Characterization results for aspirations in games with sidepayments

Mathematical Social Sciences 4 ( 1983) 229-24 1 North-Holland

229

CHARACTERIZATION RESULTS FOR ASPIRATIONS IN GAMES WITH SIDEPAYMENTS

Elaine BENNETT School of Management, State University uf New York at Buffalo, Buffalo, NY 14214, C’.S.A.

Communicated by J .C. Harsanyi Received 30 September 1981 Revised 12 November 1982

Recently several solution concepts for cooperative games have been defined on a new solution space - :he space of aspirations. This paper presents characterization results for this space. Unlike other solution spaces - the set of imputations and the set of IRPCs for a given coalition structure - the structure of the space is not trivial and vectors in the space do not have the interpretation of being the feasible payoff distributions for a given coalition structure. The results presented here provide a technique for calculating aspirations and an interpretation of aspirations as the set of all coalitionally rational payoff distributions for the game. We also show that, unlike the other solution spaces which are compact convex subsets of p, the set of aspirations is not convex

and indeed not even star shaped.

Key words: Aspirations; coalitional rationality; games with sidepayments.

Introduction

This paper presents characterizations and interpretations of the vectors in the space of aspirations.

Several solution concepts’ have be,on proposed (and reinvented) on the space of aspirations. The aspiration core was first proposed in Cross (1967), reinvented in Albers (1974) and carefully studied in Turbay (1977). It was also reinvented and applied to coalition economies in Wooders (1978). The aspiration bargaining set was proposed in Albers (1974) and reinvented in the author’s dissertation (1980). The equal gains solution was proposed in Bennett (198 1 a). Bennett ( 1980, 198 1 b) extended to the space of aspirations the solution concepts: the von Neumann-Morgenstern solution, the core, the bargaining set, the kernel and the nucleolus. The notion of aspiration and aspiration solution concepts have been extended to games without sidepayments (Albers (1979) and Bennett (1982a)).

The set of aspirations is the ‘feasible set’ from which each of these solution concepts selects payoff vectors. The set of aspirations is therefore the analog of the set of imputations for the solution concepts of the core and Shapley value and the set

’ I apologize for renaming the solution concepts which others have pioneered. A standard naming

conventioil is needed for these aspiration solution concepts.

0165-48%/83/%3.00 0 1983, Elsevier Science Publishers B.V. (North-Holland)

230 E. Bennett / Characterizations of aspirations

of IRPCs (individually rational payoff configurations) for the bargaining set and the kernel. The set of aspirations is not, however, the set of feasible payoff distributions for a given coalition structure, and the set considered as a subset of IR” does not have a ‘simple’ structure.

This paper presents results characterizing the set of aspirations and interprets the meaning of these characterizations. After introducing the aspiration approach to sidepayment games in Section 1, Section 2 provides a linear programming characterization of aspirations. This characterization shows that the set of aspirations is the union of compact convex sets. Two examples are presented to show that the space of aspirations is neither convex nor star-shaped. Section 3 shows that the set of aspirations can be interpreted as the set of all coalitionally rational payoff distributions for the given game. This result is also used to find for each game the smallest convert set containing the set of aspirations. Section 4 discusses the literature of related results on aspirations, and Section 5 contains the proofs of the theorems presented in this paper.

1. The aspiration oppro~ch

Let (A&o) denote a characteristic function game with sidepayments, where Av={l,..*, n) is the set of players and u : 2 N+ IR, is the characteristic function, which we assume is nonnegative but not necessarily superadditive. (IR, denotes the nonnegative real numbers.) S denotes a coalition and XE lRN is a payoff vector. The vector x% IRS is a pauo,ff distribution for S if x’(S) = u(S). (We use the notation x(S) in 3lace of C ,e 5 xi .) We say that S can afford x if x(S) I o(S).

?he aspiration approach corresponds to the following intuition on how characteristic %nction games are played.

The values of the players become known to the players (either before or during coalitiottal discussions). The players of various coalitions meet together to decide how the coalition’s value would be distributed *Aithin the coalition if it were to form. These ‘drsc~-,tisions’ may range from a fierce bargaining process to immediate agreement on &he ‘fair’ and ‘equitable’ distribution of payoff. After a series of such negotiations each agent selects his payoff demand (the price the player will charge for his participation in a coalition). He selects his price in the light of the prices he expects other agents to demand and in the light of his opportunities in various coalitions. After perhaps further bargaining over whether these prices are reasonable, a ‘stable’ vector of prices evolves and, given this price vector, the coalitions which can form are the coalitions that can ‘afford’ their memb& prices, and in each of these coalitions which actually forms each player receives his price.

I, 1. The feasibility arrd nco-surplus conditions

1 HIO assumptions about the way players set their prices are captured in the defini-

E. Bennett / Characterizations of aspirations 231

tion of aspiration. The first is that players will not price themselves out of the market, that is, no player will set his price so high that given the other player’s prices no coalition will be able to afford him. This condition is called the ‘feasibility’ condition. The second condition corresponds to the notion the players’ prices will rise whenever there is a surplus in any coalition. This means that whenever there is a coalition where the value of the coalition exceeds the sum of its members’ prices, then one or more of its members will raise their prices. This condition is called the ‘no-surplus’ condition.

Defhrition 1.1. The payoff vector x is an aspiration if it satisfies: (1) the feasibility condition: Vi&V, 3Ssi such that x(S)~~u(s), (2) the no-surplus condition: VS G !V, x(S)2 u(S).

Once an aspiration is selected, the coalitions which can afford the aspiration are the coalitions which are-likely to form in the game. The collection of coalitions which can afford the payoff vector x is called the generating collection of A-.

Definition 1.2. The genernting cokction of x, GC(x) is given by

GC(x)= (SI;N~X(S)~U(S)).

Notice that when x is an aspiration: (1) the generating collection of s forms a cover of the player set N, and (2) for SE GC(x), x(S) = u(S) and for S$ W(s), x(S) > U(S).

1.2. The aspiration bargaining set

One particularly appealing solution concept was first proposed in Albers (1974) and recognized in Bennett (1981 b) as the appropriate extension of the bargaining set of Aumann and Maschler (1964) to the space of aspirations. The solution concept is called the aspiration bargaining set.

The aspiration bargaining set corresponds to the following intuition on how bargaining over players’ prices might proceed. Suppose players have set their prices at X. Suppose further that in a coalition, S, which can afford .Y there are two players, call them 1 and 2, and that player 1 can threaten to form a coalition without 2 (to

convince 2 to give him a higher payoff). For player l’s threat to be believable, the alternative coalition must be able to meet l’s price while paying its other members their prices. (This requirement restricts alternative coalitions to be in U?(s).) If player 2 has no alternative coalition without 1, then presumably 2 would be willing to lower his price (so 1 can raise his) and thereby make their joint coalitions more attractive to 1. If, on the other hand, player 2 is also in an alternative coalition (in GC(x)) without 1, then presumably 2 can resist the demand for a higher payoff. If neither player has an irresistable threat, then we call the players partners. If every pair of players in every coalition which can afford s are partners, then s is in the aspiration bargaining set.

232 4% Bennett / Characterizations of aspirations

Notice that there are two ways in which a pair of players can be partners: (1) if neither player has a threat against the other (in this case, every coalition in W(x) which contains one of them actually contains both of them), or else (2) if each of them can resist the other”s threat. (In this case both players are in alternative coalitions in GC(x) without the other.)

Definition 1.3. The aspiration x is in the aspiration bargaining set if

VSeGC(x), Vi, jES: if ~S%GC(X) id’, j$S (i can threaten j) then WE GC(x) ie S”, je S” (j can resist the threat).

1.3. A voting example

Consider the following five-player game. Let the number of votes each player controls be given by w = (2,2,1,1,1). Any coalition that controls five votes or more can pass a bill. The minimal ‘winning’ coalitions are of two types. Either both players 1 and 2 form a coalition with one of players 3, 4 and 5, or else one of the @ayers 3,4 and 5. The minimal winning coalitions are respectively : [ 1,2,3], [ 1,2,4], [ 1,2,5] and [ 1,3,4,5], [2,3,4,5]. If we assign I to winning coalitions and 0 to losing coalitions, we have the following characteristic function:

1, C Wi25,

v(S) = ieS

0, otherwise.

Aspirations for this example include (1, 1, 0, 0, 0), (i, #, i, i, i) and (40, 1, 1, 1) The apiration bargaining set of this game contains a unique aspiration, (0.4,0.~, 0.2,0.2,0.2). Given that the players have selected these prices, only the coalitions that can afford these prices can actually form. In this example all of the minimr.l w%ing coalitions and only these coalitions are in its generating collection. Selectior df this aspiration ‘predicts’ that the minimal winning coalitions are the coaliticns which are likely to form. In any of these coalitions which does form, each player will receive his price. Suppose the coalition [ 1,2,3] forms. Players 1, 2 and 3 receive 0.4, 0.4 and 0.2, respectively. Since the agreement to form a coalition is viewed as a binding agreement, after [ 1,2,3] has formed, players 4 and 5 can no longer be part of a winning coalition, the highest payoff players 4 and 5 can receive is 0. Thus the final outcome of the game, given that [I, 2,3] forms, is ([l, 2,3], (41, [5]) with payoff distribution (0.4,O.q 0.2,0,0).

2,, Linear programming characterization of aspirations

This section provides a lineat programming characterization of aspirations, and

E. Bennett 1 Characterizations of aspirations 233

shows that for every game the set of aspirations is not empty. We then present two examples which shed light on the shape of the space of aspirations.

Theorem 2.1. The vector XE RN is an aspiration for the game (N, v) if and only if for some vector /3=(/l, ,..., fl,,) with OCbil1 for i= I,..., n, the vector x is a solution to the linear program:

Minimize: c P,Xi ieh:

Subject to: a constraint for each subset S in N:

x(S) 1 u(S)

for x unrestricted in sign.

For a particular vector of weights, 8, solutions of the linear program are payoff vectors x with the smallest /?-weighted sum such that the sum of the payoff demands for the agents of each coalition exceeds or equals the payoff to that coalition. As the vector of weights varies, the solutions to the linear program trace out the ‘lower’ boundary of the set of payoff demands such that the sum of the payoff demand of the agents of each coalition exceeds or equals the payoff to that coalition.

Using the linear programming characterization it is easy to show that the set of aspirations for every game is not empty.

Corollary 2.2. For each game (N, v) and each /3 for which 0 < pi I 1 for i = 1, . . . , n, the set of aspirations is not empty.

The solution space of a linear program is always a close convex set. Thus, for each 8, the set of aspirations that have least p-weighted sum is a convex set. Since each component of each aspiration is bounded by Es,, v(S), the set of aspirations is the union of compact convex sets. One can also show that for any game the set of aspirations is a connected set (see Section 4). One might also expect that the set of aspirations of each game was a convex set or at least star-shaped. However, as the following examples demonstrate, neither result is true.

Example 2.3. N=: ( 1,2,3) and v is given by

1

0, ISl=L v(S) = 12, 1s: =2,

12, S =N.

The set of aspirations of this game are the permutations of (A, 12 -A, 12 -A) for OSA 16. Consider the convex combination +(O, 12, 12) + +(12,0,12) = (6,612). The vector (6,612) is not an aspiration since there is nr> coalition containing agent 3 which can afford to pay its agents according to the aspiration. The set of aspirations for this game is therefore not convex.


Example 2.4. N = (1,2,3,4) and u is given by

u([l, 31) = 20, v([2,31) = 10, 0([3,4)) = 24,

WI = 0 for all other coalitions.

The set of aspirations of this game consists of three parts:

and

PI = {(20-a, lo-cr,cr,24-cr) for O<az~lO},

Pz = {(lo-Q,O, lO+oY, 14-a) for O<crllO}

P3 = {(0,0,20+a,4-cr) for OCar4).

We show that the set of aspirations PI U Pl U Pj is not star-shaped. Suppose to the contrary that the set were star-shaped. Then there would be an aspiration, call it X, for which every convex combination of it with any aspiration would also be an aspiration.

We hegin by showing that we must have x1 =x2 = 0. Since x is an aspiration for every i it must satisfy Xi L v([i]). We have assumed that the characteristic function is nonnegative, so Xi L 0 for every i. Now suppose that xl + 0. Then x1 > 0. We show that x1 >O leads to a contradiction. Consider another aspiration II = (0, 0,24,0) and lei y” = Ax+ (1 - i1)(0,0,24,0) represent the convex combinations of x and a. Since x1 ~0, necessarily yt > 0 for A > 0. Since we have assumed the set of aspirations to be star-shaped, y’ is an aspiration for every A. Since y” is an aspiration, there must be some coalition containing agent 1 which can afford y’. Since y: > 0 and [ 1,3] is the only coalition containing 1 with positive value, y “([ 1,3]) = 20 for all 0 < A I 1. Writing this condition in full we have

llxr+&+(l-1)24=20 for O<il=l.

Since bcjth x1 and x3 (by our previous argument) are nonnegative, this condition cannot hold for all O< A I 1. We therefore conclude that x1 could not have been positive arG therefore x1 =O. A similar argument for the same vector shows that x2=0.

We next show that x must be of the form (0, 0,24 - ar, CT) for 05 a I 24. Since x is an aspiration, x((3,4])r 24. To satisfy this condition and ensure that there exist coalitions containing 3 and 4 which can afford X, we must have x([3,4]) = 24, which means that we can write .K as (0, 0,24 - cr, a) for 0 5 a I 24. (In order for x to qualify as an aspiration, x must also satisfy CT 5 4; however, this restriction is not important in the following argument .)

We now show that x must be the vector (0, 0,20,4). Consider convex combinations of x with the aspiration a = (20, 10, 0,24). Let

y”=n(O,O,24-a,a)+(l-1)(20,10,0,24).

For all 05 A < 1, J$ > 0. Since the only positive-valued coalition containing 1 is


[ 1,3], we must have y I([ 1,3]) = 20 for 05 A < 1 in order for yA to be an aspiration. Thus,

y’([l,3])=(1 -A)20+A(24-a)=20 for O~A<l.

This equality is satisfied for OS 2 < 1 only by a! = 4. Therefore x = (0, 0,20,4). To complete our proof we show that there is a convex combination of x and the

(same) aspiration a :- (20,10,0,24) which is not as aspiration. Let y’ be given as above. Notice that, for O< A = 1, yi >O. Since the only positive-valued coalition containing agent 2 ii. f&3], for yA to be an aspiration it must satisfy: y’([2,3]) = 10 for O< A I 1. Writing out this condition we have

y”([&3])=(1 -A)lO+A(20)= 10 for OCA.5 1.

This condition, however, is not satisfied for any A > 0. We can therefore conclude that there is no aspiration for which all the convex combinations of it with any other aspiration are always an aspiration. We therefore conclude that the set of aspirations is not star-shaped.

3. Coalitional rationality of payoff distributions

3. I. Affluence, threats and payoff distributions

In cooperative games a coalition may be affluent enough that it can afford to pay its members payoffs which are so large that none of its subcoalitions could earn more on its own. Other coalitions are less affluent, so that no matter how the coalition’s payoff is distributed among its members, there is always a subcoalition which can earn more on its own.

Consider first the more affluent coalitions. Within each such coalition suppose a payoff distribution were considered in which some subcoalition was assigned less than it could earn alone. Then the subcoalition’s threat of breaking away and going off on its own should be sufficient to overturn the proposal. We can therefore expect that the payoff distributions which arise in the more affluent coalitions will at least satisfy coalitional rationality - that is, we expect a payoff distribution in which each subcoalition earns at least what it can on its own.

Less affluent coalitions cannot afford to keep internal peace by assigning coalitionally rational payoff distributions. In these coalitions it may be difficult to reach a consensus on the appropriate payoff distribution within the coalition. Even when such a distribution decision is reached, less affluent coalitions are susceptible to internal disruption when ‘underpaid’ subcoalitions decide to work on their own.

Now consider the bargaining over the payoff distributions which takes place during the play of the game. When discussing the payments the players should receive within one coalition, each player in the coalition has a basic threat, which

is not to cooperate in forming that coalition. Th e agcut’s threat is more credible if

there is another coalition in which the agent can hope to earn as much as he is

236 E. i3ennett / Characterizat%m cf aspirations

offered in the proposed coalition. Clearly, if the alternative coalition is an affluent coalition and the payment he hopes for is part of a coalitionally ratibnal payoff dis- trjbutlon, the agent’s threat is more credible than if his payment within the alternative coalition is not part of a coalitionally rational payoff distibution, since such payoff distributions are not stable and are therefore less likely to actually be realized. For this reason we expect the set of coalitionally rational payoff distributions to be relatively important in selecting the payoff distributions and coalitions which are realized in the course of the game (see also the arguments for coalitionally rational payoff distributions in the paper that originally introduced the bargaining set, Aumann and Maschler (1964)).

Next, we make precise our notion of affluence in the definition of coalitionally rational payoff distributions and present our characterization of aspirations as the set of coalitionally rational payoff distributions.

3.2. Aspirations and coalitionally rattonal payoff distributions

DefinPbn 3.1. The vector x% IF?’ is a payoff distribution for the coalition S if x’(S)= u(S), The payoff distribution x ’ for the coalition S is a coalitionally rational payoff distribution, CRPD, if the payoff distribution xs satisfies x(S’) 1 u(S’) for every S’ G S. We say that S admits a CRPD if there exists a payoff distribution xs which is a CRPD for S.

The next pair of theorems shows that the set of aspirations is, in an appropriate sense, the set of CRPDs for the game.

Theorenr 3.2. Let xs be a CRPD for some coalition, call it S, for the game (N, 1.0. There elsists an aspiration p for (N, U) for which xs =p Is.

The t rleorem above says that any CRPD of a coalition can be found as the restriction to *-he coalition of some aspiration for that game. The theorem below says that every a;&icaon is a patching-together of CRPDs.

Theorem 3.3. Let x be an aspiration for (N, u). The generating collection, GC(x), is a collection of coalitions for which USEGC(xI S = N and x 1 s is a CRPD for S for every coalition S E GC (x).

Notice that the payoff distribution xi = u([i]) is trivially a CRPD for the coalition [i). We therefore know that there is an aspiration, call it x, which assigns v[i] to agent i. Since all aspirations assign each coalition at least as much as it can earn on ito own,, we also know that no aspiration assigns i less than u[i]. The aspirations wmch assign to dndividual agents no more than they can earn alone therefore form the extreme points of the set of aspirations. Thus the set of aspirations lies within the convex hull of the set of points

(x 1 x is an aspiration and xi = u[i] for some i in N).


The proof of Theorem 3.2 also shows that the set of aspirations has the reduction property - that the set of aspirations of a reduced game is the restriction of the set of aspirations for the original game to the players remaining in the game.2

To say that xs is a CRPD for the coalition S is equivalent to saying that xs is in the core of the game (S, o IS) where o I,(S) = u(S) for X’ c S. Albers (1974) pro- vided theorems relating the core of o IS to the set of aspirations.

4. Related results in the literature

In Albers (1974) a series of results can be found concerning the shape of the space of aspirations. His Theorem 4.3 shows that the set of aspirations is the set of Pareto- miniminal points of the set of vectors which satisfy the ‘no-surplus’ condition. This result can be used to construct an indirect proof for the linear programming characterization.

In the proof of his Lemma 4.8,3 Albers provides an interesting construction which computes an aspiration extension of a given CRPD. Using my terms, his construction is as follows: Let xs be a CRPD for S. Without loss of generality, let S= (1, . . . . k) . For each i E S, set xi = xi ‘. For the remaining players define _i?; recursively. For given xj, j = 1, . . . , i - 1, define xi by

xi=min{o(S’)-_Y(S’-i) 1 kS’G {l,...,i}}.

The basic idea in this construction is that once a CRPD is fixed for a coalition, the maximum surplus generated by adding each player can be given to that player as his payoff. The resulting payoff vector is then an aspiration. Every different ordering of the players k + 1, . . . , n can produce a different aspiration which is consistent with the CRPD . ..‘. In general this calculation will not produce every aspiration which is consistent with the given CRPD.4

Recently, Bennett and Zame (1983) have constructed a transfer scheme which takes an arbitrary vector in IR”, to an aspiration. The transfer scheme corresponds to the following behavioral intuition. If a player’s price is so low (given the prices of the other players) that there is a surplus in some coalition containing him, then he will be willing to raise his price so as to absorb the surplus. If a player’s price is so high that no coalition can afford him, then he will (eventually) be willing to lower his price until some coalition can afford him. The transfer scheme is as

*For the game (N, u) for the aspiration x when the players in S are removed (S = .V- 5”). define o’(T) = maxSO z; s u(T U S’) - xs(S’).

‘Albers (1982) supplied the following corrections for the proof of his Lemma 4.8. In the first line of the proof, replace A(uK) E {XK IxEA(u)} with A(uK) 2 {xK / XEA(U)). In the last sentence, before “ja sog3r das RE B(u)” insert “falls xK E Core(uK)“.

41’he (indirect) proof given in this paper shows that the set of possible extensions of a given CRPI) is simply the set of aspirations of a particular reduced game (see Footnote 2).


follows. If at stage t (given x’) player i adjusts his price, then xft ’ becomes

Xl !+l = max I u(S)- C

jts’$ j+i xj’liES,SEN ,

1

r+l _ xi xi’ for j#i.

If this process begins at an arbitrary vector in II?: and proceeds - in any order - at least twice through the full set of players, it stops at an aspiration. As a byproduct of this transfer scheme, we show that there is a continuous map from I??“, onto the set of aspirations. In particular, this means that the set of aspirations is connected.

5. Proofs of the results

Proof of Theorem 2.1. We first prove sufficiency. For some p let x be a solution to the linear program. Using the second definition of aspirations to show that x is an aspiration, we only need to show that x(S)2 v(S) for all S and that each agent i belongs to some coalition S with x(S) = o(S). The constraints of the linear program provide the first condition. To verify the second, we only need to show that each agent is in some coalition S such that o(S) =x(S). If this does not hold for the agent j, for every coakion containing agent j, x(S) > u(S). This shows that

k= sG$gs x(S) - WI I

is greater than zero. Notice that ksxj, since one of the coalitions S containing j is the singleton [j]. Thus klxj-o([jJ)lxj.

Construct a new vector x’ as follows:

xi’ = Xi for i#j,

xi-k for i=j.

For x’ : o rnm;tructed x’(S) =x(S) L o(S) if j is not in S and by construction, if j is in S, x’+ ) =x(S) - k 2 u(S). This new vector x’ has lower @-weighted sum, and satisfies the constraints. This contradicts the hypothesis that x had minimal p-weighted sum.

This elegant proof of necessity is due to John Barrer. We need to show that every aspiration is a solution to the linear program for some /I = (/?I, . . . , /$,) where each bi satisfies OC fli 5 1. Let x be an aspiration for the given game and W(x) be 5le generating collection of x. For each i let C’i be the number of coalitions in GC(x) which contain i. Since, by property 1, GC(x) forms a covering of N, each Ci is strictly greater than zero. Let

P * Ci

i = max(Cj 1 jeN) ’

We show that for /I*, so constructed, is a to the program. By struction, OC&*S for each Now consider vector x’ satisfies the


constraints of the linear program so that x’(S) L o(S) for every coalition S. For any coalition S* in GC(x), x’(S*)zx(S*), since by definition of the generating collection x(S*) = u(S*) and x’ satisfies x’(S*) 1 o(S*). Sum these inequalities over all the coalitions in GC(x). We then obtain CleN max(Cj I&N) we have Cic,~ /?:A$= Cle,,, Bi+

C’~X;Z CieN CiXi. After dividing by Xi. We have proven that any vector

satisfying the constraints of the linear program has a /?*-weighted sum at least as large as that of x. From this we conclude that x minimizes /?+x subject to linear programming constraints and is therefore a solution to the linear program.

Proof of Corollary 2.2. Fix a game (N, U) and fix /? in the given range. Since every solution to the linear program of Theorem 2.1 is an aspiration, we need only show that there is always a solution to the linear program. Since whenever there is a feasible vector for a linear program there is an optimal vector for it, to complete the proof we only need to exhibit a vector which satisfies the set of constraints. Let x=(x),*.., x,) with xi = max[o(S) 1 S c N) for each i in N. Clearly for all SC N, x(S) 1 u(S). Thus x is feasible for the linear program.

Proof of Theorem 3.1. For the game (N,o), xs is the given CRPD for the coalition S. We construct a new game (IV’, 0’) from (N, o) by removing the players in S, in a certain way. We take aspiration for the new game, x”\‘, and show that the composition vector p = (xs, xNiS ) is an aspiration in the original game.

Without loss of generality let the coalition S have n - n’ agents for some number n’ and assume that the numbering of agents has been chosen so that the agents of S have the highest index numbers. That is, agents i = 1, . . . , n’ are not in S, and agents i=n’+ l,...,n are in S.

Define a new game, (N’, u’) as follows: Let N’= ( 1, . . . , n’) and define u : 2N’+ R, as follows: for TC N’ set

u’(T) = ~;t u(TUS’) -ss(S’).

Since u(T) L 0, and the subsets of S include the empty set, we also have v’(T) L 0. Let xN’e @’ be an aspiration for the game (N’, I/>. (Corollary 2.2 shows that aspirations exist for all characteristic function games .)

Let p = (xN; x’). We now show that p is an aspiration. First we will show that for every agent there is a coalition containing him which

can afford p. For i in S, S is a coalition such that p(S) =.&S) = u(S) since CRPDs are feasible

for their coalitions. For i not in S, we consider i in N’. Since .? is an aspiration for (.V, v), there is

a coalition T g N’ containing i such that

xN’(T) = v’(T). (1)

240 E. Bennett / Characterization of aspirations

By definition,

u’(T) = rn~ n(TU S’) - x’(S’).

Let S’ be a subcoillition llof S for which the indicated maximum is achieved. Then

NTUS’) = u’(T) +x’(S). (2)

Since, by (l), xN’(T) = v’(T), (2) becomes

but

so

v(TUS’)=x”“(T)+xS(S’),

p(TUS’)=xN*(T)+xS(S’),

v(TUS’) =p(TUS’),

and TUS’ is the desired coalition containing i. So .m we have shown that for every agent there is :a coalition containing the agent

that can afford p. To complete the proof that p is an aspiration we need to show that p(T)2 v(T) for every coalition T.

We now consider two cases. Case 1. Suppose T s S. Then p(T) =x’(T). Since xs is a CRPD for S, for every

subset T, x”(T)2 v(T). Therefore, for such a coalition T, p(T)= v(T). Case 2. Suppose T = ToU So with To E N’ and SO G S, and To + 0. Since

v’(To) = rnn (v(ToUS’)-x’(S’)),

it folloiys that

v’(To)~v(ToUSo)-xs(So).

Since I-~’ l IS an aspiration for (N’, v’), xN’( TO) 2 v’( To), we find

xN’(To)z v’(To)z v(ToUSo)-xs(So), so

xS(So)+~N’(To)~v(ToUSo).

By definition of the aspiratron p, this is just

or P(TOUSO)~ MToUSo),

P(T) 2 v(T).

This completes the proof of Case 2 and the theorem.

E%oof of Theorem 3.2. Since by the second definition of aspiration, for every i in N there exists a coalition, call it S, containing i such that x(S) = v(S) - these coalitions are all in GC(x) and clearly their union is ZV. To verify that x Is is a CRPD


for S we need to show that (1) x(S) = u(S) so that x Is is a payoff distribution for

S, and (2) x(S’) L: u(S) for every S’ in S. Since S is in GC(x), S satisfies x(S) I o(S) but since x is an aspiration, x(S) 2 v(S) for every S so clearly x(S) = o(S) and x(S’)Z u(S’) for every S’5; S.

References

Wa Albers, Zwei losungskonzepte fur kooperative mehrpersonspiele, die auf Anspruchsniveaus der Spieler basieren, OR-Verfahren (Meth. Oper. Res.) 21 (1974) l-13.

W. Albers, Core- and kernel-variants based on imputations and demand profiles, in: 0. Moeschlin and D. Palaschke, eds., Game Theory and Related Topics (North-Holland, Amsterdam, 1974).

Ws Embers, Grundzuge einiger Liisungskonzepte, die auf Forderungsnivaus der Spieler basieren, in: W. Albers, Bamberg and Selten, eds., Entscheidungen in Kleinen Gruppen (Hain, Khonigstein/Ts).

W. Albers, Private communication, 1982. R.J. Aumann, On the NTU value, Working Paper 105, Economic Series, Inst. for Mathematical Studies

in the Social Sciences, Stanford Univ., Stanford, CA, 1981. R.J. Aumann and M. Maschler, The bargaining set for cooperative games, in: M. Dresher, L.S. Shapley

and A.W. Tucker, eds., Advances in Game Theory, Annals of Mathematics Studies 52 (Princeton Univ. Press, Princeton, NJ, 1964).

E. Bennett, Coalition formation and payoff distribution in cooperative games, Ph.D. Dissertation,

Northwestern University, Evanston, IL, 1980. E. Bennett, The aspiration approach to predicting coalition formation and payoff distribution in side-

payment games, Working Paper 503, School of Management, State Univ. of New York at Buffalo,

Buffalo, NY, 1981a; Internat. J. Game Theory 12 (1) (1983). E. Bennett, The aspiration core, bargaining set, kernel and nucleolus, Working Paper 488, School of

Management, State Univ. of New York at Buffalo, Buffalo, NY, 1981b (revised 1983). E. Bennett, NTU games - the aspiration approach, Working Paper 523, School of Management, State

Univ. of New York at Buffalo, Buffalo, NY, 1982a. E. Bennett, A new approach to predicting coalition formation and payoff distribution in characteristic

function games, Working Paper 528, School of Management, State Univ. of New York at Buffalo,

Buffalo, NY, 1982b. E. Bennett and W. Zame, Aspiration transfer schemes, in progress, 1983. J. Cross, Some theoretic characteristics of economic and political coalitions, J. of Conflict Resolution

11 (2) (1967) 184-195. G. Turbay, On value theories for n-person cooperative games, Ph.D. Dissertation, Rice University,

Houston, TX, 1977. M. Woo&s, Quasi-cores and quasi-equilibria in coalition economics with transferable utility, Working

Paper 184, Economics Department, State Univ. of New York at Stony Brook, Stony Brook, NY, 1978

(with subsequent revisions).

Documents

Characterization results for aspirations in games with sidepayments