player 3. Since both players cannotsimultaneously take up their outside options, our outside option vectordoes not have the interpretation of the result of disagreement in the

w x Žcoalition 1, 2 . And it is for this reason that we prefer the term ‘‘outside.option’’ rather than ‘‘disagreement point.’’ Section 2 contains a detailed

discussion of this point, including other possible choice of outside options.It is important to keep in mind that the solution to the bargaining

problem facing coalition S depends on the outside options of each of themembers of S, which depend in turn on the agreements in other coalitions,which depend in turn on the outside options in these other coalitions,which depend in turn on the agreement in the coalition S. Thus, thisendogenous determination of outside options links the bargaining prob-lems for all coalitions.

A solution to a multilateral bargaining problem specifies an agreementŽutility for eëry potential coalition in a way that is consistent with the

3This is consistent with the Nash bargaining solution and most other proposed solutions tothe simple bargaining problem.

4 Nash uses the term ‘‘disagreement point’’ rather than ‘‘outside option;’’ we prefer thelatter term since our outside options should not be viewed as the outcome that will resultshould the bargaining break down in a particular coalition. See the discussion in Section 2.

ELAINE BENNETT154

. 5bargaining within each coalition and across coalitions, via outside options .In general, some of these agreement utilities will be feasible for thespecified coalition and some may not be. We interpret a feasible agree-ment as a prediction of the outcome of bargaining within that coalition,conditional on the assumption that the coalition forms, and an infeasibleagreement as a prediction that the coalition will not form. We do notrequire that the agreement for the grand coalition be feasible, nor thatthere be a partition of the player set into coalitions whose agreements arefeasible. An outcome consists of a partition P of the player set and aspecification of a feasible agreement z S for each coalition S g P; ofcourse, outcomes should be the end objects of any bargaining theory. We

Ž � S 4.say that the outcome P, z : S g P is consistent with the multilateral� S4 S Ssolution x if z s x for every coalition S g P which is not a singleton,

S � 4and z s 0 for singleton coalitions in P.It may be helpful to view a solution to a multilateral bargaining problem

Ž .as a set of beliefs conjectures players hold about the eventual outcomesof bargaining within and across coalitions. A set of beliefs constitutes a

Žsolution if these beliefs are consistent i.e., all players hold the same. Žbeliefs and stable i.e., given the nature of bargaining in each coalition, no

player believes he can improve his payoff in any coalition by renegotiating.an agreement . Thus, a solution is a fixed point of the process of formingŽ . Žoutside options based on conjectured agreements and agreements based

.on outside options . Of course, only certain outcomes are consistent with agiven set of beliefs.

For the four professionals, there is a unique multilateral solution:BL Ž . EL Ž . BP Ž . EP Ž .x s 12, 22 , x s 22, 22 , x s 12, 12 , x s 22, 12 . Note that the

agreements for the partnerships of the economistrlawyer and physicistrbiologist are not feasible; we interpret this as a prediction that thosepartnerships will not form. On the other hand, the agreements for theeconomistrphysicist and lawyerrbiologist are feasible; conditional on for-mation of those partnerships we predict the first partner will obtain$22,000 and the second will obtain $12,000. Thus, antagonizing the outsideopportunity vectors has a big effect on the predicted outcomes.

ŽIn Section 2 we show that under very standard and quite weak assump-tions about the sets of attainable utility vectors and the bargaining func-

.tions multilateral bargaining problems always have solutions, and everyŽ .solution corresponds to at least one outcome. Moreover, the competition

between coalitions enforces a ‘‘law of one price’’: at any solution, the

5We find it convenient to normalize so that singleton coalitions can achieve only the 0vector; we therefore suppress singleton coalitions in the solution.


Ž .agreement utilities for each player are the same in every potentialcoalition of which he is a member. As a consequence, each multilateralbargaining solution can be characterized by a vector of reservation pricesŽ .one for each player .

As the discussion above suggests, our mechanism for determining out-side options and our interpretation of solutions correspond to a particularnotion of the way in which bargaining proceeds. We make this correspon-dence explicit in Section 4 by analyzing a noncooperative model of bar-gaining in a specific class of characteristic function games, those having theproperty that at most one coalition can profitably form. Our noncoopera-tive model is an adaptation of a proposal-making model due to SeltenŽ . 61981 . We show that the stationary subgame perfect equilibria of theproposal-making model correspond precisely to the multilateral solutionsŽ .for some specification of bargaining functions . We also use the proposal-making model to offer some support and further explication of ourconstruction of outside options.7

Ž .The idea of antagonizing outside options appears in Rochford’s 1984work on matching markets. Rochford’s purpose is to identify a selectionfrom the core; to do so, she considers only candidate agreements that are

Ž .in the core. Bennett’s 1988 work on matching models is much more inthe spirit of the present work, and in fact was an outgrowth of an earlier

Ž .version of the present work. Binmore 1985 analyzes a particular class ofŽthree player bargaining problems which he called ‘‘three-playerrthree-

.cake problems’’ . Binmore presents both cooperative and noncooperativesolutions for three-playerrthree-cake problems; his cooperative solution isŽ .essentially a multilateral solution in the sense discussed here. A differentanalysis of three-playerrthree-cake problems is given by Bennett and

Ž .Houba 1992 ; their cooperative solution is again a multilateral solution inprecisely the sense described here. Less closely related analyses of multi-player bargaining problems have been presented by Kalai and SametŽ . Ž . Ž .1985 , Chatterjee et al. 1987 , and Bennett and van Damme 1991 .

Following this Introduction, Section 2 presents the formalism of multi-lateral bargaining problems, solutions, and consistent outcomes and estab-lishes the existence of solutions and their basic properties. Section 3presents a number of illustrative examples. Section 4 presents the nonco-operative model and its connections to multilateral solutions.

6Selten models the bargaining process as a recursive game; we use the more familiarstructure of a game in extensive form.

7Of course the idea of finding mutually reinforcing cooperative and non-cooperativeŽ .models is due to Nash 1951, 1953 .

ELAINE BENNETT156

2. MULTILATERAL BARGAINING PROBLEMS

We begin by giving a formal description of multilateral bargainingproblems and their solutions, and then establish some of their properties.Following the formal description we provide some additional discussion ofthe crucial point, the construction of outside options. Illustrative exampleswill be presented in Section 3.

The data of a multilateral bargaining problem consists of a triple² :N, V, f where:

v � 4N s 1, 2, . . . , n is a finite set of players

vSŽ . Ž .for each coalition S a non-empty subset of N , the set V S ; Rq

is a compact, strongly comprehensive subset of RS that contains theqorigin.8, 9

vS� 4f s f is a collection of functions, indexed by the coalitions

S ; N; for each S ; N, f S: RS ª RS is the bargaining function of theq qcoalition S.

Ž .The interpretation we intend is that V S is the set of attainable utilityvectors for the coalition S, and that the bargaining function f S summa-rizes the bargaining process within the coalition S, given outside options.10

S SŽ S .Given the outside option vector d , we interpret f d as the division ofthe proceeds, conditional on the formation of the coalition S. Throughout,we assume that each bargaining function f S satisfies the following proper-ties:

S Ž . SŽ S . S1. Individual rationality: If d g V S then f d G dS Ž . SŽ S .2. Pareto optimality: If d g V S then f d is on the Pareto

Ž .frontier of V S

3. Continuity: f S is a continuous function of the outsideoption vector dS

S Ž . SŽ S . S4. Agreeing to disagree: If d f V S then f d s d

8 S S Ž .By R we mean the nonnegative orthant of R . V S is strongly comprehensive ifqŽ . S Ž .whenever x g V S , y g R with y F x and y / x then y is in the interior of V S , relativeq

to RS . Equivalently, the weak and strong Pareto boundaries coincide.q9 ² :Note that the pair N, V is a game in characteristic function form without sidepayments.10 The assumption that bargaining within each coalition can be summarized by a function is

simply the assumption that the outcome of bargaining is determinate: the same people facingthe identical bargaining problems reach identical agreements. A coalition’s bargaining func-tion may reflect the coalition’s standards of fair division, the institutional rules governingbargaining in the coalition, or the relative bargaining skills of its members.


S Ž .If the outside option vector d belongs to V S then there are attainableutility vectors for the coalition which allocate to each member of the

Žcoalition at least the utility of his outside opportunities i.e., there are.gains from making an agreement . In this case, the first two assumptions

SŽ S .require that the agreement f d be efficient and allocate to eachmember of the coalition at least as much as he could obtain by notparticipating. The third assumption requires that small changes in players’outside options lead to small changes in the agreement. These threeproperties are familiar from classical bargaining theory and are enjoyed byvirtually every solution concept proposed for the simple bargaining prob-lem.

S Ž .The case d g V S is the only one considered in traditional bargainingtheory. As discussed earlier, however, in our setting we must allow for the

S Ž .possibility that the outside option vector d does not belong to V S ,because the outside option vector dS represents the opportunities ofmembers of S in other coalitions, and it is certainly possible that noalternative for S is as good for all of its players as their outside options. Inthis case, the last assumption says that the members of S ‘‘agree todisagree.’’ When extended to allow for infeasible outside option vectors, all

Žof the usual solutions to the simple bargaining problem including theNash bargaining solution, the egalitarian bargaining solution, and the

.Kalai]Smorodinsky solution enjoy all four properties.Ž . � 4To avoid degeneracy, we require that V S / 0 for at least one

Ž .coalition S. It is convenient and involves no loss of generality to ‘‘zero-Ž� 4. � 4normalize’’ so that V i s 0 for each player i; in what follows it is

convenient to suppress the trivial singleton coalitions. We write C for theset of nonsingleton coalitions S ; N, and CU for the subset of C consist-

Ž . � 4ing of coalitions for which V S / 0 .For notational convenience, we let Q be an index set that contains a

distinct index for each occurrence of a player in one of his coalitions, i.e.,< < ny1for each player position. Since each player position is distinct, Q s n2 .

We identify an element x g RQ with a set of vectors, one for each� S < 4 S Scoalition: x S g C , where x g R . The set of bargaining functions

� S < 4 Q Q Ž .f s f S g C defines a mapping f : R ª R , which is given by f x sq q� SŽ S .4f x .

We now turn to the construction of outside options. For each coalition Sand each player i g S, we want to use as i’s component of the outsideoption vector dS the utility he would obtain if player i broke off negotia-tions in S and took the initiative to form hisrher best alternative coalition.Ž .We return to this point shortly. Of course, i’s alternatives depend on theagreements that will be reached in other coalitions. We assume that the

Ž .players in S make accurate and therefore identical conjectures about

ELAINE BENNETT158

these agreements. To see what this implies, fix conjectured agreement xT

for each coalition T / S; given these conjectures, what are the utilities ofplayer i’s alternatives?

vT ŽIf i g T and the agreement x is feasible for the coalition T i.e.,

T Ž .. Tx g V T , then player i will obtain x if the coalition T forms.i

vTIf i g T and the agreement x is not feasible for the coalition T

Ž T Ž .. Ti.e., x f V T , player i cannot obtain x . In view of our previousidiscussion about the utility of infeasible agreements, the most that player ican obtain in T is the largest utility which allows all of the other membersof T to obtain their agreement utilities. That is, the utility to player i of

T � < T Ž .4the unattainable agreement x in the coalition T is max t x rt g V T .i i

We use xTrt to denote the vector obtained from xT by replacing theiT Ž .ith component by t . If there is no value of t for which x rt g V T , theni i i

the infeasible agreement xT has no utility for player i; by convention, wetake 0 to be the maximum in this case.11

� T < 4Formally, given agreements x T / S in all other coalitions, we defineSŽ� T < 4.the outside option vector d x T / S for the coalition S in the follow-

ing way. For each i g S and each coalition T / S with i g T , set:

xT if xT g V TŽ .iT Tu x sŽ .i T½ max 0, t : x rt g V T otherwiseŽ .� 4i i

and

S T T T<� 4d x T / S s max u x i g T and T / S .Ž .� 4Ž .i i

Note that, although we take into account coalitions T in which theconjectured agreement xT is infeasible, we only ascribe to player i theutility t he would obtain if T actually came to a feasible agreement xTrt .i i

S � T < 4 SBy definition, d is a function from collections x T / S to R , but itqis convenient to view dS as a function defined on collections of agreement

Ž S S . Svectors for all coalitions although d will not depend on x , so that dbecomes a function from RQ to RS . We refer to dS as the outside optionq q

� S < 4function for the coalition S. We sometimes write d s d S g C , andview d as a function from RQ to RQ .q q

² :A multilateral bargaining problem N, V, f specifies a set N of players,Ž . Sand a set V S of attainable utilities and a bargaining function f for each

coalition S ; N. A solution to such a multilateral bargaining problem

11 w xThis is harmless, since 0 g V i for each i.


specifies an agreement payoff vector for each coalition in such a way thatthe agreements are consistent with the bargaining in every coalition andthe determination of outside options described above.

� S < 4Formally, x s x S g C is a solution to the multilateral bargaining² : S SŽ SŽ ..problem N, V, f if x s f d x for every coalition S g C. Our inter-

pretation of solution agreements is that, if x S is feasible for the coalition SŽ S Ž .. Si.e., x g V S , then x is the agreement which will be implemented inS, provided the coalition S actually forms; if x S is not feasible for the

Ž S Ž ..coalition S i.e., x f V S , then S will not form.Ž .By an outcome P, z we mean a partition P of N and a payoff vector

z g RN such that for each S g P, the restriction of z to S, z S, is a feasibleS Ž . Ž .utility vector for S, i.e., z g V S . We say that the outcome P, z is

� S < 4 S Sconsistent with the multilateral solution x S g C if z s x for all� 4 � 4coalitions S g P l C and z s 0 for every singleton i g P. That is, fori

coalitions in the partition, the agreement is feasible and is that specified by� S < 4the multilateral solution x S g C , and remaining players do not enter

into coalitions and obtain a payoff of 0.Note that a solution is not an outcome. A solution is a set of conditional

agreements for all coalitions and some of these agreements may beinfeasible. An outcome specifies a set of agreements and coalitions so thateach player belongs to exactly one coalition of this set and every agree-ment is feasible for its coalition. Thus it is important to distinguishbetween a solution and its outcomes. In particular, note that a singlesolution may give rise to more than one outcome.

The following theorem shows the existence of solutions to multilateralŽ .bargaining problems and summarizes their basic properties: i solutions

Ž .are nonnegative and not identically 0; ii solution agreements obey thelaw of one price}all agreement payoffs for a given player are identical;Ž .iii if player i’s agreement payoff is positive in some coalition then it is

Ž . Ž .feasible in some perhaps different coalition ; iv for every player, there issome outcome consistent with the solution at which that player achieveshis agreement payoff}in particular, there are outcomes consistent witheach multilateral solution. In what follows, recall that C is the set ofnonsingleton coalitions.

² :THEOREM 1. If N, V, f is a multilateral bargaining problem, then a� S < 4solution x s x S g C exists. Moreoër, if x is any solution then:

Ž .i x G 0 and x / 0Ž . S Tii If i g S and i g T , then x s x .i i

Ž . Siii If x ) 0, then there is a coalition T g C with i g T for whichiT Ž . � S 4x g V T . In particular, at least one of the solution agreements x : S g C

is feasible.

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Ž . Ž . S Živ For eëry i g N there is an outcome P, z for which z s x fori i.each S g C .

Proof. In order to establish the existence of solutions, we show firstthat the outside option function dS of each coalition S is continuous. Inview of the definition, it suffices to show that uSX

is continuous for each SXi

with i g SX.To end this, fix T , i with i g T and define

T < Th x s max 0, t x rt g V T .Ž . Ž .� 4i i

T � n4We assert that h is a continuous function. To see this, let x be aQ T Ž n. T Ž .sequence in R converging to x; we show that h x ª h x . Note firstq

T Ž .that, since the range of h is bounded hence compact , some subsequence� T Ž n.4of h x converges. Passing to a subsequence if necessary we may

T Ž n. T Ž .assume that h x ª w for some w; we wish to show that w s h x . Fornotational convenience, renumber the players in T with player i s 0 and

n Ž n.T Tthe remaining members of T as 1, 2, . . . , k. Set z s x and z s x .n T Ž .Clearly, z ª z. If h x / w, there are two cases to consider.

T Ž . T Ž n. T Ž .Case 1. h x - w s lim h x . By definition, h y G 0 for every y,T Ž n. T Ž . T Ž n.so lim h x ) h x G 0 and hence h x ) 0 for n sufficiently large.Ž T Ž n. n. Ž . Ž .Hence, h x , z g V T for n sufficiently large. Since V T is closed,

Ž . Ž . T Ž .we conclude that w, z g V T . But since h x is the maximum feasibleT Ž .value, this implies that h x G w, a contradiction.

T Ž . T Ž n. T Ž .Case 2. h x ) w. By definition, h x G 0 for each n. Hence h xŽ T Ž . . Ž . T Ž .) 0 so h x , z g V T . Since h x ) w ) 0, strong comprehensive-

Ž . Ž .ness guarantees that w, z is in the interior of V T . This means thatŽ . Ž .there is a «-ball around w, z contained in V T for some « ) 0. In

Ž n. Ž .particular, w q «r2, z g V T for n sufficiently large. This implies thatT Ž n.h x G w q «r2 for n sufficiently large, again a contradiction.

T Ž . TWe conclude that h x s w, so that h is a continuous function foreach T. By definition,

uT xT s xT of xT g V TŽ . Ž .i i

T T T Tu x s h x otherwiseŽ . Ž .0

T T T Ž T . TSince the functions x ª x and h are both continuous and h x s xi i iT Ž . Twhenever x belongs to the boundary of V T , the function u is continu-i

ous. It follows that the outside option functions dS are continuous, asdesired.

Ž .Let m be a number sufficiently large that each V S fits inside a cubew xwith sides of length m. Let Y be the Q-fold Cartesian product of 0, m .


The functions f and d both map Y into itself. By Brouwer’s fixed point� S < 4theorem, the mapping f ( d has a fixed point x s x S g C . For each

S SŽ SŽ ..S g C, x s f d x , so x is a solution to the multilateral bargaining² :problem N, yV, f , as desired. This completes the existence proof.

Ž . S Si To see that each x G 0 note that by definition d G 0 and thatbargaining functions are required to be individually rational. To see thatx / 0, recall that to avoid degeneracy we assumed that there is a coalition

Ž . � 4S with V S / 0 . Strong comprehensiveness requires that 0 belongs toŽ . Sthe interior of V S , so an agreement x s 0 would violate Pareto

optimality of bargaining functions.Ž . Sii Fix a player i and a coalition S containing i which maximizes x ,i

and let U be any coalition containing i from which dS is calculated. ThenidS F xU F x S. There are two cases to consider. If dS - xU then dS - x S

i i i i i i iand dS / x S, so Property 3 of bargaining functions implies that x S isfeasible. For every coalition T containing i and differing from S, thedefinition of outside options implies that dT G x S, and individual rational-i iity implies that xT G dT and maximality of x S yields x S s xT. On thei i i i iother hand, if dS s xU s x S then the definition of outside options impliesi i i

U Žthat x is feasible, whence the same argument as above but with U. T Uplaying the role of S yields that x s x for every coalition T containingi i

T Ž .i. Setting p to be the common value of x establishes ii .i i

Ž . Tiii Let p be the common value of x . Since p ) 0, fix a coalition Si i icontaining i, and suppose that, for every coalition T containing i anddiffering from S, the agreement xT is not feasible for the coalition T. Thedefinition of outside options implies dS - xT s p for each T , whencei i idS / x S. But then Property 3 of bargaining functions implies that x S is

Ž .feasible for S. This establishes iii .Ž .iv Fix a player i. Let D be the collection of all coalitions S ; N for

S Ž .which x g V S . We distinguish two cases.

Ž .Case 1. p ) 0. Part iii guarantees that there is a coalition T g Diwith i g T. Let E be any subset of D which contains the coalition T ,consists of disjoint coalitions, and its maximal with respect to theseproperties. Let EU be the union of the coalitions in E. Define thepartition P of N to consist of E and the singleton coalitions in N _ EU.Define the vector z g RN by z S x S for X g EU and z s 0 for j g N _ EU.i

Ž .It is easily checked that P, z is an outcome consistent with the solution xand z s xT; that z s x S for each S g C follows from the law of onei i j iprice.

Case 2. p s 0. Let F be any subset of D which consists of disjointicoalitions, and is maximal with respect to this property. Let FU be the

ELAINE BENNETT162

union of the coalitions in E. Define the partition P of N to consist of Fand the singleton coalitions in N _ FU. Define the vector z g RN byz S s x S for S g FU and z s 0 for j g N _ FU. It is again easily checkedj

Ž .that P, z is an outcome consistent with the solution x; the construc-tion guarantees that z s 0 s x S for each S g C. This completes thei iproof. B

Ž .In view of part ii of Theorem 1, corresponding to every solutionQ Ž � S < 4. Nx g R i.e., x s x S g C we may associate the vector p g R definedq

by p s x S for any S with i g S. Note that p is the utility player i wouldi i iobtain in any feasible agreement to which he is a party. Moreover, partsŽ . Ž . Žiii and iv guarantee that p actually is feasible for player i if p ) 0i ithen there is a coalition S containing i such that x S is feasible and if

.p s 0 then player i can obtain p on his own . It is natural to interpret pi i ias a reservation price for player i; we refer to p as the reservation price

� S < 4vector of the solution x s x S g C .

Discussion: The Meaning of Outside Options

Outside options play a crucial role in our theory, so it seems appropriateto conclude this section with a more elaborate discussion of the meaningof outside options and the motivation for our choice.

If the players in the coalition S fail to reach an agreement one or moreplayers may enter into alternate coalitions; the opportunity cost for eachmember of S is thus the utility of foregone agreements in alternativecoalitions. We should therefore take as i’s outside option, dS, some utilityilevel that represents a summary of these foregone agreements. Manydifferent summary methods seem possible, each corresponding to a partic-ular view of the way in which bargaining proceeds. We have taken dS to beithe utility i would obtain from agreement in his best alternative coalition.12

Of course, this represents an optimistic assessment. Below we discuss aŽ .particular alternative ‘‘reductionist’’ scheme and argue that it corre-

sponds to quite a different view of multilateral bargaining. In Section 4 wepresent a formal noncooperative bargaining model which formalizes ourparticular view of the way in which bargaining proceeds and leads in effectto our particular method of assigning outside options.

Our construction of outside options leaves open the possibility that theoutside options of members of a coalition S may not be jointly compatible.For instance, players 1, 2 g S may each rely on a pairing with player 3, but

12 We rely on the self-interest of i’s partners in the alternative coalition to guarantee thattheir components of the agreement are at least as large as they can obtain in their alternativecoalitions. One property of the solution we present is that it is compatible with this type ofself-interest.


these pairings cannot both be realized. In particular, that means we cannotinterpret our outside option vectors as the result of disagreement in the

Žcoalition S. And it’s for this reason that we prefer the term ‘‘outside.option’’ rather than ‘‘disagreement point.’’ To motivate our interpretation

Ž .of outside options, it is useful to first recall Binmore’s 1985 analysis ofthe role of outside options in two-player simple bargaining problems.Binmore analyzes a sequential model of bargaining between two players

Ž . Ž .who must choose a utility vector x , x in the convex set V, with a given1 2Ž . Ž .feasible outside option vector d s d , d . Binmore’s analysis is based1 2

Ž .on Rubinstein’s 1982 alternating offer model: player 1 makes an offer,Ž .player 2 may accept in which case play ends or decline; if player 2

declines, he may terminate bargaining or make an offer in turn, etc.Players discount future payoffs according to the common discount factorr - 1, and infinite plays result in a payoff of 0 to both players.

Binmore analyzes two different scenarios. In the first, the result oftermination of bargaining by both players is that they obtain their compo-nent of the vector d. In the second scenario, the result of termination ofbargaining by player i is that player 1 obtains his component of d while theother player obtains nothing. In the first of these scenarios, d plays therole of the ‘‘disagreement point’’ or ‘‘status quo point’’ in the sense that

Ž .Nash and others discussed. In the second scenario, however, d plays therole of defining strategic choices for each player.

Despite the difference in interpretations, Binmore shows that two sce-Ž .narios lead to precisely the same subgame perfect equilibrium outcomes.Ž .As the discount factor r tends to 1 so that players become more patient ,

Žthis common outcome is the Nash bargaining solution see Section 3 for a.definition .

The conclusion in each of these scenarios is driven by the same intuitionthat drives the conclusion of the basic Rubinstein alternating offer model.When player 1 makes an offer, he is committed until player 2 responds;while 1’s offer is on the table, player 2 can take up his outside option, butplayer 1 cannot. And player 2’s decision to terminate bargaining and takeup his outside option is affected only by his own component of d, and notby player 1’s component.

Our interpretation of outside options corresponds to a similar view ofthe way in which bargaining proceeds: Players bargain sequentially, and an

Ž .offer binds those who make it and those who have accepted it , but notthose who have yet to respond to it. From the time that player 1 makes an

Ž .offer to player 2 for participation in some particular coalition S until thetime that the offer expires, player 1 is committed to the offer. But, untilplayer 2 responds, player 2 is not committed, so player 2 can take heroutside option but player 1 cannot take up his. In our multiplayer setting,2’s outside option represents the utility she could obtain if she broke off

ELAINE BENNETT164

negotiations with 1 in the coalition S and took the initiative to form someother coalition. In this circumstance, of course, it would be player 2 thatwould have the initiative, and not player 1. It therefore seems reasonableto take as 2’s outside option the utility of her best alternative agreement.

What utility should player i anticipate if he broke off negotiations in thecoalition S in order to enter into an agreement in the coalition T ? If itwere common knowledge that members of T would come to a feasibleagreement zT, the answer would be zT. However, if bargaining proceedsisimultaneously in all coalitions, the agreement in T cannot be common

Ž .knowledge in advance . On the other hand, player i and every otherplayer will form conjectures about the potential agreement in the coalition

Ž .T. At a perfect foresight equilibrium, these conjectures will be correctŽ .and therefore equal , so we should impute to player i the utility resultingfrom this common conjecture.

Why should player i believe that if he breaks off negotiations in S hewill be able to form his most preferred alternative coalition, say T ? Afterall, members of T will try to obtain their most preferred alter-natives}which might not include forming the coalition T. If this were thecase, player i might find that when he breaks off negotiations in S andinitiates negotiations in T , he cannot obtain the cooperation of othermembers of T. It would seem, therefore, that we should take for player i’soutside option the maximum utility he could obtain in any alternativecoalition T , subject to the additional requirement that all other membersof T find T to be their most preferred alternative. However, as we show inTheorem 1, at a multilateral solution the difference between these choicesdisappears: at a multilateral solution it will always be the case that all themembers of i’s most preferred alternative coalition T also find T to be atleast as good as any other alternative.

If the outside option vector dS for the coalition S is not feasible,Žmembers of S will not agree to any feasible division in S because for any

feasible division, at least one of them could do better by making some.other agreement , so the coalition S will not form. In this case, we adopt

the convention that the agreement coincides with the outside optionvector, keeping in mind that we interpret such an infeasible agreement asan agreement not to form the coalition S.

The possibility of infeasible agreements raises one final question. WeŽ .use as i’s outside option in the coalition S the maximum utility he or she

could obtain by entering into an agreement in some alternative coalitionT , but we allow for the possibility that the coalition T will not come to afeasible agreement; what utility should we impute to player i in this case?Depending on the point of view taken, this question could have severaldifferent answers. The most obvious is that we should impute zero utility

Ž .to player i when the conjectured agreement in T is infeasible. An


alternative view, and the one we shall adopt, is that we should impute toplayer i the maximum utility he or she could obtain in any feasibleagreement which yields all other members of T at least their utilities at

Ž . Uthe conjectured infeasible agreement. In Fig. 1 below, t is the utility weiT w ximpute to player i for the infeasible agreement x in the coalition T s i .

In the first graph xTrtU is the feasible agreement which maximizes playerji’s utility while yielding j her agreement utility xT, and in the second graphjtU s 0 because no feasible agreement for T yields j at least her agreementiutility xT.j

To see why this evaluation of infeasible agreements is quite natural inthis context, recall that if xT is infeasible then the agreement in Tcoincides with the outside option vector, dT. In that case, each member jof T could obtain at most her outside option dT by coming to agreementjin her most preferred alternative coalition U . Presumably, player j wouldjaccept any offer from player i that yielded j a payoff of dT q « , for anyj« ) 0. Hence player i can expect to obtain as much as the residual valueof the coalition T after all of his partners have been paid their outsideoption values, dT.j

Ž .Note that part ii of Theorem 1 justifies our earlier discussion aboutoutside options and most-preferred alternatives. Player i’s outside optiondS in the coalition S is the maximum he could obtain in any alternativeicoalition; why should he expect to be able to form his best alternativecoalition? If dS s 0, nothing more needs to be said, since i can obtain 0 byihimself. If dS ) 0, the definition of outside options requires that there beisome coalition T for which the vector yT s xTrd S is feasible. Were playerii to offer members of T their components of yT q « for any « ) 0, each

Ž .of them would be willing to accept, since ii guarantees that, for j g Tand j / i, yT s xT is the maximum utility player j could obtain in anj jagreement in any coalition.

Finally, our construction of outside options may be illuminated bydiscussing an alternative route we did not take.13 To calculate the outsideoption values of the players in S, determine the solution for the reducedbargaining problem in which the coalition S has no value. Formally, given

² :a multilateral bargaining problem N, V, f and a coalition S, consider the² X : XŽ . � 4 XŽ .reduced bargaining problem N, V , f such that V S s 0 and V T

Ž .s V T for all T / S. Given the solution agreements for the reduced² X :bargaining problem N, V , f , we may compute outside options for the

coalition S on the basis of this outcome. Given the outside option for thecoalition S, we may then compute the agreement in S using the bargainingfunction f S. Of course, we want to compute the predicted outcome for the

² X :reduced bargaining problem N, V , f by the obvious inductive proce-

13 We thank Martin Hellwig for stimulating discussion on this point.

ELAINE BENNETT166

² Y :dure: For each T , form a reduced bargaining problem N, V , f , compute² Y :outside options on the basis of the agreements in N, V , f , and then

² X : Tcompute agreements for N, V , f using the bargaining functions f , etc.Ž Ž . .A similar strategy has been followed by Crawford and Rochford 1986 .

Although this ‘‘reductionist’’ scheme has some appeal, it is not withoutits drawbacks. We cannot really expect to be able to predict a uniqueoutcome or even a unique solution for each multilateral bargaining prob-

² X :lem. But if the reduced bargaining problem N, V , f admits multiplesolutions, the outside option vector for the coalition S cannot be theunique vector of payoffs to members of S that would result if S were notto form. That leaves two possibilities. The first is to settle for multipleoutside option vectors for the coalition S; this seems unsatisfactory. Thesecond is to take as the outside option vector for the coalition S somesummary of outcomes of the reduced bargaining problem, but it seemsunclear what sort of summary to use.

More importantly, the ‘‘reductionist’’ scheme fails to capture what seemsto us to be an essential feature of multilateral bargaining. Suppose forsimplicity that only the coalitions S, T yield nonzero gains; to avoidtriviality, suppose also that S l T / B. The bargaining within S is influ-enced by the bargaining within T because the latter defines the opportuni-ties which members of S must forego if T forms. In our scheme, thebargaining within T is in turn influenced by the bargaining within Sbecause the latter defines the opportunities which members of T mustforego if S forms, and so forth. Thus the bargaining problems of bothcoalitions are inextricably linked. The ‘‘reductionist’’ scheme would viewthe bargaining within S as influenced by the bargaining within T , butwould then image the bargaining within T proceeding in isolation, ignoringthe influence of the bargaining within S. To make the same point dynami-cally, think about the behavior of an individual i g S l T. In our scheme,i might negotiate first with members of S; if that negotiation broke down, imight then negotiate with members of T , and later with members of S

Žagain, and so forth. The noncooperative model we present in Section 4.has exactly this feature. By contrast, the ‘‘reductionist’’ scheme would

allow for i to negotiate first with members of S, and then, if negotiationsbroke down, to negotiate with members of T but without the possibility ofreturning to members of S.

3. EXAMPLES

In this section we present a series of illustrative examples to give someinsight into the nature of multilateral solutions and their outcomes, and toprovide comparisons with other solution concepts. Recall that the data of a


multilateral bargaining problem include the specification of a bargainingfunction for each coalition. The most familiar bargaining functions are thesolutions to the simple bargaining problem, such as the Nash bargaining

Ž .solution Nash, 1950 , extended to allow for outside option vectors whichare infeasible. Formally, given the coalition S and the convex feasible setŽ . S S SV S , we define the Nash bargaining function N : R ª R in theq q

S Ž . SŽ S .following way: For d g V S , N d is the vector which maximizes theŽ S S . S Ž . S Ž . SŽ S .Nash product Ł x y d over all x g V S ; for d f V S , N dig S i i

s dS.14 If all coalitions use the Nash bargaining function, we refer to themultilateral bargaining problem as a Nash multilateral bargaining problemand to its solution as a Nash multilateral solution.15

Ž .EXAMPLE 1 Three-PlayerrThree-Cake Problems . The simplest inter-esting multilateral bargaining problems are the three-player situations inwhich individuals and the coalition consisting of all three players earnnothing, so that only the two-player coalitions are profitable to form. If the

w x w xcoalition i, j forms, it has the ‘‘cake’’ V i, j to divide, but since there areonly three players, at most one of these three cakes will actually bedivided. This class of bargaining problems has recently been the subject of

Ž . Ž .papers by Binmore 1985 and by Bennett and Houba 1992 . Dependingon the nature of the core of the underlying game we have three cases toconsider.

² :Case 1. The core of N, V is empty. In this case, there is a uniqueŽ . Svector q s q , q , q such that q is on the Pareto efficient boundary of1 2 3

Ž . w x w x w xV S for S s 1, 2 , 2, 3 , 1, 3 ; i.e., there is a unique vector q such that allthree coalitions can afford to pay their members exactly their componentsof q. For eëry choice of bargaining functions, q is the vector of reserva-tion prices of a multilateral solution.16 For the Nash, egalitarian, orKalai]Smorodinsky bargaining functions, or indeed for any bargaining

14 Recall that the Nash bargaining solution}as well as some others}are defined only forconëx attainable utility sets; in such cases we will also assume convexity.

15 Egalitarian ES and Kalai]Smorodinsky K S bargaining functions and multilateral solu-tions may be defined by similarly extending the egalitarian and Kalai]Smorodinsky bargain-

Ž .ing solutions. See Kalai 1985 for a description of these solutions for simple bargainingproblems.

16 To see why, note that in each coalition each player’s outside option is his reservationŽ .price since it is part of a feasible agreement in another coalition and since the coalition’s

outside option vector lies on the boundary of the attainable set it is the only feasible,Žindividually rational agreement. When there is no surplus to divide, the division rule is

.irrelevant.

ELAINE BENNETT168

FIG. 1. Nash multilateral solution for a game with an empty core.

functions that are strictly individually rational,17 this vector q is the vectorof reservation prices of the unique multilateral solution. See Fig. 1.

Note that none of the Nash multilateral agreements in Fig. 1 is the NashŽ .agreement for the simple bargaining problem with 0, 0 as the disagree-

ment point, i.e., competition among the coalitions forces the agreementsaway from the solutions the coalitions would have reached in isolation. For

17 S SŽ S . S S Ž .We say that f is strictly individually rational if f d 4 d whenever D g int V SŽ S .relative to R .q


this solution there are three possible outcomes:

w x w x w x w x1 , 2, 3 ; 0, q , q 1, 2 , 3 ; q , q , 04 � 4Ž . Ž .. Ž .2 3 1 2

w x w x1, 3 , 2 ; q , 0, q� Ž .Ž .1 3

i.e., any of the three cakes might be divided, but whatever cake is dividedwill be divided according to the reservation price vector q.

² : Ž .Case 2. The core of N, V contains a unique point q s q , q , q .1 2 3�w x w x4This core point is supported by a partition of coalitions, say 1 , 2, 3 . For

Ž . w xthis case q s 0 and q , q g V y2, 3 . It follows from strong compre-1 2 3w xhensiveness of V 2, 3 and the fact that q is the unique point in the core

Ž . w x Ž . w xthat q , q g V 1, 2 and q , q g V 1, 3 . As in Case 1 above, it follows1 2 1 3that, for every choice of bargaining functions, q is the reservation pricevector of a multilateral solution.

For this solution there are three possible outcomes:

w x w x w x w x1 , 2, 3 ; 0, q , q 1, 2 , 3 ; 0, q , 0� 4 � 4Ž . Ž .Ž . Ž .2 3 2

w x w x1, 3 , 2 ; 0, 0, q� 4 Ž .Ž .3

w xi.e., the payoff in the coalition 23 is according to q and if player 1Ž w x w x.succeeds in forming a coalition either 12 or 13 , he will obtain 0.

² :Case 3. The core of N, V contains more than one point. It is nothard to see that all points in the core are supported by the same partition

�w x w x4 � <Ž . w x4of coalitions, say 1 , 2, 3 . For i s 2, 3 let t s max t 0, t g V 1, i . Thei² :fact that the core of N, V contains more than one point implies that the

Ž . w x 23Ž .point t , t lies in the interior of V 2, 3 . Let N t , t be the Nash2 3 2 3w x Ž .solution in the coalition 2, 3 , given the outside option vector t , t . Then2 3

Ž 23Ž .. 30, N t , t g R is the reservation price vector of the unique Nash2 3multilateral solution. See Fig. 2. Since the agreements for the coalitionsw x w x1, 3 and 2, 3 are infeasible the Nash multilateral solution has a uniqueoutcome

w x w x 231 , 2, 3 ; 0, N t , t ,� 4 Ž .Ž .Ž .2 3

w xi.e., the cake V 23 will be according to the Nash solution for the outsideŽ .option vector t , t and player 1 obtains nothing. Similarly,1 2

��w x w x4 Ž 23Ž ... Ž .1 , 2, 3 ; 0, E t , t is the unique outcome of the unique egalitar-2 3Ž�w x w x4 Ž 23Ž ...ian multilateral solution, and 1 , 2, 3 ; 0, K t , t is the unique2 3

Ž .outcome of the unique Kalai]Smorodinsky multilateral solution. Indeed,whenever the bargaining functions f S are strictly individually rational,Ž 23Ž ..0, f t , t is the reservation price vector of the unique multilateral2 3

Ž�w x w x4 Ž 23Ž ...solution and 1 , 2, 3 ; 0, f t , t is its unique outcome.2 3

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FIG. 2. Nash multilateral solution for a game with an interval core.

The reader should not be misled. In three-playerrthree-cake problemsstrict individual rationality of bargaining functions is enough to guaranteeuniqueness of the multilateral solution. For more general multilateralbargaining problems, strict individual rationality of bargaining functionssubstantially restricts the range of possible multilateral solutions, but is notgenerally enough to guarantee uniqueness.

In the Nash bargaining solution, outside options are viewed as the basisof negotiations; an alternative view is to treat outside options as con-straints on the solution. Adopting this view leads us to define the con-strained Nash bargaining function in the following way. Given a coalition

Ž . SS and an attainable set V S ; R , the constrained Nash bargainingqsolution CN S: RS ª RS maps an outside option vector dS g RS to theq q q

SŽ S .agreement CN d defined by:

S Ž . SŽ S . SIf d f V S then CN d s dS Ž . SŽ S . Ž .If d g V S then CN d g V S is the arg max of the Nash

� Ž . S4product Ł x over the set x g V S : x G d .ig S i


Note that CN S is individually rational but not strictly individuallySŽ S . S S SŽ S . Srational. That is, CN d G d for all d , but equality CN d s d isi

S Ž .possible for some components i g S, even when d f int V S . Looselyspeaking, the distinction between individual rationality and strict individualrationality may be seen in the response to an outside offer: individual

Ž .rationality requires that outside offers be met when possible , and strictŽindividual rationality requires that outside offers be exceeded when possi-

.ble . For three-playerrthree-cake bargaining problems, Bennett and HoubaŽ .1992 characterize the constrained multilateral Nash solutions and Bin-

Ž .more 1985 characterizes the multilateral solutions of a variant of theconstrained Nash bargaining function.

Ž .EXAMPLE 2 Nash and Constrained Nash Multilateral Solutions . Tobest illustrate the difference between the Nash and constrained Nash

Žbargaining solutions and, more generally, between the multilateral solu-tions of bargaining functions that are strictly individually rational and

.those that are only weakly individual rational consider the three-Ž .playerrthree-cake transferable utility problem with attainable utility sets

given by:

2 <w xV 12 s y , y g R y q y F 50Ž .� 41 2 q 1 2

2 <w xV 23 s y , y g R y q y F 30Ž .� 42 3 q 2 3

2 <w xV 13 s y , y g R y q y F 10Ž .� 41 3 q 1 3

� 4V S s 0 for every other coalition S ; N.Ž .

This bargaining problem has a unique Nash multilateral solution gener-Ž .ated by the reservation price vector 15, 35, 0 and a unique outcome

Ž�w x w x4 Ž ..12 , 3 ; 15, 35, 0 .If, however, we consider the constrained Nash bargaining functions, the

situation is quite different: the reservation price vectors of constrained�Ž . <Nash multilateral solutions form the interval 25 y l, 25 q l, 5 y l 0 F

4 w x w xl F 5 . For all l - 5, 1, 2 and 1, 3 are the feasible coalitions. Thew xendpoint l s 0 corresponds to equal division in the coalition 1, 2 and,

intuitively speaking, is the solution for situations in which player 1 isunable to ‘‘play off’’ his two potential partners to obtain a higher payoff forhimself; while the endpoint l s 5 corresponds to the point where player 1

w xcannot obtain a higher payoff from player 3 in 1, 3 }because 3 isobtaining nothing already}and hence cannot bargain for more fromplayer 2; this corresponds to situations where player 1 has exhausted allgains from playing off his potential partners.

Cooperative solution concepts typically consider candidate payoff vec-tors with one component for each player; multilateral solutions consider

ELAINE BENNETT172

candidate payoff vectors with one component for each player in eachpotential coalition. One may wonder whether ‘‘all this additional complica-tion’’ is really necessary. To see what is gained, first recall that cooperativesolution concepts, such as the core, bargaining set, kernel, nucleolus,follow a common approach: consider a particular candidate payoff vector

Žand ask whether it is stable the notion of stability employed distinguishes.the various solution concepts . When testing for stability, the payoff vector

is treated as the ‘‘status quo’’ in the negotiations. The multilateral bargain-Ž .ing approach instead uses the conjectured agreements in other coalitions.

The following example highlights the difference this can make.

Ž .EXAMPLE 3 The Bargaining Set and The Nash Multilateral Solution .Consider the 5-player sidepayment game in which only the cyclic three-player coalitions are profitable:

U w x w x w x w x w xC s 1, 2, 3 , 2, 3, 4 , 3, 4, 5 , 4, 5, 1 , 5, 1, 2� 43 < Uw x w xV i , j, k s y , y , y g R y q y q y F 6 for i , j, k g CŽ .� 4i j k q i j k

� 4V S s 0 for all other coalitions S ; N.Ž .

�w x w x w x4 Ž .For the coalition structure 1, 2, 3 , 4 , 5 the payoff vector 3, 0, 3, 0, 0 isthe unique payoff vector in the bargaining set of Aumann and MaschlerŽ . w x1964 . Although the game is cyclically symmetric, the payoff within 1, 2, 3

w xis not. To see why, note that once the coalition 1, 2, 3 has formed and anyŽ .candidate payoff vector y , y , y , 0, 0 is selected, the symmetry of the1 2 3

situation is broken. Players 1 and 3 have alternate coalitions in which theirŽ .prospective partners players 4 and 5 are currently earning nothing}and

so presumably would accept nearly nothing. Player 2’s alternative coali-tions, however, involve either player 1 or player 3, either of whom mustreceive at his y . As a consequence, there are ‘‘justified objections’’ toievery payoff vector that allocates player 2 a strictly positive payoff. That

Ž .there are no ‘‘justified objections’’ to the payoff vector 3, 0, 3, 0, 0 seemsŽ .entirely appropriate gi en that 3, 0, 3, 0, 0 is the status quo vector.

If, however, no outcome can be viewed as the ‘‘status quo,’’ thisreasoning loses much of its force. In that case, it seems appropriate to use

w xas a basis for negotiations within the coalition 1, 2, 3 the agreements thatwould be reached through bargaining in the alternative coalitions. This ofcourse is the approach of multilateral bargaining solutions. As a conse-quence the multilateral solution respects the underlying symmetry of thegame: the unique Nash multilateral solution is generated by the reserva-

Ž . w xtion price vector 2, 2, 2, 2, 2 . If the coalition 1, 2, 3 forms, the associatedŽ .outcome has the payoff vector 2, 2, 2, 0, 0 .


Simultaneous ¨s Sequential Coalition Formation

As the next example illustrates, the problem of modeling coalitionformations may become more complicated if coalition formation may takeplace sequentially, rather than simultaneously.

ŽEXAMPLE 4. Consider again the bargaining problem from the Intro-. Ž . Ž . Ž .duction of the economist player 1 , lawyer player 2 , biologist player 3 ,

Ž . 18and physicist player 4 . The attainable utility sets for each profitablecoalition are:

2 <w xV 1, 2 s y , y g R y q y F 40Ž .� 41 2 q 1 2

2 <w xV 2, 3 s y , y g R y q y F 34Ž .� 42 3 q 2 3

2 <w xV 3, 4 s y , y g R y q y F 20Ž .� 43 4 q 3 4

2 <w xV 1, 4 s y , y g R y q y F 34Ž .� 41 4 q 1 4

� 4V S s 0 for all other coalitions S ; N.Ž .

Ž .This problem has a unique Nash multilateral solution displayed in Fig. 1Ž .with the reservation price vector p s 22, 22, 12, 12 . The only feasible

w x w xagreements are for the coalitions 1, 4 and 2, 3 . If these coalitions form,Ž�w x w x4 Ž ..the outcome will be 1, 4 , 2, 3 ; 22, 22, 12, 12 .

This outcome is entirely sensible if we imagine that coalitions formsimultaneously. To see what might happen if coalitions form sequentially,let us examine this bargaining problem in a bit more detail. Considerplayer 4. If 4 agrees to form a coalition with player 1 the division of the

w x Ž .payoff in the coalition 1, 4 will be 22, 12 . On the other hand, if 4 canwait for players 2 and 3 to form a coalition and leaë, what will remain will

w xbe a simple bargaining problem for the coalition 1, 4 in which they have34 units of utility to divide, and neither of them has a ïable outside option.In such a situation the Nash bargaining solution will yield player 4 a payoff

Ž .of 17 half of 34 . Hence, player 4 might prefer to wait rather thanŽagreeing to form a coalition initially similarly, player 3 might also prefer

.to wait .We note that this problem arises only if multiple profitable coalitions

can simultaneously form. In Section 4, we rule out this possibility byrestricting to environments for which only one coalition can profitably

Ž Uform. For instance, this would be the case if very pair of coalitions in C.had nonempty intersection. If coalition formation is sequential, and

profitable disjoint coalitions may actually form, a multistage analysis seemsnecessary; at this point, we have none to offer.

18Again we assume that utilities are linear in money and that only money matters.

ELAINE BENNETT174

4. THE PROPOSAL-MAKING MODEL

In this section we present a noncooperative analysis of multilateralbargaining problems that formalizes some of the discussion in Section 2concerning outside options and our interpretation of solutions. Our nonco-operative model is an adaptation of a proposal-making model due to

Ž .Selten 1981 . Selten considers only games with transferable utility andmodels bargaining as a recursive game; we allow for nontransferable utilityand use the more familiar structure of a game in extensive form.

² : � 4We take as given an NTU game N, V , so that N s 1, 2, . . . , n is aŽ . Sfinite set of players and for each coalition S ; N, the set V S ; R is aq

² :compact set containing the origin. As before, we assume that N, V isŽ Ž . � 4 . Žnondegenerate so that V S / 0 for some S and 0-normalized so that

w x. � 4 . Ž .V i s 0 for each i g N , and that each V S is strongly comprehensiveŽ .so that the weak and strong Pareto sets coincide . Write C for the set ofnonsingleton coalitions and CU for the subset of C consisting of coalitions

Ž . � 4S for which V S / 0 .As Example 7 demonstrates, the possibility of sequential coalition for-

mation presents special difficulties, which we wish to avoid. To this end, weconstruct an extensive form game in which bargaining terminates with the

² :formation of any coalition, and we restrict our attention to games N, Vhaving the property that every pair of profitable coalitions has nonempty

Ž Ž . � 4 Ž . � 4.intersection that is, S l T / B whenever V S / 0 and V T / 0 .Our extensive form game can be described in the following way. Players

Ž .bargain by making, accepting, and rejecting proposals. A proposal S, xspecifies a coalition S containing i and a feasible payoff distributionx g S. The game is played according to the following rules:

0. Start of game: Nature has the first move. Nature selects eachplayer i g N with probability p ) 0; the selected player has the initiativeiŽ .See a1 .

1. The initiator role: player i with the initiative can eitherŽ . Ž .a make a proposal S, x for which i g S, and name a player

Ž .j g S as the responder see a2 , orŽ .b pass the initiative to any other player j, who becomes initiator

Ž .see a1 .

Ž .2. The responder role: player j responding to a proposal S, x caneither

Ž . Ž .a accept the proposal S, x and name a player k g S who hasŽnot yet accepted to be the next responder if j is the last player in S to

.accept, see a3 orŽ . Ž .b reject the proposal and become the initiator see a1 .


3. End of game: The game ends when all members of a coalition SŽ .have agreed to a proposal S, x . In this case members of S receive their

components of x, and other players receive 0.

4. Perpetual disagreement: Infinite plays result in 0 payoff for allplayers.

Ž .A sketch of the game tree is in Bennett 1991a, p. 58 .The solution notion we adopt here is a stationary subgame perfect

equilibrium in pure strategies. By stationarity, we mean that player i,

Ž .a when the initiator, always makes the same proposal and choosesthe same responder

Ž . Ž .b when the responder, considering a proposal S, x on which play-ers in SX ; S have already acted, makes the same response and choosesthe same next responder

Of course, stationarity is a strong requirement, but it does not seemunnatural in this context. Moreover, it appears that without stationarity}orsome other requirement beyond subgame perfection}very little can be

Ž .said: Bennett 1991b shows that every individually rational feasible out-come can be supported as a subgame perfect equilibrium outcome. Bycontrast, the restriction to pure strategies is made here only for expositoryconvenience; there would be no substantial difference if we allowed for

Ž Ž . .mixed strategies with finite support. See Bennett 1991a for the proofs.ŽGiven a stationary subgame perfect equilibrium s a profile of pure

. Ž . Ž .strategies , we write PP s for the set of proposals S, x made andŽ .accepted with positive probability i.e., along the equilibrium path , and

Ž . Ž . Ž .PP s for the set of proposals S, x g PP s for which player i belongs toithe coalition S.19 Since players not in the coalition which actually forms

Ž .obtain payoffs of 0, we may in the obvious way identify PP s with the setŽof outcomes consistent with s i.e., the outcomes that occur with positive

. Ž .probability given that players follow s . Write q s for i’s payoff when heiŽ Ž . . Ž .is initiator stationarity implies that q s is well-defined , and set q s si

Ž Ž . Ž .. Ž .q s , . . . , q s ; we refer to q s as the initiation ëctor of the equilib-1 NN Ž .rium s . If q g R , we say that a proposal S, x is consistent with q if

x s q for each i g S. As the following proposition shows, only proposalsi iŽ .consistent with q s are made and accepted, so in particular it follows

Ž .that each player is indifferent among proposals in PP s , i.e., ifiŽ . Ž . Ž .S, x , T , y g PP s , then x s y .i i i

19 Ž .Keep in mind that Nature has the first move; hence PP s will generally contain morethan one proposal.

ELAINE BENNETT176

Ž .PROPOSITION. If s is a stationary subgame perfect equilibrium with q sŽ . Ž .as its initiation ëctor, then eëry proposal in PP s is consistent with q s .

Proof. Consider first the behavior of player i responding to the pro-Ž . Ž .posal S, x . If player i rejects and becomes initiator, he will obtain q s .i

Ž .If i is the last member of S to act on S, x , subgame perfection requiresŽ .that player i accept if x G q s . Backward induction implies that if i isi iŽ . Ž .not the last member to act on S, x then player i will accept if x G q si i

Ž .and x G q s for every j who has not yet acted.i jŽ . Ž .Now consider any proposal S, x g PP s . The first part of the discus-Ž . Ž .sion above shows that x G q s for each i g S. If x ) q s for some i,i i i iŽ . Ž .strong comprehensiveness of V S implies that there is a vector y g V S

Ž .with y ) q s for every j g S. The second part of the discussion abovej jŽ .shows that the proposal S, y will certainly be accepted. But then any

Ž .player j g S can obtain y ) q s when he is initiator, contradicting thej jŽ . Ž .definition of q s . We conclude that x s q s for each i g S, so thatj i i

Ž .S, x is consistent with s , as asserted. B

Ž .Although only proposals consistent with q s are made and acceptedwith positive probability, there may be other proposals consistent withŽ . 20q s which are never made at all.We are now in a position to describe the relevant connections between

the proposal-making model and multilateral bargaining problems. As we² :have noted earlier, if we begin with a game in coalitional form N, V ,

� S4then each specification of bargaining functions f leads to a multilateral² :bargaining problem N, V, f . Our first goal is to identify the initiation

vectors of stationary subgame perfect equilibria of the proposal makingmodel with those vectors RN which are reservation price vectors ofmultilateral solutions for some choice of bargaining functions.

THEOREM 2X. For a ëctor q g RN, the following are equi alent:

Ž .a there is a stationary subgame perfect equilibrium s of the proposal-Ž .making model such that q s q s , the initiation ëctor of s

Ž . � S4b there are bargaining functions f such that q is the reser ation price² :ëctor of a solution of the multilateral bargaining problem N, V, f .

Ž . NProof. Following Bennett and Zame 1988 , say that a vector q g R isq² : Ž .an aspiration for the game N, V if i for every i g N there is a coalition

S ; N with i g S such that the restriction qS of q to the players in SŽ . Ž .belongs to S q is coalitionally feasibility , and ii there does not exist a

20 If we allow for mixed strategies, and, following Selten, require that a player who isŽ .indifferent among several proposals select randomly among them, then PP s consists

Ž .precisely of those proposals which are consistent with q s .


T Ž . T Tcoalition T ; N and a vector x g V T such that x ) q for each i g Ti iŽ . Ž .q is unblocked . Bennett 1991a shows that q is an aspiration if and onlyif it is the initiation vector of some subgame perfect equilibrium of theproposal-making model. Hence to prove the desired result it suffices toshow that q is an aspiration if and only if there are bargaining functions� S4f such that q is the reservation price vector of the multilateral bargain-

² :ing problem N, V, f .One direction of this equivalence is easy. Let q be the reservation price

² :vector of a solution for the multilateral bargaining problem N, V, f . Foreach coalition S g N, qS is the agreement in the coalition S. By definition,

S Ž .q is either infeasible or on the Pareto boundary of V S ; hence q isunblocked. In view of Theorem 1, for each i g N, there is a coalition

T Ž .T g N such that i g T and q g V T ; hence q is coalitionally feasible.Ž .To obtain the reverse direction, fix an aspiration q and let GC q be the

S Ž .set of coalitions S for which q g V S . For each coalition S, define theagreement x S s qS and define the outside option vector dS using the set

� T 4 S S S Sof agreements x . Note that d F q for each S and that d s q forŽ . Ž Ž . S Ž .S f GC q . If S f GC q , then q f V S . Since q is coalitionally feasi-

T Ž .ble, for each i g S there is a coalition T with i g T and q g V T ,S T .whence d G q . For each coalition S, we need to construct a bargainingi i

S SŽ S . Sfunction f such that f d s q . It is convenient to distinguish twoŽ . S S Ž . Ž .cases. If S g GC q then d , q g V S . For y g V S , consider the ray

� Ž S S . < 4 Ž .y q l q y d 0 F l - ` . Since V S is strongly comprehensive, thisŽ . SŽ .ray meets the Pareto boundary of V S in exactly one point, call it f y .

Ž . SŽ . SFor y f V S , set f y s y. It is easily checked that the function fSŽ S . Ssatisfies our criteria to be a bargaining function, and that f d s q . If

Ž . SS f GC q , we take f to be an arbitrary bargaining function; the factS S Ž . SŽ S . S Sthat d s q g V S entails that f d s d s q . It is easily checked

� S4 � S4that x s q is a solution for the multilateral bargaining problem² :N, V, f , and that q is the reservation price vector of this solution. B

Although we have formulated this equivalence in terms of initiationvectors and reservation price vectors, it is easy to recast it in terms ofoutcomes. In particular, an outcome is consistent with a multilateralsolution if and only if it can be supported by a stationary subgame perfectequilibrium.

The proposal-making model also provides another framework in whichto think about the interpretation of outside options. Fix a multilateral

² : � S4bargaining problem N, V, f and a solution x with reservation price� S4vector q; let d be the corresponding set of outside option vectors. We

have defined dS as the highest payoff player i could obtain, given thei

ELAINE BENNETT178

� S4solution agreements s , in any coalition other than S; below we showŽ .that this definition is entirely consistent with stationary subgame perfect

equilibrium behavior in the proposal-making model.According to Theorem 3, we can find a stationary subgame perfect

equilibrium s of the proposal-making model whose initiation vector is q.Fix a player i and consider any node of the game tree at which player i hasthe initiative and is to make a proposal. The choice of this proposaldepends of course on player i’s expectations about how other players will

Ž .respond to every possible proposal assuming that other players follow s .In particular, for S any coalition containing i, the decisions to make someproposal to S and the choice of which proposal to make are influenced bythe set of acceptable proposals that could be made to coalitions other than

�Ž T .4 SS. Let T , y be the set of all such acceptable proposals, and let a beithe supremum of i’s payoffs over all these acceptable proposals; it seemsnatural to view aS as player i’s outside option relative to the coalition Siand the strategy profile s . We claim that aS s dS, player i’s outsidei i

Žoption as we have previously computed it with respect to the solution� S4.agreements x .

To establish the claim we need to show two things:

Ž . Ž .i If T / S is any coalition with i g T and T , y is a proposal forS Ž .which y ) d , then T , y will be rejected with probability 1.i i

Ž .ii for every « ) 0, there is a coalition T / S with i g T and aŽ .proposal T , y which will be accepted with probability 1 such that y )i

dS s « .i

Ž . Ž .To verify i , consider any coalition T with i g T and proposal T , ythat will be accepted with positive probability. If y - q for any j g U,j j

Ž .j / i, then player j will do better by rejecting the proposal T , y andobtaining the initiative, after which he will be able to obtain q . Hencej

T Ž T .y G q for each j g U, j / i. By definition, therefore, y F u q . Sincej j i iS � T XŽ . X S

Xd s max u q : T / S, it follows that y F d , as desired.i i T i iŽ . S � T XŽ T X . X 4To verify ii , fix « ) 0. Since d s max u q : T / S there is ai i

S T Ž T . T Scoalition T such that d s u q ; that is, the vector q rd is feasible fori i ithe coalition T. Strong comprehensiveness implies that there is a vector ywhich is feasible for the coalition T and satisfies the inequalities y ) dS

i iy « and y ) q for each j g T , j / i. Subgame perfection guaranteesj j

Ž .that the proposal T , y will be accepted with probability 1, as asserted.

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Ž .Bennett, E. 1988 . ‘‘Consistent Bargaining Conjectures in Marriage and Matching,’’ J. Econ.Theory 45, 392]407.

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