Journal of Economic Theory 136 (2007) 105–125www.elsevier.com/locate/jet

Coalitions, agreements and efficiency�

Effrosyni Diamantoudia,∗, Licun Xueb

aDepartment of Economics, Concordia University, 1455 de Maisonneuve Blvd West, Montreal, Que., Canada H3G 1M8bDepartment of Economics, McGill University, 855 Sherbrooke Str West, Montreal, Que., Canada H3A 2T7

Received 20 May 2002; final version received 21 February 2006Available online 12 July 2007

Abstract

If agents negotiate openly and form coalitions, can they reach efficient agreements? We address this issuewithin a class of coalition formation games with externalities where agents’ preferences depend solely onthe coalition structure they are associated with. We derive Ray and Vohra’s [Equilibrium binding agree-ments, J. Econ. Theory 73 (1997) 30–78] notion of equilibrium binding agreements using von Neumannand Morgenstern [Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944]abstract stable set and then extend it to allow for arbitrary coalitional deviations (as opposed to nested de-viations assumed originally). We show that, while the extended notion facilitates the attainment of efficientagreements, inefficient agreements can nevertheless arise, even if utility transfers are possible.© 2007 Elsevier Inc. All rights reserved.

JEL classification: C71; C72

Keywords: Coalition formation; Externalities; Efficiency; Agreements

1. Introduction

If agents negotiate openly and are able to form coalitions, can they reach efficient agreements?We address this issue within a class of simple coalition formation games with externalities whereeach agent’s preferences depend only on the coalition structure or partition of the agents. Arguably,if binding agreements can be written without any informational imperfections, then all the gainsfrom cooperation should be extracted. The resulting agreement must be Pareto-optimal; moreover,if utility is transferable, aggregate surplus must be maximized. Such an assertion encapsulates

� Parts of this work were conducted while the authors were visiting the Wallis Institute of Political Economy at theUniversity of Rochester and the Department d’Economia i d’Història at the Universitat Autònoma de Barcelona.

∗ Corresponding author.E-mail addresses: ediamant@alcor.concordia.ca (E. Diamantoudi), licun.xue@mcgill.ca (L. Xue).

0022-0531/$ - see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.jet.2006.02.012

106 E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125

the Coase [7] theorem. However, in Ray and Vohra [16], it is shown that when coalitions canform, efficiency can no longer be guaranteed. Indeed, the authors define the notion of equilibriumbinding agreements (henceforth EBA) for strategic form games, a more general framework thanours, and construct examples where the only agreements that can be reached are inefficient, evenwhen utility is transferable. This negative result 1 casts a shadow on the validity of Coase theorem[7] in environments where coalitions can form.

What is even more puzzling is that in Ray and Vohra’s [16] notion agents are sophisticated. Inparticular, they are assumed to be farsighted in that when contemplating a deviation, a coalitiontakes into consideration that further deviations may occur and that other deviating coalitions alsoapply similar reasoning. For farsighted agents, it is the final agreement their deviations lead tothat matters. Moreover, agents examine the credibility of the final outcome, thus, the notion isdefined consistently. One feature of EBA is the assumption of internal deviations, that is, onlya subset of an existing coalition can deviate. The negotiation process underlining EBA is asfollows: Starting with the grand coalition a subset of agents can break away anticipating thatother coalitions (subsets) may further break apart. While this feature makes a recursive definitionpossible, it precludes the possibility of coalition merging and renegotiation. As Ray and Vohra[16, p. 33] wrote,

We must state at the outset that our treatment is limited by the assumption that agreementscan be written only between members of an existing coalition; once a coalition breaks awayfrom a larger coalition it cannot forge an agreement with any member of its complement.Thus, deviations can only serve to make an existing coalition structure finer never coarser.This is also the assumption in the definition of a coalition proof Nash equilibrium. It mustbe emphasized that an extension of these notions to the case of arbitrary blocking is farfrom trivial.

It remains an interesting question whether inefficiency arises due to the restriction of inter-nal deviations. Within our framework, we extend the definition of EBA to allow for arbitrarycoalitional deviations and, hence, an open and unrestricted negotiation process. Starting with anycoalition structure (including the grand coalition), any coalition, not necessarily a subset of anexisting one, can deviate and bring about a new coalition structure. Put differently, even if agentstemporarily belong to some coalition they are not prevented from being in contact with membersof other coalitions and orchestrating a joint objection/deviation. With open negotiations it is nolonger necessary to start with the grand coalition as a coalition structure is reachable from anyother structure.

In order to formalize our extension we first reformulate Ray and Vohra’s [16] definition usingvon Neumann and Morgenstern [20] stable set. Such a reformulation is akin to that of coalitionproof Nash equilibrium (CPNE) 2 [3] by Greenberg [11] and it offers a simple definition of EBAin our framework. More importantly, the use of stable set enables us to deal with the circularitythat results from allowing arbitrary coalitional deviations while maintaining consistency as in theoriginal definition of EBA. Can inefficient outcomes emerge from an open, unrestricted negotiationas entailed by the new definition? We show that while the assumption of internal deviations hindersefficiency to some extent, inefficient agreements can still arise in open unrestricted negotiations.In particular, we identify a class of games where efficiency can be attained and construct a

1 Similar negative results also appear in Bloch [4] and Ray and Vohra [17] among others.2 Note that the difference between EBA and CPNE lies beyond the fact that the former considers binding agreements.

In the definition of EBA, agents are farsighted in that each coalition considers the final outcome its deviation leads to,while in the definition of CPNE, a deviating coalition takes its complement’s choice as given.

E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125 107

counter example where no agreement is efficient. We also analyze the consequence of transfers.Although transfers further facilitate the attainment of efficient agreements, we construct a robustexample where only inefficient agreements can emerge from open negotiations. The negativeresults reinforce the inefficiency puzzle posed by Ray and Vohra [16].

In the negotiation process underlining our extension of EBA, it is feasible for any coalition toform and object to/deviate from any coalition structure. However, being farsighted, a coalitionengages in a deviation if and only if it can ultimately benefit from doing so. Therefore, a coalitionstructure is “stable” if no coalition wishes to deviate, anticipating the final outcome its deviationmay lead to and a coalition structure is “unstable” as long as one coalition wishes to deviate,again anticipating the final outcome of its deviation. While our approach captures the foresightof selfish agents, it differs from the sequential “non-cooperative” approach of coalition formation(at least) in that we do not specify the exact order with which individual agents act. Moreover, inthe latter approach additional restrictions are often placed on the coalition formation process. Forinstance, Bloch [4] studies the same class of games as ours in a non-cooperative framework ofsequential coalition formation, while Ray and Vohra [17] 3 study a more general framework withendogenized payoff division. A common feature of their models is that once a coalition forms,the game is only played among the remaining players and established coalitions may not seek toattract new members nor break apart. The requirement for coalitions to commit plays an importantrole in determining the equilibrium coalition structures. 4 However, inefficient coalition structurescan arise in their equilibria as well.

The organization of the paper is as follows: After the presentation of the preliminaries inthe next Section we reformulate EBA in Section 3 and extend the notion in Section 4. Sec-tion 5 presents sufficient conditions for a game to admit efficient coalition structures and acounter example where no agreement is efficient. In Section 6 we discuss the impact of transfers.Section 7 concludes the paper. Detailed analysis of the first counter example is delineated inAppendix A.

2. Preliminaries

We start with some basic notations:

• Let N be a finite set of players.• A coalition S is a non-empty subset of N .• A partition of S ⊂ N is P = {S1, S2, . . . , Sk} such that

⋃kj=1 Sj = S and for all i �=

j, Si∩Sj = ∅ and P(S) is the set of partitions of S. A partition of N is called a coalitionstructure and P ≡ P(N) is the set of all coalition structures.

In the simple coalition formation games with externalities we study here, each player’s prefer-ences depend on the entire coalition structure. Formally, a simple coalition formation game withexternalities G is (N, {�i}i∈N) or (N, {ui}i∈N) where

• N is the finite set of players;

3 See also Ray and Vohra [18].4 Macho-Stadler et al. [14] analyze a Cournot oligopoly, a special case of the class of games we study here. They

assume that coalitions form by merging bilaterally, thereby relaxing the commitment of a formed coalition. They showthat if the number of firms is sufficiently large then the monopoly (grand coalition) is the equilibrium outcome.

108 E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125

• for all i ∈ N , �i is a complete, reflexive, and transitive binary relation on P , the set of coalitionstructures, i denotes the asymmetric part of �i (i.e., strict preferences) and ∼i the indifferencerelation; and

• for all i ∈ N , ui : P → � is i’s payoff function.

Throughout this paper we write P S Q for P, Q ∈ P and S ⊂ N , if P i Q or ui(P ) >

ui(Q) for all i ∈ S.The simple class of games we study here extends the class of “hedonic games” (see, e.g.,

[1,5,2,8]), where each agent’s preferences depend only on the coalition he belongs to. The classof games studied in this paper can also be viewed as a special class (with fixed payoff division) ofpartition function games introduced by Thrall and Lucas [19], which, in turn, is a special case ofnormal form TU games studied by Zhao [24]. Yi [22] studies a more restricted class of games thanours; in particular, he studies symmetric games and examines three models of coalition formation:simultaneous coalition formation model of Yi and Shin [23], sequential coalition formation modelof Bloch [4], and EBA of Ray and Vohra [16]. Moreover, Yi [22] classifies symmetric games intotwo categories, one with positive externalities (e.g., Cournot oligopoly, a class of public goodeconomies) and one with negative externalities (e.g., customs unions).

3. EBA and inefficiency

As discussed in the introduction, Ray and Vohra’s [16] notion of EBA is defined for strategicform games, a more general framework than ours. We shall first adapt their definition to oursetting. The negotiation process underlying EBA is as follows. Suppose the grand coalition N isunder consideration. A coalition S�N can break away from N and in doing so, it induces thecoalition structure {S, N\S}. This coalition structure is likely to be a temporary one since S orN\S may further break apart. More generally, given a coalition structure P ∈ P , any coalitionT �S, for some S ∈ P , can break away from S. In addition, once T breaks away from S, it cannotforge an agreement with any member of N\S; thus, deviations can only lead to finer coalitionstructures. Such an “internal” or “nested” deviation can be formalized as follows.

Internal coalitional deviation: Given a coalition structure P ∈ P and some S ∈ P , coalitionT �⊆ S, by breaking away from S, induces a temporary coalition structure given by P ′ = P \{S}∪{T �S\T }. We write P

T� P ′ in this case. Call Q a refinement of P if Q can be reached from P

through a sequence of nested coalitional deviations.

Agents are assumed to be farsighted: Each deviating coalition or perpetrator, following [16], isaware that further deviations may occur; thus, in contemplating a deviation, a coalition considersthe ultimate consequence of its deviation. Given the nature of the negotiation process, EBA canbe defined recursively.

• Start with the finest coalition structure, P̄ , of singleton coalitions. Since no further deviationsare possible, P̄ is the finest EBA.

• Now consider a coalition structure P such that PT� P̄ ; thus, P comprises all singleton coalitions

but one coalition of size 2. Let {ij} ∈ P . Then P is an EBA if and only if neither i nor j has anincentive to break away from the coalition and induce P̄ ....

E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125 109

• Consider Q ∈ P . Suppose EBAs have been defined for all refinements of Q. Then Q is anEBA if and only if it is not “blocked” by another EBA Q′. Q is blocked by Q′ if Q′ is an EBAand it can be can reached from Q via a sequence of internal coalitional deviations (i.e., Q′ is arefinement of Q) such that:

◦ each deviating coalition/perpetrator 5 belongs to Q′,◦ one of the deviating coalitions (the leading perpetrator) prefers Q′ to Q, and◦ any coalition structure attained by re-merging of the other perpetrators with their corre-

sponding residual coalitions (new coalitions in Q′ other than the perpetrators) is blockedby Q′ with one of these perpetrators as a leading perpetrator.

Given that deviations can only make a coalition structure finer, never coarser it is necessarythat the negotiation process starts from the grand coalition. The solution of a game is, therefore,defined to be the set of coarsest EBAs. Note that in the above recursive definition of EBA, thenotion of “blocking” itself is also defined recursively. To formalize this notion of blocking, wefirst introduce the following definition.

Definition 1. P ′ is RV-reachable from P via coalitions/perpetrators T 0, T 1, . . . , T k−1 if T j ∈ P ′

for all j = 0, . . . , k − 1 and P 0 T 0

� P 1 T 1

� P 2 T 2

� P 3 T 3

� P 4 · · · P k−1 T k−1

� P k where P 0 = P andP k = P ′. If, in addition, P j ≺T j P ′, for all j = 0, . . . , k−1, P ′ is said to sequentially dominate 6

P , or P ′ �Seq P , through coalitions T 0, T 1, . . . , T k−1.

Given that all perpetrators T 0, T 1, . . . , T k−1 are in the final coalition structure P ′, if P ′ isRV-reachable from P via T 0, T 1, . . . , T k−1 then P ′ is also RV-reachable via any re-orderingof T 0, T 1, . . . , T k−1. Moreover, given the set of perpetrators T 0, T 1, . . . , T k−1 in P ′ one canuniquely identify the set of residual coalitions, P ′\ (

P ∪ {T 0, T 1, . . . , T k−1

}), which are new

coalitions in P ′ but not active deviators. Therefore, any re-merging of perpetrators and theircorresponding residuals generates an intermediate coalition structure that is RV-reachable fromP via a subcollection of

{T 0, T 1, . . . , T k−1

}. Ray and Vohra’s [16] definition of EBA does not

simply employ sequential dominance as defined above for the following consideration: Suppose

that P ′ sequentially dominates P via P 0 T 0

� P 1 T 1

� P 2 T 2

� P 3 T 3

� P 4 · · · P k−1 T k−1

� P k where P 0 =P and P k = P ′, and P ′ is an EBA. To rule out P as an EBA on the basis that P ′ sequentiallydominates P implies that T 0, in contemplating its deviation, is certain/optimistic about the exactorder of other coalitional deviations. To circumvent this optimism, Ray and Vohra [16] define thefollowing dominance relation (RV-dominance, henceforth):

Definition 2. P ′ RV-dominates P , or P ′�R&V P , if there exist T 0, T 1, . . ., T k−1∈P ′ such that

(1) P ′ is RV-reachable from P via T 0, T 1, . . . , T k−1,(2) P ≺T P ′ for some leading perpetrator T ∈ {

T 0, T 1, . . . , T k−1}, and

(3) if Q = P̂ or Q is RV-reachable from P̂ via a subcollection of{T 0, . . . , T k−1

} \{T }, where

PT� P̂ , then Q is RV-dominated by P ′ with some S ∈ {T 0, . . . , T k−1}\{T } as the leading

perpetrator.

5 If a coalition in Q breaks into k coalitions in Q′ then k − 1 of these coalitions can be identified as deviating coalitionsor perpetrators while the remaining coalition is labeled as residual coalition.

6 In the original definition of sequential dominance in Ray and Vohra [16], the condition that T j ∈ P ′ for all j is notimposed.

110 E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125

Note that the above definition is recursive. In particular, condition (3) implies that no matter inwhich order coalitions in

{T 0, T 1, . . . , T k−1

} \{T } deviate, every intermediate coalition structuregenerated as a result is RV-dominated by P ′. Once RV-dominance is formulated, EBA can nowbe defined: The finest coalition structure, P̄ , of singleton coalitions is an EBA since no furtherdeviations are possible. P is an EBA if and only if it is not RV-dominated by another EBA.The fact that a coalition structure is an EBA if and only if it is not “defeated” by another EBAsignifies the consistency embedded in the definition of EBA. Such consistency is precisely whatis captured by “von Neumann and Morgenstern (vN–M) [20] stable set”. A vN–M stable set isa set of agreements, called a solution set, that is free of inner contradictions and accounts forevery element it excludes; in particular, no agreement in the solution set is defeated by anotheragreement in the same solution set and if an agreement is excluded from the solution set, it mustbe defeated by an agreement in the solution set. More formally,

vN–M stable set of (P, >): Let > be a binary relation on P and R ⊂ P . Then,

• R is vN–M internally stable for (P, >) if there do not exist P, P ′ ∈ R such that P ′ > P ;• R is vN–M externally stable for (P, >) if for all P ∈ P\R, there exists P ′ ∈ R such that

P ′ > P ; and• R is a vN–M stable set for (P, >) if it is both internally and externally stable.

Within our framework, Ray and Vohra’s [16] EBA can be reformulated as a vN–M stable set. 7

EBA (A reformulation): Let � be the vN–M stable set of (P, �R&V ). Then P is an EBA if andonly if P ∈ �. The coarsest coalition structures in � are referred to as the solution of the game. 8

Ray and Vohra [16] show that if P ′ �R&V P then P ′ �Seq P . The following lemma charac-terizes RV-dominance using sequential dominance.

Proposition 1. P ′ RV-dominates P if and only if there exist T 0, T 1, . . . , T k−1 ∈ P ′ (with T 0

being the leading perpetrator) such that

(a) P ′ sequentially dominates P via T 0, T 1, . . . , T k−1 and

(b) if Q = P̂ or Q is RV-reachable from P̂ via a subcollection of{T 1, . . . , T k−1

}, where P

T 0

� P̂ ,then P ′ sequentially dominates Q via a subcollection of {T 1, . . . , T k−1}.

Proof. Only if: Suppose P ′ RV-dominates P via a collection of perpetrators{E0, . . . , Ek−1

}.

We first show part (a). Let T 0 be the leading perpetrator and Q1 be such that P 0 T 0

� Q1. Then,by definition, P ≺T 0 P ′ and Q1 is RV-dominated via

{E0, . . . , Ek−1

} \{T 0}. Let T 1 be the

leading perpetrator and Q1 T 1

� Q2. Then by definition, Q1 ≺T 1 P ′ and Q2 is RV-dominated via{E0, . . . , Ek−1

} \ {T 0, T 1

}. Continuing in this fashion, we obtain T 0, . . . , T k−1 such that P ′

sequentially dominates P via T 0, . . . , T k−1. Part (b) can be proved by similar constructions.

7 This reformulation of EBA using a vN–M stable set cannot be directly generalized to strategic form games.8 The uniqueness of the vN–M stable set for (P, �R&V ) follows from the acyclicity of �R&V . Moreover, the vN–M

stable set of (P, �R&V ) is different from the core of (P, �R&V ) because of the presence of externalities. In the contextof cooperative games, when only internal deviations are allowed, the stable set and the core are closely related (see Ray[15] and Greenberg [12]).

E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125 111

Table 1

a b c d e

{123, 45} {234, 51} {345, 12} {451, 23} {512, 34}5, 5, 5, 9, 9 5, 5, 5, 9, 9 5, 5, 5, 9, 9 5, 5, 5, 9, 9 5, 5, 5, 9, 9

f g h i j

{12, 34, 5} {23, 45, 1} {34, 51, 2} {45, 12, 3} {51, 23, 4}4, 4, 4, 8, 8 4, 4, 4, 8, 8 4, 4, 4, 8, 8 4, 4, 4, 8, 8 4, 4, 4, 8, 8

If: The proof is by induction on �, the number of coalitions via which P ′ can be reached fromP . If � = 1, (3) in Definition 2 and (b) in Proposition 1 are superfluous and (a) in Proposition1 implies (1) and (2) in Definition 2. Now, assume that (a) and (b) imply (1)–(3) for all � < k.To show that (a) and (b) imply (1)–(3) for � = k, note that (a) implies (1) and (2) with T = T 0

as the leading perpetrator. Let PT� P̂ . Then by the induction hypothesis, P ′ �R&V P̂ via a

subcollection of{T 1, . . . , T k−1

}with some T̂ ∈ {

T 1, . . . , T k−1}

as the leading perpetrator;

moreover, for all Q such that Q is RV-reachable from P̂ via a subcollection of{T 1, . . . , T k−1

},

we have P ′ �R&V Q via a subcollection of{T 1, . . . , T k−1

}with some T̃ ∈ {

T 1, . . . , T k−1}

asthe leading perpetrator. Therefore, P ′ �R&V P . �

The following Corollary points to another alternative definition of RV-dominance that is notexplicitly recursive.

Corollary 1. P ′ RV-dominates P if and only if there exist perpetrators T 0, . . . , T k−1 such that

(1) P ′ is RV-reachable from P via T 0, . . . , T k−1,(2) P ≺T 0 P ′, and

(3′) if Q = P̂ or Q is RV-reachable from P̂ via a subcollection of{T 1, . . . , T k−1

}, where

PT 0

� P̂ , then there exists S ∈ {T 1, T 2, . . . , T k−1

} \Q such that Q ≺S P ′.

Proof. Only if: Following from Proposition 1 and the definition of sequential dominance.If: From any Q as defined in (3′) we can construct a sequence of coalitions such that P ′

sequentially dominates Q via this sequence of coalitions. Also, P ′ sequentially dominates Pgiven that P ′ sequentially dominates P̂ and P ≺T 0 P ′. Thus, by Proposition 1 P ′ �R&V P . �

The reformulation of EBA with vN–M stable set does not only offer a simple definition of EBAbut also enables us to extend it in various directions. One of the motivations for extending it wasthe inefficiency puzzle presented in [16]: although binding agreements are possible, inefficientoutcomes can, nevertheless, emerge. The following example illustrates the inability of playersto reach efficient binding agreements. 9 Table 1 shows the payoff vectors associated with 10partitions and all other partitions are assumed to yield 0 payoff vectors.

9 Ray and Vohra have a 3-player example of strategic form game where the only EBA are inefficient. Because of thesimpler framework we use, we need a 5-player game to illustrate the inefficiency of EBA.

112 E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125

It is easy to see that f, g, h, i, and j are all EBAs. So are those for which f, g, h, i, or j is nota refinement (for example, {135, 24}). On the other hand, a, b, c, d and e are not EBAs. Take

a, for example. a is not an EBA because a1

� g and a ≺1 g or a �R&V g. Therefore, all theEBAs of this game are inefficient. Indeed, f, g, h, i, and j are Pareto dominated by a, b, c, d ande, respectively and so are all other EBAs. One might ask, why are matters not renegotiated at thisstage to the dominating outcome? The assumption of nested deviations rules out the possibilityof such renegotiation. As Ray and Vohra [16, p. 51] wrote,

This is a serious issue that is neglected in our model, because we only permit “internal”deviations.

This is precisely the reason why we extend the definition of EBA to allow for arbitrary coalitionaldeviations. Equipped with the notion of vN–M stable set, we can define a notion consistently (asthe original definition) while allowing for arbitrary coalitional deviations.

4. Extended EBA

In this section we extend the notion of EBA by relaxing the assumption of internal deviationsand allowing arbitrary coalitions to deviate. Moreover, a deviating coalition is not constrainedto stay together; that is, we empower the deviating coalition with the ability to restructure itself.These features (of open negotiation) are introduced to facilitate the attainment of Pareto efficientcoalition structures. 10

Given a coalition structure, when a coalition of players, T ⊂ N , deviates by partitioning itselfin a certain way, the new coalition structure is the one consisting of the partition of T , all theunaffected coalitions, as well as all the disrupted coalitions. Formally, a coalitional deviation isdefined as follows:

Coalitional deviation: Given a coalition structure P = {S1, . . . , Sk} ∈ P , a coalition T ⊂ N canreorganize itself to some partition {T1, . . . , T�} ∈ P(T ). The resulting coalition structure, beforeany further regrouping and restructuring, is P ′ ∈ P such that:

(i) {T1, . . . , T�} ⊂ P ′, that is, the new partitioning of T is included in the new coalition structure.(ii) ∀j = 1, . . . , k, Sj ∩ T �= ∅ �⇒ Sj\T ∈ P ′, that is, the residuals of all coalitions affected

by the deviation of T are also included in the new coalition structure.(iii) ∀j = 1, . . . , k, Sj ∩ T = ∅ �⇒ Sj ∈ P ′, that is, all those coalitions that were unaffected

by the deviation of T remain members of the new coalition structure. 11

We write PT−→ P ′ to denote that “T induces P ′ temporarily from P”.

Once it is precisely defined what a coalition can (directly) induce, we proceed to define ourdominance relation. While myopic agents look only at the next step, farsighted players considerthe ultimate outcome of their actions. Thus, a coalition may choose to “deviate” to a coalitionstructure, which does not necessarily make its members better off, as long as its deviation leadsto a final coalition structure that benefits all its members; similarly, a coalition may choose not todeviate to a coalition structure it prefers if its deviation eventually leads to coalition structures that

10 Indeed, if we require the deviating coalition to stay together, the negative results remain while some of our positiveresults (e.g., Proposition 2) can no longer hold.

11 Note that points (ii) and (iii) can be written more concisely as follows: ∀j = 1, . . . , k, Sj \T �= ∅ �⇒ Sj \T ∈ P ′.

E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125 113

Table 2

a � b: b4−→ j

5−→{23, 1, 4, 5} N−→ a

a � c: c5−→ f

N−→ a

a � d: d5−→{14, 23, 5} N−→ a

a � e: e5−→ f

N−→ a

a � f : fN−→ a

a � g: g5−→{23, 1, 4, 5} N−→ a

a � h: h4−→{15, 2, 3, 4} N−→ a

a � i: i4−→{12, 3, 4, 5} N−→ a

a � j : j5−→{23, 1, 4, 5} N−→ a

a � p : pN−→ a

make its members worse off. Similar to the sequential dominance defined in the previous sectionthe following “indirect dominance” 12 captures foresight when arbitrary coalitions can deviate.

Indirect dominance: P ′ indirectly dominates P , or P ′ � P , if there exist a sequence of coalitionstructures P 1, P 2, . . . , P k ∈ P , where P 1 = P and P k = P ′, and a sequence of coalitionsT 1, T 2, . . . , T k−1 such that for all j = 1, . . . , k − 1:

(i) P j T j−→ P j+1 and(ii) P j ≺T j P ′.

An extended notion of EBA with unrestricted coalitional deviations can be defined consistentlyby using the vN–M stable set of (P, �), where the RV-dominance, �R&V , is replaced withindirect dominance, 13 �.

Extended EBA (EEBA): Let Q ⊂ P be a vN–M stable set of (P, �). P ∈ P is an extended EBAor EEBA if P ∈ Q.

Given a vN–M stable set Q, if negotiation commences with some (stable) agreement in Q, nocoalition can ultimately benefit from any deviation while starting with any (unstable) agreementin P\Q, at least one coalition can benefit ultimately by initiating a deviation. 14 Although thevN–M stable set of (P, �R&V ) has a similar interpretation, �R&V stipulates that deviations canonly be internal, and because of this the solution of a game is considered to be coarsest partitionsin the stable set. 15

Revisiting the example presented in the previous section we can see that efficiency can berestored. We will argue that a is an EEBA. Indeed, Q = {a} is a vN–M stable set of (P, �).Table 2 summarizes how all other coalition structures are indirectly dominated by a. Coalitionstructure p denotes any arbitrary coalition structure that is not listed in Table 1.

12 See also Harsanyi [13], Chwe [6] and Xue [21].13 Note that with arbitrary coalitional deviations we can no longer embed the conservative aspect of RV-dominance.14 Thus, if all agents expect Q, Q will be “self-enforcing”. In fact, subgame perfect equilibrium mapping (that specifies

the continuation equilibrium for each subgame) has a similar interpretation.15 For instance, the singleton partition is always in the stable set.

114 E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125

Table 3

j � f : f3−→{12, 3, 4, 5} N−→ j

j � g: g4−→{23, 1, 4, 5} N−→ j

j � h: h3−→{15, 2, 3, 4} N−→ j

j � i: i4−→{12, 3, 4, 5} N−→ j

j � p: pN−→ j

Due to the symmetry of the game, b, c, d and e are also EEBAs. Moreover, none of f, g, h, i andj is an EEBA. Assume in negation that there exists a vN–M stable set Q of (P, �) that supportsan inefficient coalition structure. Then, given the indirect dominance relation depicted in Table2, Q cannot contain any of the efficient outcomes due to internal stability. If, however, all a, b,c, d and e are excluded, Q must contain more than one coalition structures to account for theirexclusion. For example, if j ∈ Q, then j cannot dominate e. But, coalition structures f , g, h, i andj dominate each other (and all of them dominate p) hence they cannot coexist in Q. Table 3 showshow j indirectly dominates every other inefficient outcome. Similar arguments can be developedfor the rest of the outcomes due to the symmetry of the game.

Existence of vN–M stable set of (P, �) is not guaranteed. Case in point is a version of theroommate problem where {ij, k} i {ik, j} i {ijk} i {i, j, k} i {i, jk} and i prefers tohave j as a roommate, while j prefers to have k as a roommate and lastly k prefers to have i asa roommate. It is easy to see that no stable set exists for this game since {ij, k} � {jk, i} �{ik, j} � {ij, k}. While no two coalition structures can coexist in a stable set because of internalstability, a single coalition structure is externally unstable. In contrast, all structures involving pairsand the singletons structure are EBA since the cyclicality is assumed away through the assumptionof internal deviations: once {ij} are formed j is not allowed to collude with k. Fortunately, thepresence of cycles is not always a problem as can be seen in the example in Table 1. In thefollowing section we identify classes of games where both existence and efficiency problems areresolved.

Implicit in the definition of vN–M stable set is the optimism on the part of deviating coalitions:a coalition engages in a deviation as long as its members benefit from one of the final outcomes itsdeviation may lead to. An alternative behavioral assumption on deviating coalitions is caution: acoalition engages in a deviation if its members benefit from all the final outcomes its deviation maylead to. Under the assumption of caution, a more inclusive notion (than vN–M stable set) can bedefined and existence is guaranteed in our framework. See Chwe [6], Xue [21] and Diamantoudiand Xue [8], among others, for notions that are built on cautious behavior of deviating coalitions.

5. (In)Efficiency

5.1. Positive results

To proceed with the study of efficiency we need to distinguish between several notions ofefficiency.

Pareto efficiency: P ∈ P is Pareto efficient if there does not exist P ′ ∈ P such that P ≺N P ′.

Strong efficiency: P ∈ P is strongly efficient if there does not exist P ′ ∈ P such that∑

i∈N ui(P′)

>∑

i∈N ui(P ).

E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125 115

Table 4

Structuresize-wise

Perpetratorsize-wise

Per Firm Profit EBA

〈5〉 —— 120 ×

↓ 1〈4, 1〉 —— 1

36 , 19 ×

↓ 2〈2, 2, 1〉 —— 1

32 , 132 , 1

16 C↓ 1

〈2, 1, 1, 1〉 —— 150 , 1

25 , 125 , 1

25 ×↓ 1

〈1, 1, 1, 1, 1〉 —— 136 C

When transfer payments within each coalition are possible, we need to consider not onlycoalition structures but also payoff allocations. Consider a partition P ∈ P . A payoff allocationx ∈ �|N | is feasible for P if for all S ∈ P ,

∑i∈S xi = ∑

i∈S ui(P ). Pareto efficiency is thendefined as follows.

Pareto efficiency (with transfers): (P, x), where P ∈ P and x ∈ �|N | is feasible for P , is Paretoefficient if there does not exist (P ′, x′) such that P ′ ∈ P , x′ ∈ �|N | is feasible for P ′, and x′

i > xi

for all i ∈ N .

The following simple example from [16] illustrates the inability of players to reach stronglyefficient binding agreements.

Inefficiency puzzle—a Cournot oligopoly: The market demand is given by p = a −by, where p isthe market price and y is aggregate demand. We assume symmetric firms with constant marginalcost c. Consider a coalition structure P = {S1, S2, . . . , Sm} ∈ P . Firms within the same coalitioncooperate by maximizing their joint profit, while coalitions behave non-cooperatively across eachother. Then, the profit of each firm in Si is

�i = 1

si(m + 1)2

(a − c)2

b,

where si = |Si | for all i = 1, . . . , m.

Table 4 displays the per firm profit for the simple case of n = 5 and (a−c)2

b= 1. The last

two columns indicate which coalition structures survive the notion of Ray and Vohra [16]. Insymmetric games, coalitions need to be identified only by their sizes. For example, 〈4, 1〉 denotescoalition structures with a coalition of size 4 and a coalition of size 1.

〈2, 2, 1〉 associated with per firm profit of 132 , 1

32 and 116 is the coarsest EBA. 16 〈2, 1, 1, 1〉,

on the other hand is not an equilibrium coalition structure because the size-2 coalition will haveincentive to break apart. 〈2, 2, 1〉 is an EBA because if a coalition of size 2 breaks apart and induces〈2, 1, 1, 1〉, the other coalition of size 2 will also break apart as we already argued and they willend up at the finest structure 〈1, 1, 1, 1, 1〉 which makes the members of the first doubleton worse

16 To be more precise, all coalition structures with two size-2 coalitions and 1 singleton are EBAs.

116 E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125

off. Coalition structure 〈4, 1〉 is not an EBA since a coalition of size 2 will break away to induce〈2, 2, 1〉 which is an EBA and hence a credible deviation. Similarly, coalition structure 〈5〉 is notan EBA since one member will break away to temporarily induce 〈4, 1〉 and then a coalition ofsize 2 will break away to induce 〈2, 2, 1〉. Observe that the leading perpetrator 1 receives 1

16 under〈2, 2, 1〉 and 1

20 under 〈5〉. Similarly, 〈3, 2〉 and 〈3, 1, 1〉, that are omitted from the above table forsimplicity, are not EBAs.

Observe that none of the EBAs of the above game is strongly efficient. 17 However, thereexists a strongly efficient EEBA. Indeed, the grand coalition alone, with payoff of 1

20 per firm,constitutes a vN–M stable set of (P, �). Nevertheless, this example admits other EEBAs that arenot strongly efficient: Each permutation of the 〈2, 2, 1〉 coalition structure constitutes an EEBAas well, since 〈2, 2, 1〉 is a Pareto efficient partition that satisfies both conditions (a) and (b) ofProposition 1 below.

Proposition 2. Let P ∗ ∈ P be Pareto efficient. P ∗ is an EEBA if

(a) {1, 2, . . . , n} ≺N P ∗ and(b) for all P ∈ P such that P �= P ∗ and P �= {1, 2, . . . , n}, there is a coalition S ∈ P such that

|S| > 1 and P ≺i P ∗ for some i ∈ S.

Proof. We shall show that {P ∗} is a vN–M stable set of (P, �). Obviously, {P ∗} is internally

stable. We now need to show that P � P ∗ for all P ∈ P\P ∗. First, note that {1, 2, . . . , n} N−→ P ∗.Since {1, 2, . . . , n} ≺N P ∗, we have {1, 2, . . . , n} � P ∗. Let P 1 ∈ P be such that P 1 �= P ∗and P 1 �= {1, 2, . . . , n}. By condition (b), there is a coalition S1 ∈ P 1 such that |S1| > 1 and

P 1 ≺i1 P ∗ for some i1 ∈ S1. Thus, P 1 i1−→ P 2 ≡ {i1, S1\{i1}, P 1\S1}. Again by condition(b), there is a coalition S2 ∈ P 2 such that |S2| > 1 and P 2 ≺i2 P ∗ for some i2 ∈ S2. Thus,

P 2 i2−→ P 3 ≡ {i2, S2\{i2}, P 2\S2}. Continuing in this fashion, we can identify a sequence of

agents i1, i2, . . . , ik such that P 1 i1−→ P 2 i2−→ P 3 i3−→ · · · ik−→{1, 2, . . . , n} and P i� ≺i� P ∗ for all

� = 1, . . . , k. Given that {1, 2, . . . , n} N−→ P ∗ and {1, 2, . . . , n} ≺N P ∗, we have P 1 � P ∗. �

In a symmetric game, the grand coalition constitutes an EEBA under similar conditions.

Corollary 2. In a symmetric game {N} is an EEBA if:

(i) {N} is Pareto efficient,(ii) 〈1, 1, . . . , 1〉 is not Pareto efficient and

(iii) for all P ∈ P such that P �= {N} and P �= 〈1, 1, . . . , 1〉 there is a coalition S ∈ P suchthat |S| > 1 and P ≺i {N} for some i ∈ S.

An alternative corollary applies when agents’ payoffs are comparable.

Corollary 3. Consider a symmetric game (N, {ui}i∈N) where agents’ payoffs are comparable.{N} is an EEBA if:

(i) for all P, P ′ ∈ P , if |P | >∣∣P ′∣∣ then

∑i∈N ui(P ) <

∑i∈N ui(P

′) and(ii) for all P ∈ P , if S, T ∈ P and |S| > |T |, then ui(P ) < uj (P ) for i ∈ S and j ∈ T .

17 They are, however, Pareto efficient.

E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125 117

Condition (i) in Corollary 3 states that as coalition structures become coarser aggregate payoffincreases. Condition (ii) in Corollary 3 states that in a given coalition structure smaller coalitionsyield higher per member payoffs, implying thus, that coalition formation has positive externalities.Games with positive externalities were defined in Yi [22] and one property is that when any twocoalitions merge, others benefit. Note that Yi’s [22] definition of positive externalities is strongerthan conditions (i) and (ii) in Corollary 3.

The above results apply to the symmetric oligopoly model studied earlier in this section as wellas to a public good economy (see [16,22]).

5.2. A counter example

In the previous section, we identified sufficient conditions for a game to admit an efficient EEBA.However, as the example in Table 5 shows, it is possible for a game to have only inefficient EEBAs.This result reinforces the inefficiency puzzle posed by Ray and Vohra [16].

Observe that the first row is Pareto efficient while every other cell is inefficient. In particular,the first entry in each column Pareto dominates all other entries in the same column.

Table A1 in Appendix A shows how A (indirectly) dominates all other outcomes except D andd. Therefore, we cannot construct a stable set containing A alone since it cannot account for theexclusion of D and d . If A is included in an stable set, according to internal stability none ofthe outcomes it dominates can be included in the same stable set. Moreover, D and d cannot beincluded in the same stable set either, since they (indirectly) dominate A as illustrated in TableA1. By the symmetry of the game the same arguments extend to all the other efficient outcomesB, C, D and E. Hence no stable set can support (contain) efficient outcomes.

Table 5

A B C D E{512, 34} {123, 45} {234, 51} {345, 12} {451, 23}

19, 27, 32, 17, 22 19, 27, 32, 17, 22 19, 27, 32, 17, 22 19, 27, 32, 17, 22 19, 27, 32, 17, 22

a b c d e{512, 3, 4} {123, 4, 5} {234, 5, 1} {345, 1, 2} {451, 2, 3}

18, 26, 31, 16, 21 18, 26, 31, 16, 21 18, 26, 31, 16, 21 18, 26, 31, 16, 21 18, 26, 31, 16, 21

f g h i j{5, 12, 3, 4} {1, 23, 4, 5} {2, 34, 5, 1} {3, 45, 1, 2} {4, 51, 2, 3}

15, 25, 30, 15, 20 15, 25, 30, 15, 20 15, 25, 30, 15, 20 15, 25, 30, 15, 20 15, 25, 30, 15, 20

k � m n o{51, 23, 4} {12, 34, 5} {23, 45, 1} {34, 51, 2} {45, 12, 3}

15, 25, 30, 15, 20 15, 25, 30, 15, 20 15, 25, 30, 15, 20 15, 25, 30, 15, 20 15, 25, 30, 15, 20

p q r s t{5, 14, 2, 3} {1, 25, 3, 4} {2, 31, 4, 5} {3, 42, 5, 1} {4, 53, 1, 2}

15, 25, 20, 30, 15 15, 25, 20, 30, 15 15, 25, 20, 30, 15 15, 25, 20, 30, 15 15, 25, 20, 30, 15

u v w x y{5, 142, 3} {1, 253, 4} {2, 314, 5} {3, 425, 1} {4, 531, 2}

15, 25, 20, 30, 15 15, 25, 20, 30, 15 15, 25, 20, 30, 15 15, 25, 20, 30, 15 15, 25, 20, 30, 15

z{1, 2, 3, 4, 5}

15, 15, 15, 15, 15

118 E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125

Table 6

�124−→ u ∼

Nk

�124−→ f ∼

Nk

�124−→ p ∼

Nk

�12−→ h ≺

12k h

12−→ �

Table 7

�124−→ s ≺

12k s

12−→ f ∼N

k

12 ↙�

124−→ z ≺124

k z124−→ u ∼

Nk

z124−→ f ∼

Nk

z124−→ p ∼

Nk

z124−→ s �� k

Next we argue that Q = {k, �, m, n, o, z} is stable. For external stability we need to showthat all outcomes not in Q are (indirectly) dominated by some outcome in Q. Table A2 inAppendix A lists all such paths of dominance. To prove the internal stability of Q we haveto show that no element of Q indirectly dominates another element of Q. We start with k and�. First we argue that k �� �. Note that k 124 � and from � coalition 124 and its subsets caninduce u, f, p, h, s and z. But, u, f and p have the same payoffs for all players as k does, thus,once at u, f or p, no sequence will initiate with destination k. Note that although k 12 h fromh coalition 12 can only induce �. Table 6 summarizes the above information.

We still have to show that s and z will not lead to k. Observe that k 12 s and from s coalition12 can induce f and z. While f does not lead to k, z is still to be checked. But k 124 z and fromz coalition 124 can induce u, f, p and s which are all examined already. Table 7 summarizes thisinformation.

Next we show that � �� k in Table A3 in Appendix A. Similarly, we show that k does notdominate and is not dominated by o, m, and n, respectively, by outlining the blocked paths intables D, E and F in Appendix A. Moreover, it is already shown, in Table 7 above, that k �� z. It isalso easy to see that z cannot indirectly dominate k, �, m, n and o since no player strictly prefersz to any of them. The rest of the proof of internal stability follows from symmetry.

6. Transfers and (in)efficiency

When intra-coalitional transfers are possible, there are two ways of specifying a game with ex-ternalities. One is to start with our simple coalition formation game with externalities (N, {ui}i∈N)

and then allow for transfers, thereby treating ui(P ) as the “primitive payoff” of agent i for a givencoalition structure P . For example, when two firms with convex cost functions collude and max-imize their joint profits, their individual outputs and profits are also determined. In this case,

E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125 119

Table 8

{12}10, 5

{1, 2}6, 6

transfers enable them to reallocate their “primitive profit”. The second way of specifying a gameis to use the partition function game of Thrall and Lucas [19] with the joint payoff of each coali-tion in a coalition structure as the primitive. The main example (Table 9) in this section can bespecified in either way and our main conclusion remains valid. For convenience, however, weshall follow the first approach.

It is easy to construct examples where transfer payments within coalitions are necessary toachieve strongly efficient agreements. Consider a simple example with N = {1, 2} and the fol-lowing payoffs (Table 8).

{1, 2} is the only EBA and EEBA and it is Pareto efficient but not strongly efficient. If transfersare possible, then there exist strongly efficient EEBAs. For example, {12} with (8, 7) payoffallocation constitutes an EEBA.

When intra-coalitional transfers are possible, each agreement comprises two components:a coalition structure and a payoff allocation feasible for this coalition structure as defined inSection 5.1. Let (P, x) be an agreement (i.e., P ∈ P and x is feasible for P). If a coalition S devi-ates from this agreement, it can re-partition itself as in Section 4 and at the same time agree on howto share the proceeds within each of its coalitions, but it remains to be specified how the payoffsof agents in N\S are affected. One approach is to reset these payoffs to their primitive levels asspecified by (N, {ui}i∈N). The second approach is to allow each coalition to make conditionalstatements with respect to transfers in the sense of Ray and Vohra [17]—each coalition, uponformation, specifies transfers to be made for every coalition structure that contains it. In otherwords, when a coalition forms, its members not only agree on how to share their present payoffbut also agree on a contingency plan in the event their total payoff changes due to externalitiesimposed by the restructuring of its complement. For simplicity, we shall follow the first approachhere. Our main conclusion regarding the example in Table 9 still holds even if we take the secondapproach.

Transfers do facilitate the attainment of efficient agreement. For the counter example in Table 5,the following is a stable set for (P, �):

{({512, 34} , 30, 30, 18, 21, 18) , ({234, 51} , 18, 30, 30, 18, 21)} .

However, there is also a stable set with only inefficient elements. In fact, {k, l, m, n, o} with theoriginal payoffs remains a stable set with transfers. Some alternative assumptions as to whatensues a deviation also yield efficient stable sets.

It is then natural to ask the following question: Does there exist at least one efficient agreementfor every game when transfers are possible? Unfortunately, as the next example illustrates, itis possible for a game to admit only inefficient agreements (i.e., no stable set contains efficientelements).

Consider a game with five players, N = {i, j, k, l, m}. Table 9 shows the payoff vectorsassociated with 21 partitions and all other partitions are assumed to yield 0 payoff vectors.

120 E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125

Table 9

{12345} {ijk, lm} {ijk, l, m}10, 10, 10, 10, 10 15, 15, 15, 1, 1 15, 15, 15, 1, 1

It is easy to see that Pareto efficiency requires the formation of the grand coalition. However,the example admits the following stable set:

{({ijk, lm}, 15, 15, 15, 1, 1) , ({ijk, l, m}, 15, 15, 15, 1, 1) |for all distinct i, j, k, l, m } .

Note that all elements in the stable set are inefficient. Moreover, as the following Propositionestablishes there does not exist a stable set that contains efficient agreements.

Proposition 3. The above example does not admit an efficient EEBA if transfers are possible.

Proof. The proof is conducted in several steps. Let � be a stable set for (P, �).

Lemma 4. Let (P, x) ∈ �. Then there does not exist an agent i ∈ N such that xi �0.

First we show that if (P, x) is such that xi �0 for some i ∈ N , then there does not exist (Q, y)

such that yk �0 for all k ∈ N and (Q, y) � (P, x). There are two cases. First, if P is a partitionwith 0 payoff vector (that is, P is not one of the partitions in Table 9), there does not exist S ⊂ N

such that (Q, y) ≺S (P, x) and therefore no coalition will initiate a sequence from (Q, y) to(P, x). If P is a partition in Table 9, i must have participated actively in the inducement of (P, x)

from (Q, y) for otherwise, i gets a payoff of at least 1. Now, let S be the first deviating coalitioncontaining i in this process and suppose that (Q′, y′) was the status quo when S deviated. Theny′i �0 and (Q′, y′) ≺S (P, x), which imply that xi > 0, a contradiction.

Now, we proceed to show that if (P, x) is such that xi �0 for some i ∈ N , then (P, x) /∈ �. To

this end, let (Q, y) be such that (P, x)i→(Q, y) and assume in negation that (P, x) ∈ �. Then

the payoff structure of the game implies that either yj = 0 for all j ∈ N or yi = 1. If yj = 0for all j ∈ N , then we claim (Q, y) /∈ �. Indeed, if (Q, y) ∈ �, then by internal stability, �does not contain any (Q′, y′) such that y′

j > 0 for all j ∈ N . Then by the result in the precedingparagraph, � is not externally stable. Thus, (Q, y) /∈ �. Then there exists (P ′, x′) ∈ � such that

x′j > 0 for all j ∈ N and (P ′, x′) � (Q, y). Since (P, x)

i→(Q, y) and xi �0 but x′i > 0, we

have (P ′, x′) � (P, x), a contradiction. Now, consider the case that yi = 1. By internal stability,(Q, y) /∈ � since (P, x) � (Q, y). This leads to a similar contradiction as the previous case.

Lemma 5. � cannot contain two elements that are efficient and give strictly positive payoff toeach agent.

Let (N, x) and (N, y) be two elements in �. Then, there exists i such that xi �= yi . Without loss

of generality, let yi > xi . Then, (N, y) � (N, x) by the following sequence: (N, x)i→(P, z)

N→(N, y) where zj = 0 for j ∈ N .

Lemma 6. A single efficient element is not externally stable.

Assume in negation that (N, x) is the only element in �. By external stability, (N, x) �{ijk, l, m}. Then, some player among i, j and k gets more than 15 in (N, x). Without loss of gen-

E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125 121

Table 10

Structure Per-person Payoff EEBA

〈3〉 4 ×〈2, 1〉 5, 11 ×〈1, 1, 1〉 6 C

erality, let xi > 15. Now consider {jkl, m, i}, it must also be the case that (N, x) � {jkl, m, i}.Then some player among j, k and l gets more than 15 in (N, x). Without loss of generality, letxj > 15. Finally consider {klm, i, j} again it must be the case that (N, x) � {klm, i, j} andxk > 15. Thus, xl + xm < 5. Now consider ({klm, i, j}, y) with yk > xk, yl > xl , and ym > xm.Such an allocation is feasible because xi > 15 and xj > 15 (so xk < 20). It is easy to see that({klm, i, j}, y) �� (N, x).

Now we conclude the proof of Proposition 3 by showing that � does not contain any efficient el-ement. Assume otherwise that (N, x) ∈ �. By Lemmata 4, 5 and 6, we know that � contains (P, y)

such that P �= N and yj > 0 for all j ∈ N and xj > 0 for all j ∈ N . Obviously, P = {ijk, lm} orP = {ijk, l, m}. By internal stability it must be the case that xi = yi, xj = yj , and xk = yk . Thenxl + xm = 5. Assume w.l.o.g. that xk = min{xi, xj , xk} and consider ({klm, i, j}, z) ≡ (Q, z)

with zk > xk, zl > xl , and zm > xm. Then, (N, x) � (Q, z). By internal stability, (Q, z) /∈ �.By external stability, there exists (Q′, z′) ∈ � such that (Q′, z′) � (Q, z). Consequently, someagent, say i′, in {klm} strictly prefers (Q′, z′) to (Q, z), i.e., z′

i′ > zi′ . It follows that z′i′ > xi′ .

Then (Q′, z′) � (N, x) by the following sequence: (N, x)i′→(P ′, x′) N→(Q′, z′) where x′

j = 0for all j ∈ N . �

Lastly, the following example, a perturbed roommate problem, illustrates how transfers acrosscoalitions may be necessary to achieve strong efficiency (Table 10).

〈1, 1, 1〉 is the only EBA and EEBA; it is Pareto efficient but not strongly efficient. Inter-coalitional transfers are necessary to achieve strong efficiency in this case. A similar phenomenonwas identified as beneficial altruism in Greenberg [10] and Dréze and Greenberg [9].

7. Conclusion

In this paper we extend Ray and Vohra’s [16] notion of EBA to allow for arbitrary coalitionaldeviations. We show that despite this extension, the new notion, EEBA, while facilitating theattainment of efficient agreements, does not guarantee efficiency. In particular, we identify suffi-cient conditions for a game to admit an efficient EEBA and construct a counter example wherenone of the EEBAs is efficient. Moreover, we illustrate that the introduction of transfers doesnot guarantee efficiency either. A second counter example is provided where there is no efficientEEBA, despite the possibility of transfers. The negative results strengthen Ray and Vohra’s [16]inefficiency puzzle. Therefore, this subject warrants further study to explore solution conceptsand negotiation processes, among agents who exercise their decision power freely, that lead toefficient agreements.

Acknowledgments

We would like to acknowledge both institutions for their hospitality and the seminar participantsfor helpful comments. We would also like to thank Hubert Kempf, Per Krussel, two anonymous

122 E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125

referees and an associate editor for their comments and suggestions. Part of this research wasfunded by SSHRC and FQRSC.

Appendix A.

Table A1 shows how A (indirectly) dominates all other outcomes except D and d. Table A2shows external stability of Q.

Table A1

A � a : aN−→ A A � B : B

2−→�1N−→ A

A � f : fN−→ A A � b : b

12−→ fN−→ A

A � k : kN−→ A A � g : g

12−→ fN−→ A

A � p : pN−→ A A � � : �

4−→ fN−→ A

A � u : uN−→ A A � q : q

12−→ fN−→ A

A � z : zN−→ A A � v : v

124−→�2N−→ A

A � C : C125−→ A A � E : E

4−→ kN−→ A

A � c : c125−→ A A � e : e

24−→�3N−→ A

A � h : h125−→ A A � j : j

24−→�3N−→ A

A � m : m125−→ a

N−→ A A � o : o4−→ f

N−→ A

A � r : r125−→ a

N−→ A A � t : t24−→�8

N−→ A

A � w : w125−→ A A � y : y

24−→�4N−→ A

A � i : i13−→�1

N−→ A A �� D while

A � n : n123−→ f

N−→ A D � A : A345−→ D

A � s : s12−→ f

N−→ A A �� d while

A � x : x13−→�5

N−→ A d � A : A45−→ o

2−→ iN−→ d

Table A2

A � � : A5−→ � B � m : B

1−→ m

a � � : a35−→�6

N−→ � b � m : b14−→�9

N−→ m

f � � : f35−→�6

N−→ � g � m : g14−→�9

N−→ m

p � � : p35−→�7

N−→ � q � m : q14−→�10

N−→ m

u � � : u35−→�2

N−→ � v � m : v14−→�11

N−→ m

C � n : C2−→ n D � o : D

3−→ o

c � n : c25−→�12

N−→ n d � o : d13−→�1

N−→ o

h � n : h25−→�12

N−→ n i � o : i13−→�1

N−→ o

r � n : r25−→�13

N−→ n s � o : s13−→�15

N−→ o

w � n : w25−→�14

N−→ n x � o : x13−→�5

N−→ o

E � k : E4−→ k t � k : t

24−→�8N−→ k

e � k : e24−→�3

N−→ k y � k : y24−→�4

N−→ k

j � k : j24−→�3

N−→ k

E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125 123

Table A3

� �� k while � 35 k k5−→ g ∼

N�

k3−→ j ≺

23� j

23−→ k

k35−→ t ≺

23� t

23−→ g ∼N

�

23 ↙k

35−→ z ≺235

� z235−→ g ∼

N�

q ∼N

�

v ∼N

�

t �� �

Table A4

k �� o while k 24 o o4−→ f ∼

Nk

o24−→ z �� k see Table 7

o24−→ s �� k see Table 7

o2−→ i ≺

12k i

12−→ o

o �� k while o 135 k k3−→ j ∼

No

k135−→ y ∼

No

k135−→ t ∼

No

k15−→ g ≺

15o g

15−→ k

k135−→ r ≺

15o r

15−→ j ∼N

o

15 ↙k

135−→ z ≺135

o z135−→ y ∼

No

z135−→ j ∼

No

z135−→ t ∼

No

z135−→ r �� o

Let �i , where i = 1, . . . , 15, represent coalition structures that do not appear in Table 5 andwhose payoff is 0 for all the players. Obviously all �i’s are directly dominated by any elementof Q. The �i’s are provided below.

�1 = {13, 45, 2}, �4 = {135, 24}, �7 = {14, 35, 2}, �10 = {14, 25, 3}, �13 = {13, 25, 4},

�2 = {124, 53}, �5 = {245, 13}, �8 = {24, 35, 1}, �11 = {235, 14}, �14 = {134, 25},

�3 = {24, 51, 3}, �6 = {12, 35, 4}, �9 = {14, 23, 5}, �12 = {34, 25, 1}, �15 = {13, 24, 5}.

124 E. Diamantoudi, L. Xue / Journal of Economic Theory 136 (2007) 105–125

Table A5

k �� m m12−→ o ≺

24k o �� k see Table A4

while k 12 m m12−→ i ≺

12k i

12−→ o

m �� k k34−→ n ≺

134m n

134−→ w ∼N

m

while m 34 k n134−→ r ∼

Nm

n134−→ h ∼

Nm

n34−→ j ≺

34m j

34−→ n

n134−→ p ≺

34m p

34−→ h ∼N

m

34 ↙n

134−→ z ≺134

m z134−→ w ∼

Nm

r ∼N

m

h ∼N

m

p �� m

k34−→ j ≺

34m j

34−→ n �� m

Table A6

k �� n n12−→ � ≺

124k � �� k Tables 6 & 7

while k 12 n n12−→ h ≺

12k h

12−→ � �� k Tables 6 & 7

n �� k k45−→ m ≺

25n m

25−→ i ∼N

n

while n 45 k m25−→ g ≺

45n g

45−→ m

m25−→ q ≺

45n q

45−→ i ∼N

n

45 ↙m

25−→ z ≺245

n z245−→ x ∼

Nn

z245−→ s ∼

Nn

z245−→ i ∼

Nn

z245−→ q �� n

k45−→ g ≺

45n g ��n

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