Unique stability in simple coalition formation games

Games and Economic Behavior 48 (2004) 337–354 www.elsevier.com/locate/geb

Unique stability in simple coalition formation games Szilvia Pápai Department of Finance, Mendoza College of Business, University of Notre Dame, Notre Dame, IN 46556, USA Received 10 January 2001 Available online 21 December 2003

Abstract We investigate the uniqueness of stable coalition structures in a simple coalition formation model, for which specific coalition formation games, such as the marriage and roommate models, are special cases that are obtained by restricting the coalitions that may form. The main result is a characterization of collections of permissible coalitions which ensure that there is a unique stable coalition structure in the corresponding coalition formation model. In particular, we show that only single-lapping coalition formation models have a unique stable coalition structure for each preference profile, where single-lapping means that two coalitions cannot have more than one member in common, and coalitions do not form cycles. We also give another characterization using a graph representation, explore the implications of our results for matching models, and examine the existence of strategyproof coalition formation rules.  2003 Elsevier Inc. All rights reserved. JEL classification: C71; C78; D71 Keywords: Coalition formation; Matching; Stability; Core; Strategyproofness

1. Introduction We study simple coalition formation games, where society is partitioned into coalitions, and players rank the coalitions that they are a member of. The most well-known examples of such simple coalition formation games are the matching problems, and in particular the marriage and roommate problems.1 Further coalition formation problems that may be examined in this simple form are team formation problems, group formation in hierarchies, partnership formation, etc. In this study, we consider all these coalition formation games in E-mail address: [email protected]. 1 For an introduction to this literature, see Roth and Sotomayor (1990).

0899-8256/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/j.geb.2003.09.004

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one general model, for which the specific coalition formation games can be seen as special cases that are obtained by restricting the coalitions that may form. This general model allows us to systematically explore the various coalition formation games, and obtain general results that pertain to all specific examples of coalition formation. Among the first proponents of restricting permissible coalitions is a paper by Kalai et al. (1979), in which they state that “Specifying a set T of coalitions which may form in order to block or improve, instead of assuming that all logically possible groups may actually cooperate, offers interesting possibilities for modeling different notions of stability and yields a variety of solution concepts of differing strengths” (p. 18). Kalai, Postlewaite, and Roberts go on to describe several examples that validate this approach. For example, one can represent a cooperation structure by a graph in which vertices represent players, where players who can cooperate with one another are linked in the graph, and then specify that the permissible coalitions are simply the connected sets of vertices in the graph— an approach due to Myerson (1977). Another example is given by syndicates of traders, where any permissible coalition contains either all or none of the members of a syndicate. Several further examples are related to the size of coalitions. A simple requirement one may impose is that only coalitions of less than a certain size form. This requirement had already been used and discussed in the contexts of large economies and bilateral trade, and led Leo Hurwicz to suggest that it would provide a way of recognizing the costs of coalition formation, even if in an ad hoc fashion. In the simple but general coalition formation model that we study, players rank the coalitions, which means that the value of a coalition to a member depends solely on the identity of the other members of the coalition,2 the “hedonic aspect” (Drèze and Greenberg, 1980), and players only care about the coalition they join. This model, which is also known as the hedonic coalition formation model, was first examined by Banerjee et al. (2001), Bogomolnaia and Jackson (2002), and Cechlárová and Romero-Medina (2001).3 The first two papers deal with the existence of stable coalition structures, and provide, among other results, restrictions on preference profiles that ensure the non-emptiness of the core.4 In the third paper preferences over coalitions are a priori assumed to be based on the first-ranked or last-ranked member of the coalition. Our study differs from these papers in three main aspects. One important difference, as discussed above, is that we consider this model as a generalization of more specific coalition formation models, such as the marriage and roommate models, and thus our focus 2 In most of the previous literature on coalition formation, personal preferences for membership in specific coalitions are usually not taken into account. Instead, an objective worth of each coalition is specified, which may be split among the members of a coalition; or each coalition chooses an alternative on its own for its members. The current model may be viewed, however, as a simplified version of earlier models. In particular, if the payoffs within each coalition are a priori determined (e.g., by some solution concept), or may be predicted in advance, then we can simply specify players’ preferences over coalitions, without making explicit either the potential alternatives for the coalitions or the payoff divisions among the members of each coalition. 3 More recent papers that study the hedonic coalition formation problem are Alcalde and Revilla (2001), Alcalde and Romero-Medina (2000), Barberà and Gerber (2003), Burani and Zwicker (2003), and Haeringer (2000). 4 In this model the core and stable outcomes are the same, since all coalitional deviations are ruled out by both concepts.

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is on the permissible coalitions, over which we allow all preferences. This is in contrast to the above papers which focus on restricting the available preferences. Another main difference is that, while the other papers primarily investigate the existence of stable coalition structures, we are interested in the uniqueness of stable coalition structures. The issue of a unique stable coalition structure has not been directly addressed previously; Banerjee et al. (2001) do, however, present a condition on preference profiles, called the top-coalition property, which is sufficient for the existence of a unique stable coalition structure. We discuss the relationship of their result to our results in Section 3.3. Finally, our paper differs from the above papers in that we are able to provide a complete characterization of the existence of a unique stable coalition structure in specific coalition formation models that are obtained by restricting the coalitions that may form. By contrast, the above papers provide sufficient conditions only for the existence of a stable coalition structure in a coalition formation problem. The primary goal of this paper is to identify coalition formation models that have a unique stable coalition structure. We provide a characterization of coalition formation models that guarantee the existence of a unique stable coalition structure. In particular, we show that a simple property of collections of permissible coalitions, the single-lapping property, is necessary and sufficient for unique stability (Section 3). This property is easy to understand, and it has an intuitively clear graph representation (Section 4). Implications of our characterization results for matching models, and in particular, for the marriage and roommate models, are explored in Section 5. Furthermore, we show that the uniqueness of stable coalition structures is very closely linked to the existence of strategyproof coalition formation rules, and investigate this relationship in Section 6.

2. Definitions There is a finite set of players N = {1, . . . , n}. We will refer to each nonempty subset of N as a coalition. Let Π = {S: S ⊆ N, S = ∅} denote the set of all coalitions in N .5 Furthermore, for all coalitions S ⊆ N , let [S] = {{i}: i ∈ S} denote the set of singletons for the members of S. A collection of permissible coalitions Π ∗ ⊆ Π is a subset of Π such that [N] ⊆ Π ∗ . Thus, any collection of permissible coalitions Π ∗ contains all singletons. It is natural to restrict our attention to these collections of coalitions, since in most contexts players cannot be coerced to join any coalition. A coalition formation model is defined by a collection of permissible coalitions Π ∗ . For example, Π ∗ = {S: S ⊆ N , |S| ∈ {1, 2}} is the roommate model. Each player i ∈ N has a set of preferences Ri . We assume that players only care about the coalition they join. Player i’s preferences Ri ∈ Ri strictly order all nonempty subsets S ⊆ N of N containing i. Preferences are strict, complete, and transitive. We will write S Pi S  to indicate strict preferences, and S Ri S  to indicate that either S Pi S  or S = S  . R = (R1 , . . . , Rn ) is a preference profile, where R ∈ R and R = "i∈N Ri . We will use 5 Since the set of players N is fixed throughout this paper, there is no need to use the index N .

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the notation top(Ri ) to indicate the top-ranked coalition according to Ri : top(Ri ) = S if S ⊆ N , i ∈ S, and for all T ⊆ N with i ∈ T , S Ri T . Given R ∈ R and S ⊆ N , we denote (Ri )i∈S by RS . We also write R−i = RN−{i} and R−S = RN−S . For a given coalition formation model Π ∗ ⊆ Π and Ri ∈ Ri , let Ri |Π ∗ ∈ Ri denote the restriction to Π ∗ , i.e., Ri |Π ∗ strictly orders the coalitions in Π ∗ that contain i such that the ordering of these coalitions by Ri is preserved, and all coalitions not in Π ∗ are ranked below {i}. We will also use the notations R|Π ∗ , Ri |Π ∗ , and R|Π ∗ , which denote similar restrictions of R, Ri , and R, respectively. A coalition formation problem is defined by a pair (Π ∗ , R|Π ∗ ). That is, a coalition formation problem consists of a coalition formation model Π ∗ and a preference profile which is restricted to the permissible coalitions Π ∗ .
kA coalition structure σ = {S1 , . . . , Sk }, with n
k
1, is a partition of N , i.e., t =1 St = N , where all St are pairwise disjoint. For all i ∈ N , σi is the coalition in σ that contains i. Let Σ denote the set of all coalition structures. Given a preference profile R ∈ R, a coalition S ⊆ N blocks σ ∈ Σ if for all i ∈ S, S Pi σi . A coalition structure σ is stable at R if there is no coalition that blocks σ , given R. We will say that a coalition formation problem (N, R) has a stable coalition structure σ ∈ Σ if σ is stable at R. Stable coalition structures can also be referred to as core coalition structures since the two notions are identical in this context. A coalition structure σ ∈ Σ is stable at R|Π ∗ if no coalition S ∈ Π ∗ blocks σ , given R|Π ∗ . Hence, a coalition structure that is stable at some R|Π ∗ is individually rational, since no singleton blocks it, which means that none of the players that are in a nonsingleton coalition prefer to stay on their own. A coalition formation model Π ∗ has a stable coalition structure if for all R ∈ R|Π ∗ , there exists a coalition structure σ ∈ Σ which is a stable coalition structure at R. A coalition formation model Π ∗ has a unique stable coalition structure if for all R ∈ R|Π ∗ , there exists a coalition structure, σ ∈ Σ, which is the unique stable coalition structure at R.

3. The single-lapping property: a characterization of unique stability 3.1. The single-lapping property Now we can state the property of coalition formation models, i.e., the property of collections of permissible coalitions, that characterizes unique stability. A coalition formation model Π ∗ ⊆ Π is single-lapping if the following two conditions hold. Condition (a) For all S, T ∈ Π ∗ such that S = T , |S ∩ T |  1. Condition (b) For all {S1 , . . . , Sm } ⊆ Π ∗ such that m
3 and for all t = 1, . . . , m, |St ∩ St +1 |
1, where we let Sm+1 := S1 , there exists i ∈ N such that for all t = 1, . . . , m, St ∩ St +1 = {i}. Condition (a) says that if there is an overlap between any two coalitions, there may be no more than one player who is a member of both coalitions, and hence every two coalitions may be at most single-lapping. Condition (b) is a cyclical single-lapping condition: it

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requires that if coalitions form a cycle in which every two neighbors have a common member, then all these coalitions have the same single member in common. Note that these two conditions can be combined by allowing m = 2 in Condition (b). 3.2. A characterization of unique stability Theorem 1 (Single-lapping property). A coalition formation model has a unique stable coalition structure if and only if it is single-lapping. Proof. Part 1. If a coalition formation model is single-lapping then it has a unique stable coalition structure. First we specify an algorithm that leads to the unique stable coalition structure for each preference profile R ∈ R|Π ∗ , given a single-lapping collection of permissible coalitions Π ∗ . Then we verify that the algorithm indeed always selects the unique stable coalition structure. Algorithm (Selection of the unique stable coalition structure).6 Let Π ∗ be a singlelapping coalition formation model. Fix R ∈ R|Π ∗ . We will identify the unique stable coalition structure σ ∗ (R) ⊆ Π ∗ . First we show that there exists S ∈ Π ∗ such that for all i ∈ S, top(Ri ) = S. Suppose, by contradiction, that such an S does not exist. Fix i ∈ N . Then [N] ⊆ Π ∗ implies that top(Ri ) = {i}, and thus there exists j ∈ top(Ri ) such that top(Rj ) Pj top(Ri ). Then there exists l ∈ top(Rj ) such that top(Rl ) Pl top(Rj ). Note that l = i, by Condition (a). Then there exists h ∈ top(Rl ) such that top (Rh ) Ph top(Rl ). Then h = j , by Condition (a), and thus h = i, by Condition (b). Continuing similarly, we get to a contradiction, since there is a finite number of players. Note that there may be several coalitions S such that for all i ∈ S, top(Ri ) = S, and that all these coalitions are disjoint. Let    and T1∗ := S. M1∗ := S ∈ Π ∗ : for all i ∈ S, top(Ri ) = S S∈M1∗

Furthermore, let Π2∗ := {S ∈ Π ∗ : S ∩T1∗ = ∅}. It is easy to verify that Π2∗ is single-lapping. Hence, we can apply the above argument to find   M2∗ := S ∈ Π2∗ : for all i ∈ S, top(Ri ) = S . In general, let Π1∗ := Π ∗ , and define iteratively, for t = 1, . . . , m,   Mt∗ := S ∈ Πt∗ : for all i ∈ S, top(Ri ) = S , Tt∗ :=



S,

S∈M1∗ ∪···∪Mt∗

and for t = 1, . . . , m − 1, Πt∗+1 := {S ∈ Π ∗ : S ∩ Tt∗ = ∅},

where m  n satisfies Tm∗ = N .

6 This algorithm exhibits some similarity to the top trading cycle algorithm due to David Gale, which selects

the unique core allocation in indivisible goods markets in which each individual owns one good (Shapley and Scarf, 1974; Roth and Postlewaite, 1977).

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S. Pápai / Games and Economic Behavior 48 (2004) 337–354 ∗ is a partition of N such that (M ∗ ∪ · · · ∪ M ∗ ) ⊆ Π ∗ . Let Note that M1∗ ∪ · · · ∪ Mm m 1 ∗ ∪ · · · ∪ Mm .

σ ∗ (R) = M1∗

Now we show that σ ∗ (R) is the unique stable coalition structure at R. Step 1a. σ ∗ (R) is the only conceivable stable coalition structure at R. For all σ ⊆ Π ∗ such that σ is a stable coalition structure at R, it must be the case that M1∗ ⊆ σ , since otherwise each coalition T ∈ M1∗ blocks σ . If m
2, this implies that for all σ ⊆ Π ∗ such that σ is a stable coalition structure at R, we must have M2∗ ⊆ σ , since, given that M1∗ ⊆ σ , each coalition T ∈ M2∗ would block σ otherwise. Continuing this way, we can show that the ∗. only coalition structure that is possibly stable at R is σ ∗ (R) = M1∗ ∪ · · · ∪ Mm Step 1b. σ ∗ (R) is a stable coalition structure at R. Suppose there exists S ∈ Π ∗ such that S blocks σ ∗ (R). Then S ∩ T1∗ = ∅, since for all i ∈ T1∗ and for all T ∈ Π ∗ with i ∈ T , σi∗ (R) Ri T . Therefore, S ∈ Π2∗ . Then S ∩ T2∗ = ∅, since for all i ∈ T2∗ and for all T ∈ Π2∗ with i ∈ T , σi∗ (R) Ri T . Continuing iteratively we can show that for all t = 1, . . . , m, S ∩ Tt∗ = ∅. Given that Tm∗ = N , we have S = ∅, which is a contradiction. This implies that σ ∗ (R) is stable at R. Part 2. If a coalition formation model has a unique stable coalition structure then it is single-lapping. Let Π ∗ ⊆ Π be a coalition formation model that has a unique stable coalition structure. Suppose it is not single-lapping. Step 2a. Suppose Condition (a) is violated. Then there exist S, T ∈ Π ∗ such that S = T and |S ∩ T |
2. Let i, j ∈ S ∩ T such that i = j . Let R ∈ R satisfy the following. For all l ∈ S − T , let Rl rank S first and {l} second. For all l ∈ T − S, let Rl rank T first and {l} second. Let Ri rank S first, T second, and {i} third. For all l ∈ (S ∩ T ) such that l = i, let Rl rank T first, S second, and {l} third. Note, in particular, that Rj ranks T first, S second, and {l} third. (v) Finally, for all l ∈ N − (S ∪ T ), let Rl rank {l} first.

(i) (ii) (iii) (iv)

Then both σ = {S} ∪ [N − S] and σ  = {T } ∪ [N − T ] are stable at R. Note, however, that R ∈ R|Π ∗ , and thus there is a unique stable coalition structure at R. Therefore, we have a contradiction. Step 2b. Suppose Condition (b) is violated. Then there exists {S1 , . . . , Sm } ⊆ Π ∗ with m
3 such that for all t = 1, . . . , m, |St ∩ St +1 |
1, where Sm+1 = S1 , and there does not exist i ∈ N such that St ∩ St +1 = {i} for all t = 1, . . . , m. Then Step 2a implies that for all t = 1, . . . , m, |St ∩ St +1 | = 1, and that there exists {T1 , . . . , Tk } ⊆ {S1 , . . . , Sm } with 3  k  m such that for all t = 1, . . . , k, there exists {it } = Tt ∩ Tt +1 , where we let Tk+1 = T1 , and all it are distinct.

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Let R ∈ R satisfy the following. (i) For all t = 1, . . . , k and for all l ∈ Tt +1 such that l = it , it +1 , where we let ik+1 = i1 , let Rl rank Tt +1 first and {l} second. (ii) For all t = 1, . . . , k, let Rit rank Tt +1 first, Tt second, and {it } third.
(iii) Finally, for all l ∈ N − kt=1 Tt , let Rl rank {l} first. First note that R ∈ R|Π ∗ , and thus there is a unique stable coalition structure at R. Let σ ∈ Σ be the unique stable coalition structure at R. Then there exists t = 1, . . . , k such that Tt ∈ σ ,
say T1 ∈ σ , since for all t = 1, . . . , k, and for all l ∈ Tt , Tt Pl {l}, and for all T  ∈ Π ∗ − kt=1 {Tt } and for all l ∈ T  , {l} Rl T  . Then T2 ∈ / σ , since T1 ∩ T2 = {i1 }. Then T2 blocks σ unless T3 ∈ σ , which in turn implies that T4 ∈ / σ . Continuing this way we / σ , given get a contradiction if k is odd, since in this case Tk ∈ σ , which implies that T1 ∈ Tk ∩ T1 = {ik }. Therefore, k is even, and σ ⊇ {T1 , T3 , . . . , Tk−1 }. In this case, however, a similar argument implies that there exists another stable coalition structure at R with respect to Π ∗ , σ  , such that σ  ⊇ {T2 , T4 , . . . , Tk }. Thus, we have a contradiction, since there is a unique stable coalition structure at R. 2 3.3. Restrictions on preference profiles versus restrictions on coalitions It is important to understand the relationship of restricting coalitions to restricting preferences, especially since the literature on hedonic coalition formation has so far only focused on the latter. Restricting coalitions is formally equivalent to restricting preferences in a certain form (whereas preference restrictions may take any form). Namely, coalition restrictions are restrictions on preferences in the form of ranking coalitions that are not permissible “low” in the preference ordering, which would prevent them from being chosen by reasonable coalition formation rules. One convenient way of imposing this restriction, assuming that individual rationality is required (i.e., no player is forced to join some coalition), is that not permissible coalitions cannot be ranked above {i} (i.e., staying alone) for any player i in the coalition. Thus, if a coalition is ranked above {i} for some player i, then it can be ranked anywhere by all players in this coalition, since in this case it is a permissible coalition in the coalition restriction context. We can adopt the definition of the single-lapping property to a coalition formation problem in a natural way. First, for all R ∈ R, let   A(R) = S ⊆ N: there exists i ∈ S such that S Ri {i} denote the set of permissible coalitions at R. Then a coalition formation problem R ∈ R is single-lapping if A(R) is single-lapping. Theorem 1 has an immediate implication for the existence of a unique stable coalition structure for a coalition formation problem. Namely, that if a coalition formation problem is single-lapping then it has a unique stable coalition structure. This follows from Part 1 of Theorem 1. The only property that has been considered in the literature so far which concerns unique stability is the top-coalition property proposed by Banerjee et al. (2001). They show that it

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is a sufficient condition for the existence of a unique stable coalition structure in a coalition formation problem. Given a coalition S ⊆ N , a coalition T ⊆ S is a top-coalition of S if for all i ∈ T and all T  ⊆ S with i ∈ T  , T Ri T  . A coalition formation problem satisfies the top-coalition property if for every coalition S ⊆ N there exists a top-coalition of S. The top-coalition property is proposed in the preference restriction framework, and cannot be trivially adopted to the coalition restriction framework, since it is not a condition in the form described above (i.e., restricting certain coalitions to be ranked “low”). Nonetheless, the top-coalition property turns out to be closely related to the single-lapping property. This relationship is described in the following lemma. Lemma 1. Fix a coalition formation model Π ∗ ⊆ Π . Then Π ∗ is single-lapping if and only if all R ∈ R such that A(R) = Π ∗ satisfy the top-coalition property. The two parts of this lemma are proved using the proofs of the two parts of Theorem 1. Proof sketch. Part 1. If Π ∗ is single-lapping then all R ∈ R such that A(R) = Π ∗ satisfy the top-coalition property.7 Let Π ∗ ⊆ Π be single-lapping. Fix R ∈ R such that A(R) = Π ∗ and fix S ⊆ N . Let Π be the set of those permissible coalitions at R which only contain members of S:   Π = T ⊆ S: there exists j ∈ T such that T Rj {j } . Note that Π ⊆ A(R) = Π ∗ . Then, since Π ∗ is single-lapping, Π is also single-lapping. Then we can show that there exists T ∈ Π such that for all i ∈ T , top(Ri |Π ) = T , using a similar argument as in the algorithm in the proof of Part 1 of Theorem 1. This means that T is a top-coalition of S, which is what we needed to show. Part 2. If all R ∈ R such that A(R) = Π ∗ satisfy the top-coalition property then Π ∗ is single-lapping. The proof of this part of the lemma runs very similarly to the proof of Part 2 of Theorem 1. We suppose, in turn, that Conditions (a) and (b) in the definition of the single-lapping property for coalition formation models is violated, and specify a preference profile R in each case such that A(R) = Π ∗ and R does not satisfy the top-coalition property. In particular, we can use the preference profiles specified in Steps 2a and 2b of the proof of Theorem 1, respectively, with the following modification. When Condition (a) is violated, for all S  ∈ Π ∗ such that |S  |
2 and S  ∈ / {S, T }, let S  Pi {i} and {j } Pj S  , ∗ where i, j ∈ S. Then A(R) = Π and S ∪ T does not have a top-coalition. Similarly, when  / {T , . . . , T }, let Condition (b) is violated, for all S  ∈ Π ∗ such that |S  |
2 and 1 k
k S ∈   ∗ S Pi {i} and {j } Pj S , where i, j ∈ S. Then A(R) = Π and t =1 Tt does not have a top-coalition. Therefore, R does not satisfy the top-coalition property in either case, which is a contradiction. 2 7 Given that if Π ∗ is single-lapping then all of its subsets are single-lapping, this result also holds with ⊆ instead of =.

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Given Lemma 1, we can state Theorem 1 using the top-coalition property instead of the single-lapping property. Theorem 1 (Top-coalition property). A coalition formation model Π ∗ has a unique stable coalition structure if and only if all R ∈ R such that A(R) = Π ∗ satisfy the top-coalition property. One direction of Theorem 1 , which says that if all R ∈ R such that A(R) = Π ∗ satisfy the top-coalition property then Π ∗ has a unique stable coalition structure, follows from the sufficiency result of Banerjee et al. (2001, Theorem 2), which says that if R ∈ R satisfies the top-coalition property then it has a unique stable coalition structure. This means that the Banerjee et al. (2001) result together with Part 1 of Lemma 1 implies Part 1 of Theorem 1. The other direction of Theorem 1 and Part 2 of Theorem 1, however, are not related to the Banerjee et al. (2001) result. Indeed, the top-coalition property is only a sufficient condition for the existence of a unique stable coalition structure for a given coalition formation problem,8 while the single-lapping property is a necessary and a sufficient condition for the existence of a unique stable coalition structure for a given coalition formation model. The single-lapping property is both more intuitive and more appropriate than the topcoalition property in the coalition restriction framework studied in this paper. The singlelapping property gives explicit restrictions on coalitions; i.e., no more than one common member for any two permissible coalitions, and no coalition cycles are allowed. These are intuitive and explicit restrictions on permissible coalitions. By contrast, the top-coalition property says that, for a given preference profile, for each set of players there is always a best (“top”) coalition, in the sense that it is everyone’s first choice, and this is required to hold for each preference profile that is restricted to appropriate permissible coalitions. While the single-lapping property is more clear and appropriate in the coalition restriction context, the top-coalition property, which imposes restrictions on preference profiles, is a useful property for the algorithm that selects the unique stable coalition structure at each preference profile, as can be seen from the algorithm presented in the proof of Part 1 of Theorem 1.

4. The tree structure representation: an alternative characterization of unique stability 4.1. The tree structure representation Coalition formation models can be represented by a graph. The graph representation of single-lapping coalition formation models, which we call tree structure representation, 8 In fact, for the Banerjee et al. (2001) result, the top-coalition property may be replaced by the following weaker property which is also sufficient for the existence of a unique stable coalition structure at a preference profile R: there exists a partition of the players, {S1 , . . . , Sk }, such that for all k, Sk is a top-coalition of
N \ ( t
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provides further insights into coalition formation models that have a unique stable coalition structure. A graph G = (N, E) is given by the set of vertices N , representing the set of players, and a set of edges E, where each edge is an unordered pair of distinct players in N . Thus, a pair {i, j } ∈ E is an edge between i and j . Since we allow for multiple edges between a pair of players, identical edges (e.g., two edges between i and j ) are indexed (e.g., {i, j }1 , {i, j }2 , etc.). A path of G is a sequence {i1 , i2 }l1 , {i2 , i3 }l2 , . . . , {ik−1 , ik }lk−1 of distinct edges in G, such that all players i1 , . . . , ik are distinct, except, possibly, for i1 and ik . We say that this path connects players i1 , . . . , ik . If i1 = ik and the path connects at least three distinct players then it is a cycle. A graph is connected if there is a path between every pair of distinct players in N . A graph that is not connected is the union of two or more connected subgraphs, where different connected subgraphs have no vertex in common. These disjoint subgraphs are called the connected components of the graph. Every path {i1 , i2 }l1 , {i2 , i3 }l2 , . . . , {ik−1 , ik }lk−1 represents a coalition S ⊆ N , where S = {i1 , . . . , ik }. For simplicity, we will also say that the edge set ES = {{i1 , i2 }l1 , {i2 , i3 }l2 , . . . , {ik−1 , ik }lk−1 } represents S. For each coalition formation model Π ∗ ⊆ Π , we construct its associated graph G = (N, E) as follows. If Π ∗ = [N], we have E = ∅, that is, the edge set is empty in G. Otherwise, let Π ∗ = [N] ∪ {S1 , . . . , Sm }, without loss of generality, where coalitions S1 , . . . , Sm are distinct and non-singleton. Then the edge set E is such that it can be partitioned into {E1 , . . . , Em }, where for each t = 1, . . . , m, Et represents St . It is important to note that such a representation of an arbitrary collection of coalitions may require multiple edges between a pair of players. Thus, in a graph associated with Π ∗ there is a distinct path representing each non-singleton coalition in Π ∗ , in the sense that each edge belongs to exactly one coalition. A graph associated with a collection of permissible coalitions may not be unique, since it depends on which members of a coalition are connected by edges. We can ensure the uniqueness of the graph G by following the convention that two players i, j ∈ S, where i < j , are connected by an edge in G to represent the coalition S ∈ Π ∗ only if there is no i  ∈ S such that i < i  < j . In order to simplify the exposition, we will assume in the following that this convention is used for all graphs associated with some coalition formation model Π ∗ ⊆ Π . Thus, for each coalition formation model Π ∗ there exists a unique graph G = (N, E) associated with Π ∗ . We will say that a coalition formation model has a tree structure representation if its associated graph contains no cycles and no multiple edges between any two vertices. Next, we give an example of a single-lapping coalition formation model which has a tree structure representation. Example 1. Let S1 = {1, 2, 13}, S2 = {2, 4}, S3 = {3, 5, 6}, S4
= {5, 7, 8, 9, 12}, S5 = {9, 10, 11}, and S6 = {9, 14}. Let Π ∗ = {S1 , S2 , S3 , S4 , S5 , S6 } ∪ ( 14 i=1 {i}). We shall illustrate that both Condition (a) and Condition (b) of the single-lapping property are satisfied by Π ∗ , and thus the collection of permissible coalitions in the example is single-lapping. Note first that pairs of coalitions which have a non-empty intersection have the following members in common: S1 ∩ S2 = {2},

S3 ∩ S4 = {5},

S4 ∩ S5 = {9},

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Fig. 1. The associated graph of Π ∗ in Example 1.

S4 ∩ S6 = {9},

S5 ∩ S6 = {9}.

Since every other pair of coalitions has an empty intersection, and since every pair listed above has one member in common, we can conclude that Condition (a) holds. Furthermore, notice that the cyclical condition in (b) is only satisfied by the set of coalitions {S4 , S5 , S6 }. Then one can easily check that Condition (b) is satisfied, given that S4 ∩ S5 = S4 ∩ S6 = S5 ∩ S6 = {9}. The associated graph of Π ∗ appears in Fig. 1. Since this graph contains no cycles and no multiple edges between any two vertices, Π ∗ has a tree structure representation. 2 4.2. An alternative characterization of unique stability Theorem 2 (Tree structure representation). A coalition formation model has a unique stable coalition structure if and only if it has a tree structure representation. We will prove the following lemma, which, together with Theorem 1, yields Theorem 2. Lemma 2. A coalition formation model is single-lapping if and only if it has a tree structure representation. Proof. Part 1. If a coalition formation model is single-lapping then it has a tree structure representation. Fix a single-lapping coalition formation model Π ∗ ⊆ Π . Let G = (N, E) be the graph associated with Π ∗ . Suppose G contains a set of edges C ⊆ E which either consists of three or more vertices that form a cycle or two vertices that have at least two edges between them. Recall that E can be partitioned into {E1 , . . . , Em } such that for each z = 1, . . . , m, Ez represents Sz ∈ Π ∗ . Then C can be partitioned into {C1 , . . . , Ck } such that for all t = 1, . . . , k, there exists zt ∈ {1, . . . , m} with Ct ⊆ Ezt . Note that k
2. Suppose k = 2. Then there exist i, j ∈ N , i = j , such that {i, j } ⊆ Sz1 ∩ Sz2 . This contradicts Condition (a) in the definition of the single-lapping property. Hence, k > 2. Then, for all t = 1, . . . , k − 1, |Szt ∩ Szt+1 |
1, |Sz1 ∩ Szk |
1, and there does not exist i ∈ N such that for all t = 1, . . . , k − 1, Szt ∩ Szt+1 = {i} and Sz1 ∩ Szk = {i}. This contradicts Condition (b) in the definition of the single-lapping property. Hence, G contains no cycles and no multiple edges between any two vertices, which means that Π ∗ has a tree structure representation.

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Part 2. If a coalition formation model has a tree structure representation then it is singlelapping. Fix a coalition formation model Π ∗ ⊆ Π that has a tree structure representation. Let G = (N, E) be the graph associated with Π ∗ . Then G contains no cycles and no multiple edges between any two vertices. Suppose Π ∗ is not single-lapping. Assume first that Condition (a) does not hold. Then there exist S, T ∈ Π ∗ such that S = T and |S ∩ T |
2. This implies that there exist ES ⊂ E and ET ⊂ E such that ES ∩ ET = ∅, ES represents S, and ET represents T . Then, since |S ∩ T |
2, G contains either a cycle or multiple edges between two vertices, which is a contradiction. Thus, Condition (a) holds, and hence Condition (b) must be violated. We can show similarly for this case that G contains a cycle, which is a contradiction. Therefore, Π ∗ is single-lapping. 2 A special class of coalition formation problems is called consecutive (Greenberg and Weber, 1986; Bogomolnaia and Jackson, 2002). A coalition formation problem is consecutive if there exists an ordering of the players such that every permissible coalition is made up of players who are indexed consecutively with respect to this ordering. The above papers prove, in the preference restriction context, that this property is sufficient for the existence of a stable coalition structure. Noting that the permissible coalitions in a consecutive coalition formation problem are represented by arbitrary paths in a linear graph,9 it is easy to compare consecutive and single-lapping coalition formation models. Singlelapping coalition formation models are both more and less restrictive than consecutive coalition formation models. On the one hand, in single-lapping coalition formation models permissible coalitions are represented by paths in a tree rather than in a linear graph, which is more general. On the other hand, single-lapping coalition formation models are more restrictive since two coalitions that have more than one member in common are not allowed, and thus not all coalitions are permissible which are represented by a path. Demange (1994) considers cooperation structures that are given by a tree, i.e., a model in which all permissible coalitions are consecutive (connected) on a tree. She shows that the existence of a stable coalition structure is guaranteed for this model. Uniqueness does not hold in general for a coalition formation model in which all permissible coalitions are connected coalitions on a tree, since this model may allow for two coalitions with two or more common members.

5. Implications for matching models Two special cases of simple coalition formation models are the well-studied models of marriage and roommate markets. For these models Π ∗ ⊆ Π is such that for all S ∈ Π ∗ , |S|  2, that is, any non-singleton permissible coalition is of size two. In the case of the marriage model it is also required that for all S ∈ Π ∗ with S = {i, j }, i = j , we have i ∈ W and j ∈ M, where N = W ∪ M and W and M are the disjoint sets of women and men. The existence of a stable marriage matching (i.e., a stable coalition structure) for every 9 The consecutive coalition formation model is a special case of the Myerson (1977) model, where the cooperation structure is a linear graph.

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preference profile is shown by Gale and Shapley (1962) in their classical paper (for an alternative non-constructive proof see Sotomayor (1996)), and they also show in the same paper that for some roommate problems there does not exist a stable roommate matching. It is also well known that stable marriage matchings are typically not unique. Alcalde (1995) gives a sufficient condition for the existence of stable roommate matchings. His sufficiency property, α-reducibility, is the two-person equivalent of the top-coalition property of Banerjee et al. (2001). A complete characterization of preference profiles for which a stable roommate matching exists is provided by Tan (1991), and a stronger sufficiency condition, which allows for indifferences, the no-odd-rings condition, is identified by Chung (2000). Our characterization results, Theorems 1 and 2, shed some light on these matching models as well, by providing necessary and sufficient conditions for the existence of a unique stable marriage matching or roommate matching. Our results also apply to the more complex models of multi-sided matching (as opposed to the marriage model which is a two-sided matching model), for which the permissible coalitions are singletons and coalitions of size k (k
3), and every coalition of size k has one player from each of the k sides of the market (see Alkan (1988) for a non-existence result for three-sided matching markets). Theorems 1 and 2 do not directly apply, however, to two-sided matching models in which players on one side of the market may be matched to more than one player on the other side of the market (many-to-one matching problems), since it is usually assumed that the players do not care about who else is matched to the same player as they are.10 Furthermore, our results do not apply to many-to-many stable matching problems (see, for example, Alkan (1999), Sotomayor (1999), and Martinez et al. (2003)), primarily because these problems are not hedonic coalition formation problems, given that many-to-many matchings do not lead to a partition of the players. In the case of roommate models (and marriage models as a special case), unique stability may be best understood by considering the graph representation characterization given in Theorem 2. In this special case there is a unique graph associated with any coalition formation model (i.e., a set of permissible pairs), regardless of any convention, since in this case there is an edge between two players if and only if they constitute a permissible pair. Note also that the graph associated with a roommate problem unambiguously indicates the permissible coalitions. The only difference between the associated graphs of marriage and roommate models is that for roommate models there may be an edge between any two players, whereas for marriage models the graph is a priori restricted to be bipartite11 with respect to the partition of women and men, and thus an edge is ruled out a priori between players of the same sex. Given that there is an edge between two players in a graph representation of collections of coalitions for marriage and roommate models if and only if the pair is permissible, in the 10 Some work has been done to understand many-to-one stable matchings when players care about who else is matched to the same player on the other side of the market (see Dutta and Massó (1997)). Our results apply in this framework. 11 A graph is bipartite if the set of players can be split into two disjoint sets such that each edge connects a player from one set with a player in the other set. Note that if G is a bipartite graph then each cycle of G is even (i.e., consists of an even number of edges).

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following we will call a graph associated with such a special model an acceptability graph. It follows from Lemma 2 that a marriage or roommate model is single-lapping if and only if its acceptability graph has no cycles. Therefore, the following corollary is implied by Theorem 2. Corollary 1. A marriage or roommate model has a unique stable matching if and only if its acceptability graph contains no cycles. It is easily seen that Theorem 1 leads to the same implication. Since the requirement that two distinct coalitions cannot have more than one member in common is trivially satisfied for roommate models, the definition of the single-lapping property reduces to Condition (b), which, in turn, is immediately seen to be equivalent to the requirement that there are no cycles in the acceptability graph. Let us also remark that, for marriage and roommate models, unique stability is consistent with every consecutive coalition being permissible (whether consecutiveness is meant on a line or on a tree), unlike for general coalition formation models, since Condition (a) holds vacuously for these models. Thus, for these matching models the single-lapping property may be regarded as a consecutiveness property on a tree. A result that is related to Corollary 1, but deals with the existence rather than the uniqueness of stable coalition structures, is due to Abeledo and Isaak (1991). They show that, in our terminology, a roommate model has some stable matching, which is not necessarily unique, if and only if its acceptability graph is bipartite. This is a finding that explains the gap between the existence result for marriage models and the non-existence result for roommate models, namely, that marriage models rule out odd cycles (cycles that consist of an odd number of edges). Our condition is more demanding, since it rules out every cycle, not only odd cycles. Acceptability graphs without cycles may arise naturally or may be constructed easily, based on a priori information. If it is publicly known that some mates are not acceptable to some players (the most obvious example of this is players of the same sex for each player in marriage markets), then it is natural to restrict E accordingly, and then remove further edges from the graph, if necessary, in order to arrive at a spanning forest (i.e., a graph with a maximum number of edges such that it has no cycles). For example, if links are costly in a communication network, then an efficient network may not contain any cycle, and one would naturally restrict the acceptability to pairs of players with a communication link. Furthermore, the acceptability graph for roommate problems is a linear graph if we order the players according to a one-dimensional characteristic, and any pair of players is permissible if and only if they are adjacent in this ordering.12

6. Strategyproofness and unique stability Our characterization results, Theorems 1 and 2, have some further implications, as unique stability is closely related to the existence of strategyproof coalition formation 12 An example of such a characteristic for choosing roommates, due to Chung (2000), is tidiness.

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rules. A rule is strategyproof if players cannot misrepresent their preferences profitably, regardless of what preferences the others have. Most results on strategyproof coalition formation and matching rules are negative.13 Sönmez (1999) shows that there is no strategyproof, Pareto efficient, and individually rational rule when all coalitions are permissible. We will show here that, for a singlelapping coalition formation model, not only the existence of a unique stable coalition structure is guaranteed for every preference profile, but also the coalition formation rule that chooses the unique stable coalition structure at each profile possesses desirable properties, namely, strategyproofness, Pareto efficiency, and individual rationality. This is the first part of the next theorem. Furthermore, Sönmez (1999) proved for a general indivisible goods allocation model, which includes our model as a special case, that if there is a strategyproof, Pareto efficient, and individually rational allocation rule, whenever the core is nonempty, the allocation rule must select an allocation in the core, and the core must be a singleton when preferences are strict.14 Therefore, the coalition formation rule that chooses the unique stable coalition structure at each preference profile is the only rule that has these three desirable properties. While uniqueness, the second part of the next theorem, follows from Sönmez (1999), we provide an alternative direct proof of uniqueness for our model. ∗ Fix a coalition formation model Π ∗ ⊆ Π . A coalition formation rule f Π : R|Π ∗ → Σ selects for each preference profile R ∈ R|Π ∗ a coalition structure σ ∈ Σ.15 We use the notation fi (R) = σi , where f (R) = σ . A coalition formation rule f is strategyproof if for i ∈ Ri |Π ∗ , fi (R) Ri fi (R i , R−i ). A coalition formation rule all R ∈ R|Π ∗ , i ∈ N , and R f is Pareto efficient if for all R ∈ R|Π ∗ , there does not exist σ ∈ Σ such that for all i ∈ N , σi Ri fi (R), and for some j ∈ N , σj Pj fj (R). A coalition formation rule f is individually rational if for all R ∈ R|Π ∗ and for all i ∈ N , fi (R) Ri {i}. Theorem 3. Let Π ∗ ⊆ Π be a single-lapping coalition formation model. Then there exists a unique coalition formation rule f : R|Π ∗ → Σ that is strategyproof, Pareto efficient, and individually rational, namely, the rule that selects the unique stable coalition structure of Π ∗ at every preference profile. Proof. Fix a single-lapping coalition formation model Π ∗ ⊆ Π . Part 1. Existence. Let f : R|Π ∗ → Σ be the coalition formation rule that selects the unique stable coalition structure of Π ∗ at every preference profile. We will show that f is strategyproof, Pareto efficient, and individually rational. First, suppose that f is not individually rational. Then there exist i ∈ N and R ∈ R|Π ∗ such that {i} Pi fi (R). This means that {i} blocks f (R), given R, and f (R) is not 13 There is, however, a parallel result with Theorem 3 for Shapley–Scarf indivisible goods markets (Shapley

and Scarf, 1974), which is proved by Roth (1982) and Ma (1994). 14 Another related paper is Ledyard (1977), which studies strategyproof rules that select a core allocation, and finds that the existence of a best unblocked allocation for all the participants is a necessary and sufficient condition for the existence of a strategyproof rule in core selecting organizations. 15 Since it will not cause any confusion, in the following we will use simply f instead of f Π ∗ .

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stable at R|Π ∗ , given that {i} ∈ Π ∗ . This is a contradiction, and hence f is individually rational. Next, suppose f is not Pareto efficient. Then there exist R ∈ R|Π ∗ and σ ∈ Σ such that for all i ∈ N , σi Ri fi (R), and for some j ∈ N , σj Pj fj (R). Let S = σj . Then for all i ∈ S, S = fi (R) and thus σi = S implies that S Pi fi (R). This means that S blocks f (R). Furthermore, for all i ∈ S, S Pi fi (R) Ri {i}, by individual rationality. This implies that S ∈ Π ∗ , which contradicts the fact that σ is stable at R|Π ∗ . Hence, f is Pareto efficient. It remains to show that f is strategyproof. First note that for all R ∈ R|Π ∗ , f (R) ⊆ ∗ Π , by individual rationality, and thus for all i ∈ N and R ∈ R|Π ∗ , fi (R) ∈ Π ∗ . i ∈ Ri Suppose f is not strategyproof. Then there exist R ∈ R|Π ∗ , i ∈ N , and R i , R−i ) Pi fi (R). Since fi (R i , R−i ) does not block f (R), there exists such that fi (R i , R−i ) such that fj (R) Pj fj (R i , R−i ). Since both fj (R) and fi (R i , R−i ) are j ∈ fi (R i , R−i ) and j ∈ fj (R) imply that in Π ∗ , and since Π ∗ is single-lapping, i, j ∈ fi (R i∈ / fj (R), otherwise Condition (a) would be violated. Hence, fj (R) = fi (R). Then, i , R−i ), there exists l ∈ fj (R) such that fl (R i , R−i ) Pl since fj (R) does not block f (R i , R−i ) and fj (R) are in fl (R). Note that l = i, since i ∈ / fj (R). Since both fl (R i , R−i ) imply that Π ∗ , and since Π ∗ is single-lapping, j, l ∈ fj (R) and l ∈ fl (R i , R−i ), otherwise Condition (a) would be violated. Hence, fl (R i , R−i ) = j ∈ / fl (R i , R−i ), since j ∈ fi (R i , R−i ). Then, since fl (R i , R−i ) does not block f (R), there fi (R i , R−i ) such that fh (R) Ph fh (R i , R−i ). Note that h = j , since j ∈ exists h ∈ fl (R / i , R−i ). Furthermore, Condition (b) implies in this case that h = i. Continuing fl (R similarly, we reach a contradiction, given that the set of players is finite. Therefore, f is strategyproof. Part 2. Uniqueness. Let f : R|Π ∗ → Σ be a coalition formation rule that satisfies strategyproofness, Pareto efficiency, and individual rationality. We will show that f selects the unique stable coalition structure of Π ∗ at every preference profile. Fix R ∈ R|Π ∗ . Let σ ∗ ∈ Σ denote the unique stable coalition structure of Π ∗ at R. We will prove that f (R) = σ ∗ . i rank σ ∗ first, and, provided σ ∗ = {i},  ∈ R|Π ∗ as follows. For all i ∈ N , let R Specify R i i ∗ ∗ ∗ i rank {i} second. Let σ = M ∪ · · · ∪ Mm be the partition of the players as defined let R 1 in the algorithm described in the proof of Part 1 of Theorem 1. For all t = 1, . . . , m,
/ let Tt∗ = S∈Mt∗ S. Suppose there exist S ∈ M1∗ and R −S ∈ R−S |Π ∗ such that S ∈ f (RS , R −S ). Let S = {1, . . . , s}, without loss of generality. Note that s
2, by individual 1 , R2 , . . . , Rs , R −S ) = rationality. Since top(R1 ) = S, strategyproofness implies that f1 (R  S. Then f2 (R1 , R2 , . . . , Rs , R −S ) = S. Then, since top(R2 ) = S, strategyproofness implies 1 , R 2 , R3 , . . . , Rs , R −S ) = S. Continuing iteratively, we get fs (R S , R −S ) = S, that f2 (R  which means that S ∈ / f (RS , R −S ). Then, since s
2, individual rationality implies that S , R −S ), which contradicts Pareto efficiency. Therefore, for all S ∈ M ∗ and [S] ⊆ f (R 1 R −S ∈ R−S |Π ∗ , we have S ∈ f (RS , R −S ). This implies that for all R −T1∗ ∈ R−T1∗ |Π ∗ , M1∗ ⊆ f (RT1∗ , R −T1∗ ). Let Π2∗ = {S ∈ Π ∗ : S ∩ T1∗ = ∅}. Suppose that there exist S ∈ M2∗ and R −(T1∗ ∪S) such that S ∈ / f (RT1∗ , RS , R −(T1∗ ∪S) ). Then, given that for all i ∈ S, top(Ri |Π2∗ ) = S,

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and that for all R −T1∗ ∈ R−T1∗ |Π ∗ , M1∗ ⊆ f (RT1∗ , R −T1∗ ), a similar argument to the above implies a contradiction. Therefore, for all R −(T1∗ ∪T2∗ ) ∈ R−(T1∗ ∪T2∗ ) |Π ∗ , M2∗ ⊆ f (RT1∗ , RT2∗ , R −(T1∗ ∪T2∗ ) ). Continuing iteratively, we can establish that for all t = 1, . . . , m, Mt∗ ⊆ f (R). This means that f (R) = σ ∗ , as desired. 2 The coalition formation rule f which always selects the unique stable coalition structure of Π ∗ (given that Π ∗ is single-lapping) is also group-strategyproof. A coalition formation rule f : R|Π ∗ → Σ is group-strategyproof if there exist no S ⊆ N , R ∈ R|Π ∗ , and S , R−S ) Ri fi (R), and for some j ∈ N , S ∈ RS |Π ∗ such that for all i ∈ S, fi (R R  fj (RS , R−S ) Pj fj (R). It is a stronger property than strategyproofness since it also rules out joint manipulations by coalitions. The group-strategyproofness of f as specified in Theorem 3 can be seen as follows. Suppose, by contradiction, that there exists a coalition S which can manipulate as described in the definition of group-strategyproofness. For all t = 1, . . . , m, let St = S ∩ Tt∗ , where the unique stable coalition structure of Π ∗ at R ∗ (as defined in the algorithm in the proof of Theorem 1), and is σ ∗
= M1∗ ∪ · · · ∪ Mm ∗ S , R−S ) = fi (R), and Tt = T ∈M ∗ ∪···∪Mt∗ T for all t = 1, . . . , m. Then for all i ∈ S1 , fi (R 1 thus M1 ⊆ σ ∗ , given the algorithm in the proof of Part 1 of Theorem 1. Then for all i ∈ S2 , S , R−S ) = fi (R), and M2 ⊆ σ ∗ . Repeating iteratively we obtain that for all i ∈ Sm , fi (R S , R−S ) = fi (R). Since Sm = S ∩ Tm∗ = S ∩ N = S, this is a contradiction. fi (R A possibility result reported previously is the finding by Alcalde and Barberà (1994) that if preferences satisfy a so-called top dominance condition for one side of a marriage market, then it is sufficient, and, under a richness condition on the preference domain, also necessary that there exists a stable and strategyproof matching rule.16 In a recent paper Alcalde and Revilla (2001) identify a strategyproof coalition formation rule, which is characterized by stability and strategyproofness, under the assumption that agents’ preferences satisfy a so-called tops responsiveness condition. This restriction on preferences ensures that the core is non-empty, but it may not necessarily be a singleton.17

Acknowledgments I would like to thank Kim-Sau Chung, Tom Gresik, Guillaume Haeringer, Eduardo Zambrano, and participants at GAMES 2000, the World Congress of the Econometric Society in Seattle, and ASSET 2000 in Lisbon for their comments. I am grateful to two associate editors and two referees for their insights and thoughtful suggestions. The support of the Universidade Nova de Lisboa, where the first version of this paper was written, is also acknowledged. 16 See also Roth and Sotomayor (1990) for a discussion of the incentives in matching problems and further references. 17 The tops responsiveness condition does not satisfy the preference domain requirements used in Sönmez (1999), which explains why the singleton core result of the latter does not apply to the context of Alcalde and Revilla (2001).

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