Int J Game Theory (2009) 38:431–452 DOI 10.1007/s00182-009-0162-9 ORIGINAL PAPER

Strategy-proof coalition formation Carmelo Rodríguez-Álvarez

Accepted: 15 December 2008 / Published online: 9 June 2009 © Springer-Verlag 2009

Abstract We analyze coalition formation problems in which a group of agents is partitioned into coalitions and agents’ preferences only depend on the coalition to which they belong. We study rules that associate to each profile of preferences a partition of the society. We focus on strategy-proof rules on restricted domains of preferences, as the domains of additively representable or separable preferences. In such domains, the only strategy-proof and individually rational rules that satisfy either a weak version of efficiency or non-bossiness and flexibility are single-lapping rules. Single-lapping rules are characterized by severe restrictions on the set of feasible coalitions that are consistent with hierarchical organizations. These restrictions are necessary and sufficient for the existence of a unique core-stable partition. In fact, single-lapping rules always select the associated unique core-stable partition. Thus, our results highlight the relation between the non-cooperative concept of strategyproofness and the cooperative concept of uniqueness of core-stable partitions. Keywords Coalition formation · Strategy-proofness · Single-lapping property · Core-stability JEL Classification

C71 · C78 · D71

1 Introduction We study simple coalition formation problems in which a group of agents is partitioned into coalitions and agents have preferences over the coalitions to which they

C. Rodríguez-Álvarez (B) Departamento de Fudamentos del Análisis Económico II, Facultad CC. Económicas y Empresariales, Universidad Complutense de Madrid, Campus de Somosaguas, 28223 Madrid, Spain e-mail: [email protected]

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belong. Hence, we exclude externalities among coalitions. Following the terminology proposed by Drèze and Greenberg (1980), we focus on problems characterized by the “hedonic” aspect of coalition formation. Examples of such problems are matching problems such as marriage and roommate problems, or the formation of social clubs, teams, and societies. The formation of coalitions is a relevant phenomenon in a wide variety of social and economic environments. The rationale behind the formation of coalitions is that agents form groups in order to exploit the joint benefits of cooperation. The literature on Coalitional Game Theory has extensively analyzed the existence of stable partitions in hedonic coalition formation problems.1 Instead, we propose a social choice and implementation approach. We study coalition formation rules that associate to each profile of agents’ preferences a partition of the group of agents. A coalition formation rule can be interpreted as a recommendation for the society that represents an optimal compromise between the conflicting preferences of the agents. Since preferences are not observable, they must be elicited from the agents. Thus, given a coalition formation rule, a fundamental concern is whether agents have the incentive to reveal their true preferences. In this paper, we analyze the possibility of devising coalition formation rules that always give agents such an incentive. Hence, we are interested in rules that satisfy strategy-proofness. Strategy-proofness is the strongest decentrability property. It implies that it is a dominant strategy for the agents to straight-forwardly reveal their preferences. Indeed, each agent needs to know only her own preferences to compute her best choice. It is well known that strategy-proofness is hard to satisfy. In the abstract model of social choice, Gibbard (1973) and Satterthwaite (1975) show that—provided there are more than two alternatives at stake– every strategy-proof social choice rule is dictatorial. Reasonable strategy-proof rules exist however, if appropriate restrictions are imposed on agents’ preferences. In coalition formation problems, such domain restrictions arise naturally. On the one hand, coalition formation rules select a partition of the society for each preference profile but each agent only cares about the coalition she is a member of. On the other hand, additional restrictions on how an agent may compare different coalitions can be easily justified. For instance, an interesting problem arises when there are no complementarities among the members of a coalition. That is, if an agent i prefers joining agent j rather than being alone, then for each coalition C such that i ∈ C and j ∈ / C, agent i prefers C ∪ { j} to C.2 Then, agents’ preferences are additively representable or separable. These domains of preferences have been studied in the general context of abstract social choice by Barberà et al. (1991) and Le Breton and Sen (1999), among others, and positive results have been obtained. Yet, the possibility of constructing strategy-proof coalition formation rules for general

1 For further references, see the recent works by Banerjee et al. (2001), Barberà and Gerber (2003, 2007),

Bogomolnaina and Jackson (2002), and Pápai (2004). 2 Think, for example, in the preferences of a senior member of an Economics Department about the job–

candidates for two tenure–track positions that are available (but that need not to be filled). Suppose that there are two candidates, a macroeconomist and an econometrician. If the senior economist prefers hiring the macroeconomist rather than not hiring anybody, then the senior economist should also prefer hiring the macroeconomist and the econometrician rather than hiring the econometrician alone.

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problems when agents’ preferences are additively representable or separable has not been addressed in the literature. We characterize a family of rules, the family of single-lapping rules, that satisfy strategy-proofness, individual rationality, and a weak version of efficiency in minimally rich domains of preferences (as the domain of additively representable preferences).3 Single-lapping rules are characterized by strong restrictions over the set of feasible coalitions—the single-lapping property—that can be justified by the initial existence of a hierarchical structure of the society. Pápai (2004) shows that the single-lapping property is a necessary and sufficient condition for the existence of a unique core-stable partition for every profile of preferences. In fact, single-lapping rules always select the unique core-stable partition, in the sense that no feasible coalition of agents unanimously prefer joining each other rather than staying in the coalition they are assigned to. Hence, our results provide further evidence on the relation between the non-cooperative game theory concept of strategy-proofness and the cooperative game theory concept of uniqueness of core-stable partitions. In addition, since we do not impose initial restrictions on feasible coalitions and we do not insist on full efficiency, our characterization contrasts with other impossibility results regarding the construction of strategy-proof coalition formation rules in specific problems.4 Before proceeding with the formal analysis, we review related literature. Pápai (2004) and Sönmez (1999) analyze the relation between strategy-proofness and uniqueness of the core.5 Pápai’s main focus is on finding necessary and sufficient conditions on the set of feasible coalitions for the uniqueness of core-stable partitions. Additionally, this author shows that, given an initial set of coalitions that satisfy the singlelapping property, its associated single-lapping rule is the unique rule that satisfies strategy-proofness, individual rationality, and (constrained) efficiency when agents are restricted to prefer any coalition in the initial set to any other coalition. Our analysis complements Pápai’s results in several directions. We show that the single-lapping structure of the set of feasible coalition is directly implied by Pápai’s axioms. Moreover, the characterization also holds in more restricted domains of preferences over coalitions. On the other hand, Sönmez (1999) proposes a general model of allocation of objects which includes coalition formation as a special case.6 In this general framework and assuming the non-emptiness of the core, Sönmez (1999) shows that there exists a strategy-proof rule if the core is essentially single-valued (or unique). Having said that, our results cannot be obtained from Sönmez’s because this author introduces assumptions on agents’ preferences that are incompatible with separable preferences. Finally, we refer to the related works by Alcalde and Revilla (2004), Cechlárová and Romero-Medina (2001), and Dimitrov et al. (2006). Alcalde and Revilla (2004) and 3 Individual rationality requires that no agent is forced to join some coalition. 4 See for instance Alcalde and Barberà (1994), Roth and Sotomayor (1990), and Sönmez (1999). 5 The line of research that investigates the existence of strategy-proof rules in core selecting organizations was initiated by Ledyard (1977). 6 Takamiya (2003) further analyzes Sönmez’s framework under additional assumptions on agents’ prefer-

ences.

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Cechlárová and Romero-Medina (2001) analyze the manipulability of coalition formation rules when preferences over coalitions are based on the best or the worst group of agents in each coalition. In their frameworks, these authors prove the existence of strategy-proof rules that always select core–stable partitions. Finally, Dimitrov et al. (2006) study coalition formation problems where agents are equipped with separable preferences and they consider the remaining members of the society as friends or enemies. These authors however, focus on the existence and computability of core–stable partitions. The remainder of the paper is organized as follows. In Sect. 2, we present the model and basic notation. In Sect. 3, we present different domains of preferences over coalitions and the notion of minimally rich domain. In Sect. 4, we introduce the main axioms and in Sect. 5 we present single-lapping rules and the main result. In Sect. 6, we propose an alternative characterization and discuss the role of the domain restrictions. In Sect. 7, we finally prove the theorems. In the Appendix, we collect omitted proofs.

2 Basic notation Let N ≡ {1, . . . , n} be a society consisting of a finite set of at least 3 agents, (n ≥ 3). We call a non-empty subset C ⊆ N a coalition. Let N denote the set of all nonempty subsets of N . For each C ⊆ N , let [C] ≡ {{i} | i ∈ C}. A collection of coalitions is a set of coalitions
⊆ N that contains all singleton sets, [N ] ⊆
. Let σ be a partition of N and let  denote the set of all partitions of N . Analogously, for each C ⊆ N , C denotes the set of all partitions of coalition C. For each i ∈ N and each σ ∈ , we denote by σi ∈ σ the coalition in σ to which i belongs. For each i ∈ N , let Ci ≡ {C ⊆ N | i ∈ C}. That is, Ci is the set of all coalitions to which i belongs. A preference for i,
i , is a complete order on Ci .7 For each i ∈ N , we denote by Di the set of all preferences for i. Note that preferences are strict. Hence, for each i ∈ N , each
i ∈ Di , and each C, C  ∈ Ci , we write C i C  to indicate that i strictly prefers C to C  , and C
i C  to indicate that either C i C  or C = C  . We assume that agents only care about the coalition to which they belong. Then, preferences over partitions are completely defined by preferences over coalitions. Thus, abusing notation, we say that for each i ∈ N , each
∈ Di , and each σ, σ  ∈ , σ is at least as good as σ  , σ
i σ  , if and only if σi
i σi . For each i ∈ N , each set of coalitions X ⊆ N with X ∩ Ci = ∅, and each
i ∈ Di , let top(X ,
i ) be the coalition in X ∩ Ci that is ranked first according to
i . Let D ≡ ×i∈N Di . We call
∈ D a preference profile. For each C ⊂ N , DC = ×i∈C Di , while for each
∈ D,
C ∈ DC denotes the restriction of profile
to the preferences of the agents in C.

7 An order is a reflexive, transitive, and antisymmetric binary relation.

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Let D˜ ⊆ D, we say that D˜ is a cartesian domain if for each i ∈ N , there is D˜ i ⊆ Di such that D˜ = ×i∈N D˜ i . We are interested in rules that associate a partition of the society to each profile of agents’ preferences. Let D˜ ⊂ D be a cartesian domain. A (coalition formation) rule defined on the domain D˜ is a mapping ϕ : D˜ → . ˜ ϕi (
) denotes the coalition in ϕ(
) to Naturally, for each i ∈ N and each
∈ D, which i belongs. Finally, R ϕ denotes the range of ϕ, that is, the set of feasible partitions, ˜ ϕ(
) = σ }, R ϕ ≡ {σ ∈  | there is
∈ D, while, F ϕ denotes the set of feasible coalitions, F ϕ ≡ {C ∈ N | there is σ ∈ R ϕ , C ∈ σ }. 3 Restricted domains of preferences over coalitions We start by introducing two classes of preferences over coalitions—top and bottom preferences—that play a crucial role in our analysis. These preferences are obtained by extending orders over single agents to orders over coalitions. The underlying idea is that each agent i divides the society into two groups according to some order over the set of agents: her friends—those agents ranked above i− and her enemies—those agents ranked below i−. According to top preferences, agent i ranks coalitions by (lexicographically) comparing the highest ranked friends in each coalition. If the sets of friends coincides for two coalitions, then agent i (lexicographically) compares the lowest ranked enemies. Bottom preferences apply the reverse logic. According to bottom preferences, agent i ranks coalitions by (lexicographically) comparing the lowest ranked enemies. If the sets of enemies coincides for two coalitions, then agent i (lexicographically) compares the highest ranked friends. Let P be the set of all complete orders over N . For each P ∈ P, R denotes the weak order associated to P. It is defined in the standard way. For each C ⊆ N and each P ∈ P, max(C, P) and min(C, P) denote, respectively, the first-ranked and the last-ranked agent of C according to P. Next, for each i ∈ N , each P ∈ P, and each C ∈ Ci , let Ci+ (P) ≡ { j ∈ C, s.t. j R i} , and Ci− (P) ≡ { j ∈ C s.t. i R j} . Now, define Ci+ (1, P) ≡ max(Ci+ (P), P) and Ci− (1, P) ≡ min(Ci− (P), P) . Once Ci+ (t, P) and Ci− (t, P) are defined for some t ≥ 1, iteratively, let
   Ci+ (t + 1, P) ≡ max Ci+ (P)\ ∪tk=1 Ci+ (k, P) , P , and
   Ci− (t + 1, P) ≡ min Ci− (P)\ ∪tk=1 Ci− (k, P) , P . For each i ∈ N and each P ∈ P, the preference
i ∈ Di is the top preference associated to P by i,
i =
i+ (P), if for each two distinct coalitions C, C  ∈ Ci , C i C  if and only if

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• Ci+ (P) = Ci+ (P) and Ci+ (t, P) P Ci+ (t, P), where t is the first integer such that Ci+ (t, P) = Ci+ (t, P).8 • Ci+ (P) = Ci+ (P) and Ci− (t  , P) P Ci− (t  , P), where t  is the first integer such that Ci− (t  , P) = Ci− (t  , P) . For each i ∈ N and each P ∈ P, the preference
i ∈ Di is the bottom preference associated to P by i,
i =
i− (P), if for each two distinct coalitions C, C  ∈ Ci , C i C  if and only if • Ci− (P) = Ci− (P), and Ci− (t, P) P Ci− (t, P), where t is the first integer such that Ci− (t, P) = Ci− (t, P). • Ci− (P) = Ci− (P) and Ci+ (t  , P) P Ci+ (t  , P), where t  is the first integer such that Ci+ (t  , P) = Ci+ (t  , P). For each i ∈ N , let Di+ ≡ {
i ∈ Di such that for some P ∈ P,
i =
i+ (P)}, Di− ≡ {
i ∈ Di such that for some P ∈ P,
i =
i− (P)}, Di∗ ≡ Di+ ∪ Di− and, D∗ ≡ ×i∈N Di∗ .

¯ Let D¯ ⊆ D. We say that D¯ is minimally rich if D¯ is cartesian and D∗ ⊆ D. We consider that a domain of preferences over coalitions is minimally rich if it contains top and bottom preferences. Minimal richness also requires that the domain be cartesian. That is, an agent’s set of admissible preferences does not depend on the preferences of the other agents. The following remark shows that in minimally rich domains agents’ preferences over being alone and two other different coalitions are not restricted. Remark 1 For each i ∈ N and each two distinct C, C  ∈ Ci \{i}, there exist
i ,
i ,
i ∈ Di∗ such that: {i} i C i C  , C i {i} i C  , and C i C  i {i}. It can be argued that top and bottom preferences reflect rather extreme preferences over coalitions. The domains of additively representable and separable preferences however, are minimally rich. These domains exclude the possibility of (negative or positive) complementarities among the members of a coalition. 8 The fact that C + (t, P) = ∅ for some integer t is irrelevant for this definition. Given that i i ∈ Ci+ (P) ∩ Ci+ (P) , if t is the first integer such that Ci+ (t, P) = Ci+ (t, P) , then both Ci+ (t, P) = ∅ and Ci+ (t, P) = ∅. This line of argument also applies to the following possibility C + (P) = C + (P)

and to the definition of bottom preferences.

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Let i ∈ N . A utility function for agent i is a mapping u i : N → R such that u i (i) = 0 . A preference for agent i,
i ∈ Di is additively representable  if there is a  ∈ C , C
C  if and only if such that for each C, C utility function u i i i c∈C u i (c) ≥   ). For each i ∈ N , A denotes the set of all i’s additively representable u (c   i i c ∈C preferences for agent i and let A ≡ ×i∈N Ai . A preference for i,
i ∈ Di , is separable if for each j ∈ N and each C ∈ Ci such that j ∈ / C, {i, j} i {i} if and only if (C ∪ { j}) i C. Let Si be the set of all agent i’s separable preferences and let S ≡ ×i∈N Si . The following remark shows that the domain of additively representable preferences and the domain of separable preferences are indeed minimally rich domains. Moreover, for small societies both domains coincide with the smallest minimally rich domain. Remark 2 Let i ∈ N . (a) If n = 3, then Di∗ = Ai = Si . (b) If n ≥ 4, then Di∗ ⊂ Ai ⊂ Si . 4 Axioms This section introduces three properties that rules may satisfy. Let D˜ ⊆ D be a carte˜ sian domain and let ϕ be a rule defined on D. Our main axiom is an incentive constraint. A rule should never provide an incentive for an agent to misreport her preferences. Only if a rule elicits the true preferences from the agents, the social choice will be based upon the correct information. Of course, this property refers to the specific domain in which the rule is defined. Strategy-Proofness. For each i ∈ N , each
∈ D˜ , and each
i ∈ D˜ i , ϕi (
)
i ϕi (
N \{i} ,
i ). We also consider a minimal participation constraint. Agents should not prefer to stay on their own rather than to belong to the coalition that the rule assigns them. ˜ ϕi (
)
i {i}. Individual Rationality. For each i ∈ N and each
∈ D, We introduce a weak version of efficiency. This notion of efficiency corresponds to efficiency restricted to the set of feasible coalitions. It is the standard notion of efficiency for coalition formation problems.9 ˜ there is no σ ∈  such that for each C ∈ σ , Constrained efficiency. For each
∈ D, C ∈ F ϕ , and for every i ∈ N , σi
i ϕi (
), and for some j ∈ N , σ j  j ϕ j (
). Constrained efficiency does not imply that every conceivable coalition is feasible. In fact, it allows for very restricted sets of feasible coalitions. Indeed, an imposed rule that always selects the same partition of the society regardless of agents’ preferences satisfies constrained efficiency.

9 See Sönmez (1999) and Pápai (2004).

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5 Main characterization In this section we analyze the implications of the axioms listed above over rules defined on rich domains. First, we introduce additional notation due to Pápai (2004). A collection of coalitions
satisfies the single-lapping property if Condition (a): For each C, C  ∈
, C = C  implies #(C ∩ C  ) ≤ 1. Condition (b): For each {C1 , . . . , Cm } ⊆
with m ≥ 3 and for each t = 1, . . . , m, #(Ct ∩ Ct+1 ) ≥ 1 (where m + 1 = 1), there is i ∈ N such that for each t = 1, . . . , m, Ct ∩ Ct+1 = {i}. Condition (a) states that if two coalitions in the collection overlap, they cannot have more than one agent in common. Condition (b) is a non-cycle condition. It requires that if a set of coalitions in the collection form a cycle in which every two neighbor coalitions have a common member, then all these coalitions have the same common member. There is an interesting property that all single-lapping collections of coalitions satisfy. For each single-lapping collection of coalitions and each preference profile, there is a coalition in the collection such that all its members think that this coalition is the best coalition in the collection. Remark 3 (Pápai 2004, Theorem 1’). Let
be a single-lapping collection of coalitions. For each
∈ D there is C ∈
such that for each i ∈ C, C = top(
,
i ). The single-lapping property implies severe restrictions in the set of admissible coalitions. Indeed, Pápai (2004) shows that single-lapping collections of coalitions can be associated to a non-directed graph endowed with a tree structure. Tree structures are characteristic of many hierarchical societies in which only members of adjacent levels in the hierarchy are connected and can form a coalition.10 On the other hand, the single-lapping property ensures the existence and uniqueness of a core-stable partition for every preference profile. That is, given a single-lapping collection of coalitions
, for each preference profile
∈ D there is a (unique) partition σ ∈  such that for no C ∈
, for each j ∈ C, C  j σ j . We use the single-lapping property to define a class of rules. For each
∈ D and each single-lapping collection of coalitions
⊂ N , the corestable partition associated to
at profile
, σ¯
(
), can be identified by the following algorithm: Algorithm: (Pápai, 2004). Let
∈ D and let
be a single-lapping collection of coalitions. Find C ∈
such that for each i ∈ C, top(
,
i ) = C . By Remark 3, such coalition exists. Note that there may be several such coalitions, and all these coalitions are disjoint. Let
(1,
) ≡
, M (1,
) ≡ {C ∈
| ∀ i ∈ C, top(
,
i ) = C},


T
(1,
) ≡ ∪C∈M
(1,
) C. 10 See Demange (2004, 2009) for more on the relation of hierarchical structures and coalitional stability.

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Hence, M
(1,
) denotes the set of all the coalitions that are formed in this first stage and T
(1,
) denotes the set of agents that are matched in the first stage. Once
(t,
), M
(t,
), and T
(t,
) are defined for some t ≥ 1, let,
(t + 1,
) ≡ {C ∈
| C ∩ T
(t,
) = {∅}}, M
(t + 1,
) ≡ {C ∈
(t + 1,
) | ∀ i ∈ C, top(
(t + 1,
),
i ) = C}, T
(t + 1,
) ≡ ∪C∈M
(1,
)∪...∪M
(t+1,
) C. Note that, for each t = 1, . . . , m,
(t,
) ⊂
,
(t,
) is a collection of coalitions for the reduced society N \T
(t,
). Moreover,
(t,
) satisfies the singlelapping property. Let m ≤ n be the smallest integer such that T
(m,
) = N . Then, the algorithm identifies a unique partition, σ¯
(
) ≡ {C ∈
| ∃ t ≤ m, C ∈ M
(t,
)}. For each single-lapping collection of coalitions and each preference profile there is a unique core-stable partition. Thus, each single-lapping collection of coalitions immediately defines a unique rule. ˜ Let D˜ ⊆ D be a cartesian domain of preferences and let ϕ be a rule defined on D. The rule ϕ is a single-lapping rule if there is a single-lapping collection of coalitions ˜ ϕ(
) = σ¯
(
).
such that for each
∈ D, It turns out that, given a fixed single-lapping collection of coalitions
, if agents are restricted to prefer standing on their own to any other coalition C ∈ /
, then the single-lapping rule associated to
is the unique rule that satisfies strategy-proofness, individual rationality, and constrained efficiency (Pápai 2004, Theorem 3). This implies that single-lapping rules satisfy strategy-proofness even on the unrestricted domain of preferences over coalitions D. This fact is not surprising because, by restricting the set of feasible coalitions, single-lapping rules eliminate agents’ opportunities for profitable misrepresentation of preferences. Our first theorem significantly extends Pápai’s Theorem 3. The single-lapping structure of the set of feasible coalitions is a direct implication of the three axioms on any minimally rich domain. Hence, reducing the set of admissible preferences does not allow for additional rules. Theorem 1 Let D¯ be a minimally rich domain. A rule ϕ : D¯ →  satisfies strategy-proofness, individual rationality, and constrained efficiency if and only if ϕ is a single-lapping rule. We present the proof of Theorem 1 in the concluding section. The intuition behind necessity runs as follows. For every rule that satisfies our axioms, whenever the members of a feasible coalition of individuals agree that this coalition is the most preferred feasible coalition, this coalition is formed. Then, using Remark 3, we only have to check that the set of feasible coalitions satisfies the single-lapping property. This step is far from being immediate and constitutes the bulk of the proof. The analysis is relatively simple for three-agent societies but requires the careful study of different possible configurations of the set of feasible coalitions. An iterative argument allows us to extend the result to arbitrary societies.

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Theorem 1 presents a new version of the classical trade-off between strategy-proofness and full efficiency. Strategy-proofness implies that the set of social alternatives is severely restricted. On the other hand, Theorem 1 shows that only rules that select the unique core-stable partition given an initial set of feasible coalition satisfy our list of axioms. Hence, this result provides further evidence on the relation between strategy-proofness and uniqueness of core-stable partitions. This relation has been already presented in previous works as Sönmez (1999) and Pápai (2004). Our first theorem however, introduces several novelties. Sönmez (1999) and Pápai (2004) assume restrictions on preferences or on feasible coalitions that ensure the existence of core-stable partitions.11 We obtain instead that the rule selects the unique core-stable partition directly from our axioms. Moreover, since we do not introduce any assumption on the set of feasible coalitions, we obtain a characterization that applies to every kind of coalition formation problems, including marriage problems, room–mate problems, and college–admission problems when students have preferences over colleagues. In addition, our result applies to rules defined on extremely restricted domains of preferences. The domains of additively representable and separable preferences are minimally rich domains. Hence, we obtain the following corollaries to Theorem 1. Corollary 1 A rule ϕ : A →  satisfies strategy-proofness, individual rationality, and constrained efficiency if and only if ϕ is a single-lapping rule. Corollary 2 A rule ϕ : S →  satisfies strategy-proofness, individual rationality, and constrained efficiency if and only if ϕ is a single-lapping rule. 6 Discussion 6.1 An alternative characterization We start by examining the role of constrained efficiency in Theorem 1. Our weak version of efficiency seems uncontroversial. Having said that, one may however feel that our axioms drive directly the result because both individual rationality and constrained efficiency are part of the definition of the core (no profitable deviation for single agents and for the whole society). With this problem in mind, we think that it is of interest to analyze the implications of alternative axioms that may be appropriate in coalition formation problems. Hence, we substitute constrained efficiency for two axioms: non-bossiness and flexibility. 11 In fact, Sönmez (1999) assumes the existence of certain preferences that need not exist in a minimally rich domain. Basically, in Sönmez’s framework for each i ∈ N , and each C ∈ (F ϕ ∩ Ci ), if there is an admissible preference
i such that C i {i}, then there is another admissible preference
i such that for each C  ∈ (F ϕ ∩ Ci )\{i}, C 
i C if and only if C 
i C, while C
i C  if and only if C
i C  and C
i {i}
i C  . There are minimally rich domains, namely the domain of additively representable, for which such preferences are not admissible. Let i, j, k ∈ N , and assume {i, j}, {i, k}, {i, j, k} ∈ F ϕ . Let
i ∈ Ai be such that {i, j, k} i {i, j}  {i, k}  {i} , but there is no
i ∈ Ai such that {i, j, k} i {i}, {i} i {i, j}, and {i} i {i, k}.

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Non-Bossiness is a condition frequently used in contexts of strategy–proof allocation where agents only care about private features of the social choice.12 Non-Bossiness says that whenever a change in an agent’s preference does not change the coalition she is assigned to, then the assignment for the remaining agents does not change. ˜ and each
 ∈ D˜ i , ϕi (
) = ϕi (
N \{i} , Non-Bossiness. For each i ∈ N , each
∈ D, i  
i ) implies ϕ(
) = ϕ(
N \{i} ,
i ). Flexibility is a technical condition on the range of the rule. It implies that the range of a rule is determined by the set of feasible coalitions. Thus, it avoids situations in which a coalition may form only if other (disjoint) coalition simultaneously forms. Flexibility. For each σ = {C1 , . . . , Cm } ∈ , Ct ∈ F ϕ for each t = 1, . . . , m, implies σ ∈ R ϕ . In many environments, strategy-proofness and non-bossiness directly imply constrained efficiency. This is not the case in our framework.13 On the other hand, flexibility is implied by constrained efficiency. Our next result shows that, when applied to strategy-proof and individually rational rules, non-bossiness and flexibility are equivalent to constrained efficiency. Theorem 2 Let D¯ be a minimally rich domain. A rule ϕ : D¯ →  satisfies strategyproofness, individual rationality, non-bossiness, and flexibility if and only if ϕ is a single-lapping rule. The intuition behind the proof of Theorem 2 runs parallel to the proof of Theorem 1. The crucial difference is on proving that whenever the members of a feasible coalition of individuals agree that this coalition is their most preferred feasible coalition, this coalition is formed. Then, the arguments in the proof of Theorem 1 are used (with minor modifications) to show that the set of feasible coalitions satisfies the single-lapping property. 6.2 Independence of the Axioms We devote this section to present some examples of rules that show the independence of the axioms. Theorem 1 is tight, but Theorem 2 is tight only if there are at least four agents. When there are only three agents, flexibility is directly implied by individual 12 See for instance Satterthwaite and Sonnenschein (1981) and Pápai (2000). 13 Consider a society formed by four agents N = {i, j, k, l}. Define the rule ϕ¯ in the domain of separable

preferences. Let ϕ¯ : S → . Agents i and j are the founding members of a club and they are always together. Then, for each
∈ S, {i} ∈ ϕ¯ j (
), { j} ∈ ϕ¯i (
). Preferences of agents k and l are irrelevant for the social choice. Agent k enters the club if i likes agent k. Thus, {k} ∈ ϕ¯i (
) if {i, k}
i {i}. Agent l enters the club if j likes l. Thus, {l} ∈ ϕ¯ j (
) if { j, l}
l { j}. The rule ϕ¯ satisfies strategy-proofness and non-bossiness, but ϕ¯ violates individual rationality and constrained efficiency. Let
∈ S be such that {i, j, k} i {i, k} i {i, j} i {i} i C for each C ∈ Ci \({i, j, k}, {i, k}, {i, j}, {i}) , {i, j, l}  j { j, l}  j {i, j}  j { j}  j C  for each C  ∈ C j \({i, j, l}, { j, l}, {i, j}, { j}) , {k} = top(N ,
k ), and {l} = top(N ,
l ). Basically, i likes j and k but strongly dislikes l, j likes i and l and strongly dislikes k, whereas k and l would rather stay alone. Note that ϕ(
) ¯ = {i, j, k, l}, but for each i  ∈ N , ({i, j}, {k}, {l}) i  ϕ(
). ¯

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rationality. The following examples are stated for 3 and 4-agent societies but are easily generalized to arbitrary numbers of agents. ¯ let Example 1 (Strategy-proofness) Let N = {i, j, k}. For each
∈ D,   I Ri (
) ≡ C ∈ Ci , such that for each j ∈ C, C
j { j} . ¯ ϕ −S (
) ≡ top(I Ri (
),
i ) and for each Let ϕ −S be such that for each
∈ D, i −S j ∈ / top(I Ri (
),
i ), ϕ j (
) ≡ { j}. Note that ϕ −S satisfies individual rationality, constrained efficiency, non-bossiness, and flexibility, however, ϕ −S violates strategyproofness.14 Example 2 (Individual rationality) Let N = {i, j, k}. Let ϕ −I be such that for each ¯ ϕ −I (
) = top(N ,
i ), and for each j ∈
∈ D, / top(N ,
i ), ϕ −I i j (
) = { j}. The rule ϕ −I is dictatorial. Note that ϕ −I satisfies strategy-proofness, constrained efficiency, non-bossiness, and flexibility, however, ϕ −I violates individual rationality. Example 3 (Constrained efficiency and non-bossiness) Let N = {i, j, k}. Let ϕ −E be ¯ such that for each
∈ D, ⎧   {i, j, k} if for each i  ∈ N , {i, j, k}
i  i  , ⎪ ⎪ ⎪
 ⎨ ϕ −E
= ({i, j} , {k}) if {i, j} i {i}, {i, j}  j { j} and top(N ,
k ) = {k}, ⎪ ⎪ ⎪ ⎩ otherwise. [N ] Note that ϕ −E satisfies strategy-proofness, individual rationality, and flexibility, but ϕ −E violates both constrained efficiency and non-bossiness.15 ¯ Example 4 (Flexibility) Let N = {i, j, k, l}. Let ϕ −F be such that for each
∈ D, ϕ

−F

(
) =

({i, j}, {k, l}) if for each m ∈ N , ({i, j}, {k, l})
m [N ] , [N ]

otherwise.

Note that ϕ −F satisfies strategy-proofness, individual rationality, and non-bossiness, However, ϕ −F violates both constrained efficiency and flexibility. 14 In order to check that ϕ −S violates strategy-proofness, let N = {i, j, k},
∈ D ∗ , and
 ∈ D ∗ be such j j

that {i, j} i {i, j, k} i {i}, {i, j, k}  j {i, j}  j { j, k}  j { j}, and {i, k} k {i, j, k} k {k}; while { j, k} j {i, j, k} j { j}. Note that ϕ −S (
) = ({i, j}, {k}), while ϕ −S (
N \{ j} ,
j ) = {i, j, k}. Then,

−S  ϕ −S j (
N \{ j} ,
j )  j ϕ j (
). 15 In order to check that ϕ −E violates constrained efficiency and non-bossiness, let
∈ D ∗ ,
 ∈ D ∗ k k be such that {i, j} i {i}, {i, j}  j { j}, top(N ,
k ) = {k}, while { j, k} k {k} k {i, j, k}. Note that ϕ −E (
) = ({i, j}, {k}) and ϕ −E (
N \{k} ,
k ) = ({i}, { j}, {k}).

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6.3 On minimal domains We address finally the issue of whether Theorems 1 and 2 hold for domains of preferences strictly contained in D∗ . As D∗ consists of the union of the domains of bottom and top preferences, it is natural to check whether there exist non-single-lapping rules that satisfy our axioms on those domains. It turns out that new possibilities arise in both domains. The rule ϕ −S presented in Example 2 satisfies strategy-proofness when defined on the domain of bottom preferences ×i∈N Di− . On the the other hand, the domain of top preferences is included in the domain of top-responsive preferences proposed by Alcalde and Revilla (2004). These authors provide an algorithm—the top-covering algorithm—that always select a core-stable partition if preferences are top-responsive. Basically, this algorithm looks iteratively for a minimal (with respect to inclusion) coalition such that for all its members their preferred group of agents is contained in the coalition. Alcalde and Revilla (2004) show that the top-covering algorithm defines the unique efficient rule that satisfies our axioms in their domain. Interestingly, in the top-responsive domain, there are preference profiles with multiple core stable partitions. This fact highlights the key role of bottom preferences in obtaining the relation between strategy-proofness and unique core stability.16 In the light of these examples, we can interpret Theorems 1 and 2 as minimal domain results. The smallest minimally rich domain D∗ is a minimal domain for which the single-lapping rules are the unique rules that satisfy strategy-proofness, individual rationality, and either constrained efficiency, or non-bossiness and flexibility.

7 Proofs of the theorems We begin this section by introducing a property implied by our axioms. This property incorporates the idea that a rule cannot be against the preferences of the members of the society. Whenever the members of a coalition consider this coalition as the best coalition, this coalition should form, independently of the preferences of the remaining agents in society. Of course, the following axiom refers to rules defined on a minimally ¯ rich domain D. ϕ ϕ ¯ Top-Coalition. Let
C ∈ F and
∈ D. If for each i ∈ C, top(F ,
i ) = C , then for each i ∈ C, ϕi
= C. Constrained efficiency and top-coalition are logically independent. Note that topcoalition is a property of rules. Banerjee et al. (2001) use the term top-coalition to name a property of preference profiles. These authors say that a preference profile satisfies the top-coalition property if for every group of agents V ⊆ N there is a coalition C ⊆ V that is the best coalition for all the members of C. Basically, our top-coalition implies that at a preference profile that satisfies Banerjee et al.’s top-coalition property,

16 Indeed, in the domain of top preferences, any agent prefers to join the whole society rather than staying alone unless her best preferred coalition is staying alone. Finally, we have to note that Alcalde and Revilla’s results depend crucially on the fact that every conceivable coalition is feasible. Their top-covering algorithm works if there are no restrictions in the set of feasible coalitions.

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then the rule selects a partition in which all the coalitions such that all their members consider as the best feasible coalition are formed. The proof of Theorem 1 follows from a series of lemmata. From now on, D¯ refers ¯ First, we prove to an arbitrary minimally rich domain and ϕ to a rule defined on D. that our axioms imply top-coalition. Lemma 1 If ϕ satisfies strategy-proofness, individual rationality, and constrained efficiency, then ϕ satisfies top-coalition. Proof Let C ∈ F ϕ . Let
∈ D¯ be such that for each i ∈ C, top(F ϕ ,
i ) = C. If #C = 1, then by individual rationality ϕC (
) = C. If C = N , then by constrained efficiency ϕC (
) = C. Hence, assume that 1 < #C < n. For each i ∈ C, let Pi ∈ P be such that Ni+ (Pi ) = C. Let
 ∈ D¯ be such that for each i ∈ C,
i =
i− (Pi ) , and for each j ∈ N \C,
j =
j . Note that by the definition of bottom preferences, for each i ∈ C, C = top(F ϕ ,
i ) . Moreover, for each S ⊆ N , S
i {i} if and only if S ⊆ C. By individual rationality, for each i ∈ C, ϕi (
 ) ⊆ C. Hence, by constrained efficiency, for each i ∈ C, ϕi (
 ) = C. Next, let i  ∈ C. By strategy-proofness, ϕi  (
N \{i  } ,
i  )
i  ϕi  (
 ) = C = top(F ϕ ,
i  ) . Then, ϕi  (
N \{i  } ,
i  ) = C . Repeating the argument, with the remaining agents in C, we obtain that for each i ∈ C, ϕi (
) = C.   The crucial step in the proof of necessity side of Theorem 1 relies on showing that the set of feasible coalitions is endowed with the single-lapping property. The following lemmata show the result sequentially. First, we introduce a definition and a well–known result adapted to the coalition formation framework that will prove helpful in the proofs. A rule ϕ : D˜ →  is dictatorial if there is i ∈ N (a dictator) such that for each ˜ ϕi (
) = top(F ϕ ,
i ) .
∈ D, Gibbard–Satterthwaite Theorem Every strategy-proof rule on an unrestricted domain either is dictatorial or its range contains only two elements. Lemma 2 If a ϕ satisfies strategy-proofness, individual rationality, and constrained efficiency, then F ϕ satifies Condition (a) of the single-lapping property. Proof We start by introducing a useful piece of notation.  ¯ = {i}. Now for each C ⊆ N , ¯ ∈ D− be such that for each i ∈ N , top N ,
Let
i C :D ¯ C → C in the following way. For each
C ∈ D¯ C , let define the auxiliary rule ϕ
 ¯ ϕ C be defined by ϕ C (
C ), [N \C] ≡ ϕ(
C ,
N \C ). Note that, since ϕ satisfies strategy-proofness, individual rationality, and constrained efficiency, ϕ C satisfies strategy-proofness, individual rationality, and constrained efficiency. By Lemma 1, ϕ and ϕ C satisfy top-coalition. Since ϕ satisfies C top-coalition, F ϕ ≡ {X ∈ F ϕ , X ⊆ C}. We prove first that there cannot be coalitions in F ϕ involving three agents that violate Condition (a) of the single-lapping property. Claim 1 For each C ⊆ N with #C ≤ 3, F ϕ satisfies the single-lapping property. C

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The result holds trivially for each C with #C ≤ 2. Thus, let C = {i, j, k}. Assume C to the contrary that F ϕ does not satisfy Condition (a). Then, there are S, S  ∈ F ϕ  such that #(S ∩ S ) ≥ 2. Without loss of generality, assume {{i}, { j}, {k}, {i, j}, {i, j, k}} ⊆ F ϕ . C

1 ∈ D ∗ be such that,17 Let
C C


i1 :


1j :


1k :

{i, j} {i} {i, j, k} {i, k}

{i, j} { j} {i, j, k} { j, k}

{ j, k} {i, j, k} {k} {i, k}


1 = ({i, j}, {k}). By top-coalition, ϕ C
C 2 ∗ 2 1 Let
C ∈ DC be such that
C\{i} =
C\{i} and {i, j, k}
i2 {i, j}
i2 {i, k}
i2 2 2 1 2 ) is either {i, j, k} or {i}. By strategy-proofness, ϕi (
C )
i ϕi (
C ). Then, ϕiC (
C 2 ) = {i, j}. Then, {i, j}. By individual rationality, because { j} 2j {i, j, k}, ϕiC (
C 2 1 C C ϕ (
C ) = ϕ (
C ) . 3 ∈ D ∗ be such that
3 2 3 3 Let
C C j} =
C\{ j} and {i, j}
j {i, j, k}
j
{ j}.By
3  3 C\{ C C C 2 3 3 = C strategy-proofness, ϕ j
C
j ϕ j (
C ). Then, ϕ j (
C ) = {i, j} and ϕ
C
 2 . ϕ C
C 4 ∈ D ∗ be such that
4 3 4 4 Now, let
C C\{i} =
C\{i} and {i, k}
i {i, j, k}
i {i}. Then,
i4 :


4j :


4k :

{i, k} {i, j, k} {i} {i, j}

{i, j} {i, j, k} { j} { j, k}

{ j, k} {i, j, k} {k} {i, k}

4 ) = {i, j, k}. By individual rationality and constrained efficiency, we have ϕ C (
C
3 C C 4 3 Note that ϕi (
C ) = {i, j, k} i {i, j} = ϕi
C , which violates strategy-proofC ness, a contradiction. Then, F ϕ satisfies Condition (a), which concludes the proof of Claim 1. Note that if n = 3, ϕ C = ϕ and Claim 1 directly implies that F ϕ is a single-lapping collection of coalitions. We extend the preceding lemma to arbitrary sizes of coalitions.

Claim 2 For each C ⊆ N , F ϕ satisfies the single-lapping property. C

We prove first the result for coalitions formed by four agents. Thus, let X = X {i, j, k, l}. Assume, to the contrary, that there are S, S  ∈ F ϕ such that #(S ∩ S  ) ≥ 2. 17 From now on, in the description of agents’ preferences we omit the comparisons involving coalitions

that are not contained in C because they do not affect our arguments.

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C. Rodríguez-Álvarez C

Note that for each C, C  ⊆ N such that C ⊂ C  , F ϕ ⊂ F ϕ . Then, by Claim 1, S ∪ S  = X . We have two cases. Case (i) There are S, S  = X with #(S ∩ S  ) ≥ 2. Without loss of generality assume that S = {i, j, k} and S  = {i, j, l}.  ˜ ¯ Let
X \(S∩S  ) ∈ D X \(S∩S  ) be such that for k ∈ S\S , there is Pk ∈ P with + −  ¯ =
(Pk ) , l ∈ S \S, there is Pl ∈ P with N + (Pl ) = Nk (Pk ) = S and
k k l ¯ =
− (Pl ) . Define the rule ϕ˜ S∩S  : D¯ S∩S  →  in such a way that for S  and
l l  X ˜
S∩S  ∈ D¯ S∩S  , ϕ˜ S∩S (
S∩S  ) ≡ ϕ X (
S∩S  ,
X \(S∩S  ) ) . Since ϕ satisfies strategy proofness, ϕ˜ S∩S satisfies strategy-proofness. By top-coalition, for each i  ∈ S ∩ S  ,   S∩S  ∩ Ci = {i}, S, S  . Moreover, for each
S∩S  ∈ D¯ , ϕ˜i (
S∩S  ) = {i} if and F ϕ˜ only if ϕ˜ j (
S∩S  ) = { j}. By Remark 1, the preferences of the agents in S ∩ S  C

S∩S 

over the coalitions they may belong to in F ϕ˜ are not restricted. By the Gib bard–Satterthwaite Theorem, ϕ˜ S∩S is dictatorial. Without loss of generality, let i ∈  S∩S  S ∩ S  be a dictator for ϕ˜ S∩S . Let
S∩S  ∈ D¯ S∩S  be such that top(F ϕ˜ ,
i ) = S, S∩S 

and for each j ∈ (S ∩ S  )\{i} , top(F ϕ˜ ,
j ) = { j} . Then, for each j ∈ S ∩ S  , X ¯ ϕ Xj (
S∩S  ,
X \(S∪S  ) ) = S , which violates ϕ ’s individual rationality. Case (ii) X ∈ F ϕ and for each S, S  ∈ F ϕ \X , #(S ∩ S  ) ≤ 1. X In this case, {i, j, k, l} ∈ F ϕ and without loss of generality either {i, j} or {i, j, l} X X also belong to F ϕ . Let C ∈ F ϕ \X with #C ≥ 2. Let j ∈ C and define C¯ ≡ C\{ j}. ¯ there is P 1 ∈ P with j = max(N , P 1 ), Let
1X ∈ D∗X be such that for each i ∈ C, i i + − 1 1 1 1 Ni (Pi ) = C, and
i =
i (Pi ), for j there is P j1 ∈ P with N + j (P j ) = C, and
1j =
−j (P j1 ), while for each k ∈ X \C, there is Pk1 ∈ P with Nk+ (P) = { j, k}, and 1 X 1
1k =
+ k (Pk ). By top-coalition, for each i ∈ C, ϕi (
X ) = C. Next, let
2X ∈ D∗X be such that
1X \C¯ =
2X \C¯ , while for each i ∈ C¯ there is Pi2 ∈ P X

such that j = max(X, Pi2 ), Ni+ (Pi2 ) = X , and
i2 =
i+ (Pi2 ). By Claim 1, for each i ∈ ¯ top(F ϕ X ,
2 ) = X and top(F ϕ X \N ,
2 ) = C . Let i ∈ C. ¯ By strategy-proofness, C, i i ϕiX (
1X \{i} ,
i2 )
i2 ϕiX (
1X ) = C . By individual rationality, ϕ Xj (
1X \{i} ,
i2 ) = X . ¯ we get Then, ϕiX (
1X \{i} ,
i2 ) = C . Repeating the same argument with each i ∈ C, X 2 X 1 that for each i ∈ C, ϕi (
X ) = ϕi (
X ). Let
3X ∈ D∗X be such that
2X \{ j} =
3X \{ j} and
3j =
+j (P j1 ). Note that

top(F ϕ ,
3j ) = C . By strategy-proofness, ϕ Xj (
3X )
3j ϕ X (
2X ). Then, ϕ Xj (
3X ) = C, and for each i ∈ C, ϕiX (
3X ) = ϕiX (
2X ). ¯
3 =
4 , and there is P 4 ∈ P Let
4X ∈ D∗X be such that for some i ∈ C, X \{i} X \{i} i + ¯ and
4 =
+ (P 4 ). Note that for each and k¯ ∈ X \C such that Ni (Pi4 ) = {i, k}, i i i ¯ if j ∈ C. ˜ By S ⊆ X , S i4 {i} if and only if k¯ ∈ S. Analogously, S 4k¯ {k} ¯ ⊆ ϕ X (
4 ) 4 {i}. Note individual rationality, either ϕiX (
4X ) = {i}, or {i, j, k} i X i 3 X 3 X that X i ϕi (
X ). Then, by strategy-proofness, ϕi (
4X ) = X . Note also that for each C  = X with {i, j, k} ⊆ C  , #(C ∩ C  ) ≥ 2, and by Case (i) and Claim 1, X / F ϕ . Therefore, ϕiX (
4X ) = {i}. A similar reasoning implies that ϕ Xj (
4X ) = { j}. C ∈ X

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By individual rationality, for each k ∈ X \C, ϕkX (
4X ) = {k}. Finally, by Claim 1, X there is no C  ∈ F ϕ such that #C  ≥ 2 and C  ⊂ C. This implies that for each i  ∈ C\{i, j}ϕiX (
4X ) = {i  }. Thus, ϕ X (
4X ) = [X ], which contradicts constrained

efficiency, because X ∈ F ϕ and for each i ∈ X , X i {i}. Cases (i) and (ii) exhaust all the possibilities. Then, for each X ⊆ N with # X ≤ 4, X ϕ F satisfies Condition (a) of the single-lapping property. Repeating the same arguments iteratively,18 we can prove that the result holds for coalitions of arbitrary sizes, which suffices to prove the lemma.   X

Lemma 3 If ϕ satisfies strategy-proofness, individual rationality, and constrained efficiency, then F ϕ satisfies Condition (b) of the single-lapping property. Proof Assume, to the contrary, that ϕ satisfies strategy-proofness, individual rationality, and constrained efficiency and F ϕ does not satisfy Condition (b). Then, there is a list of coalitions {C1 , . . . , Cm }, with m ≥ 3 such that for each t = 1, . . . , m, (m + 1 = 1), (Ct ∩ Ct+1 ) = {∅} , and there is no i ∈ N such that for each t = 1, . . . , m, {i} = (Ct ∩ Ct+1 ) . By Lemma 2, for each t = 1, . . . , m , #(Ct ∩ Ct+1 )=1 . For each t = 1, . . . , m , let i t ≡ (Ct ∩ Ct+1 ) . Let
∈ D∗ be such that for each t = 1, . . . , m and each j ∈ (Ct \{i t−1 , i t }) , there is P j ∈ P with N + j (P j ) = C t , i t = max(N , P j ) and
j =
− (P j ) , and for each t = 1, . . . , m , there is Pit ∈ P with Ni+t (Pit ) = Ct ∪ Ct+1 , max(N , Pit ) = i t−1 and
it =
− (Pit ) . By individual rationality and Lemma 2, for each t = 1, . . . , m; ϕit (
) is either Ct , Ct+1 or {i t }. By Lemma 3, ϕ satisfies top-coalition. By strategy-proofness and top-coalition, for each t = 1, . . . , m ; ϕit (
)
it Ct . Assume first that m is odd. Then, there is t  ∈ {1, . . . , m} such that ϕit  (
) = {i t  } , a contradiction with ϕit (
)
it Ct for each t = 1, . . . , m . Assume now that m is even. Without loss of generality, assume that for each t odd, ϕit (
) = Ct+1 and for each t  even, ϕit  (
) = Ct  . Let t¯ be even. Let Pit¯ ∈ P be such that Ni+¯ (Pit¯ ) = Ct¯+1 . Let
i ¯ =
i−¯ (Pit¯ ) ∈ Di∗¯ . Note that top(F ϕ ,
i ¯ ) = Ct¯+1 and t t t t t for each T  Ct¯+1 , {i t¯} i ¯ T . By individual rationality, ϕit¯ (
N \{it¯} ,
i ¯ ) = Ct¯ . Let t t
i ¯ ∈Di∗¯ be such that top(F ϕ ,
i ¯ ) = Ct¯−1 .19 By top-coalition, ϕit¯−1 (
N \{it¯−1 ,it¯} , t −1 t −1 t −1
{i ¯ ,i ¯} ) = Ct¯−1 .By strategy-proofness,ϕit¯−1 (
N \{it¯} ,
{i ¯} )
it¯−1 ϕit¯−1 (
N \{it¯−1 ,it¯}, t −1 t t
{i ¯ ,i ¯} ). Therefore, ϕit¯−1 (
N \{it¯} ,
i ¯ ) = Ct¯−1 , and ϕit¯−2 (
N \{it¯} ,
i ¯ ) = Ct¯−1 . t −1 t t t Repeating the argument as many times as necessary, for each t odd, ϕit (
N \{it¯} ,
i ¯ ) = t Ct , while for each t  even ϕit  (
N \{it¯} ,
i ¯ ) = Ct  +1 , and ϕit¯ (
N \{it¯} ,
i ¯ ) = Ct¯+1 . t t Then, ϕit¯ (
N \{it¯} ,
i ¯ ) it¯ ϕit¯ (
) , which violates strategy-proofness, a contradict   tion. Then, F ϕ satisfies Condition (b) of the single-lapping property. Proof of Theorem 1 Pápai (2004, Theorem 3) shows that every single-lapping rule satisfies strategy-proofness, individual rationality, and constrained efficiency when agents preferences over feasible coalition is unrestricted (and preferred to any unfea18 Note that the reasoning in Case (i) applies even if the sets S\S  and S  \S contain more than one agent.

Analogously, the arguments in Case (ii) apply directly for larger societies.

+ −    19 For instance, let P  i t¯−1 ∈ P be such that Ni t¯−1 (Pi t¯−1 ) = C t¯−1 and
i t¯−1 =
i t¯−1 (Pi t¯−1 ).

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sible coalition). As single-lapping rules do not take into account preferences over unfeasible coalitions, sufficiency follows immediately. Thus, we focus on necessity. Let ϕ : D¯ →  satisfy strategy-proofness, individual rationality, and constrained efficiency. By Lemma 1, ϕ satisfies top-coalition. By Lemmata 2–3, F ϕ satisfies the single-lapping property. Let
∈ D¯ . By Remark 3, there is C ∈ F ϕ such that for each i ∈ C, top(F ϕ ,
i ) = C . By top-coalition, for each i ∈ C, ϕi (
) = C and  , ϕ (
 ) = C . Define now the auxiliary social for each
 ∈ D¯ such that
C =
C i N \C : D¯ N \C →  N \C in such a way that for each
N \C ∈ D¯ N \C , choice function ϕ¯ (ϕ¯ N \C (
N \C ), C) ≡ ϕ(
N \C ,
C ) . Clearly, ϕ¯ N \C satisfies strategy-proofness, indiN \C vidual rationality, and constrained efficiency. Moreover, F ϕ¯ = {C  ∈ F ϕ , C ∩C  = N \C ϕ ¯ {∅}}, and F satisfies the single-lapping property. Repeating the same arguments ϕ   as many times as necessary, we get ϕ(
) = σ¯ F (
) . Before following with the proof of Theorem 2, we provide a unanimity property that will prove helpful. It says that whenever there is a partition that each agent considers at least as good as every other partition, a rule should choose that best-preferred partition. Unanimity. Let σ = {C1 , . . . , Cm } ∈  be such that for each t = 1, . . . , m , ¯ each t = 1, . . . , m , and each i ∈ Ct , top(F ϕ ,
i ) = Ct Ct ∈ F ϕ . For each
∈ D, implies ϕ(
) = σ . The following lemmata show that the alternative axioms imply unanimity and topcoalition. Then, we can prove that F ϕ satisfies the single-lapping property and we conclude the proof of Theorem 2. Lemma 4 If ϕ satisfies strategy-proofness, non-bossiness, and flexibility, then ϕ satisfies unanimity. Proof Let σ = {C1 , . . . , Cm } ∈  be such that for each t = 1, . . . , m, Ct ∈ F ϕ . Let
∈ D¯ be such that for each t = 1, . . . , m and each i ∈ Ct , top(F ϕ ,
i ) = Ct . By ¯ such that ϕ(
 ) = σ . Let i  ∈ N , by strategyflexibility, σ ∈ R ϕ . Then, there is
 ∈ D,      proofness, ϕi (
N \{i  } ,
i )
i ϕi (
 ) = top(F ϕ ,
i  ) . Then, ϕi  (
N \{i  } ,
i  ) = ϕi  (
 ). By non-bossiness, ϕ(
N \{i  } ,
i  ) = ϕ(
 ). Repeating the argument as many   times as necessary, we obtain ϕ(
) = ϕ(
 ). Lemma 5 If ϕ satisfies strategy-proofness, individual rationality, non-bossiness, and flexibility, then ϕ satisfies top-coalition. Proof Let C ∈ F ϕ . Let
∈ D¯ be such that for each i ∈ C, top(F ϕ ,
i ) = C. If #C = 1, then by individual rationality, ϕC (
) = C. If C = N , then by unanimity, ϕC (
) = C. Hence, assume that 1 < #C < n. Let
 ∈ D¯ be such that for each / C, i ∈ C, there is Pi ∈ P such that Ni+ (Pi ) = C and
i =
i− (Pi ), and for each k ∈
k =
k . By individual rationality, for each i ∈ C, ϕi (
 ) ⊆ C. Let
 ∈ D− be such that for each i ∈ C,
i =
i and for each k ∈ N \C, top(F ϕ ,
k ) = ϕk (
 ) . Let k  ∈ N \C By strategy-proofness, ϕk  (
N \{k  } ,
k  )
k  ϕk  (
 ) = top(F ϕ ,
k  ) , then ϕk  (
N \{k  } ,
k  ) = ϕk  (
 ) . By non-bossiness, ϕ(
N \{k  } ,
k  ) = ϕ(
 ) . Repeating the arguments for each k ∈ N \C, we get ϕ(
 ) = ϕ(
 ) . By unanimity, for each i ∈ C, ϕi (
 ) = C . Then, for each i ∈ C, ϕi (
 ) = C . Finally, let i ∈ C. By strategy-proofness, ϕi (
N \{i} ,
i )
i ϕi (
 ) = top(F ϕ ,
i ) . Then,

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Strategy-proof coalition formation

449

ϕi (
N \{i} ,
i ) = C. Repeating the argument as many times as necessary, we obtain   that for each i ∈ C, ϕi (
) = C. Lemma 6 If ϕ satisfies strategy-proofness, individual rationality, non-bossiness, and flexibility, then F ϕ satisfies the single-lapping property. Proof The proof of Condition (a) is parallel to the proof of Lemma 2 and it is left to the appendix. Once Condition (a) is proved, note that in the proof of Lemma 3 we only use strategy-proofness, individual rationality, and top-coalition. Then, Condition (b) follows immediately from the same arguments.   Proof of Theorem 2 In order to prove sufficiency we only need to check that singlelapping rules satisfy non-bossiness and flexibility. Let ϕ be a single-lapping rule defined ¯ Let F ϕ =
. By the definition of a single-lapping rule, on a minimally rich domain D.
is a single-lapping collection of coalitions. First, let us check that ϕ satisfies nonbossiness. Let i ∈ N ,
∈ D¯ , and
i ∈ D¯ i be such that ϕi (
) = ϕi (
N \{i} ,
i ) . Let i ∈ T
(t,
) . By the definition of a single-lapping rule, for each j ∈ ∪t  ≤t T
(t  ,
) , ϕ j (
) = ϕ j (
N \{i} ,
i ) . Moreover, because ϕi (
) = ϕi (
N \{i} ,
i ) , for each k ∈ ∪t  ≥t T
(t  ,
), we have that ϕk (
) = ϕk (
N \{i} ,
i ) . Then, ϕ(
) = ϕ(
 ) , which proves non-bossiness. Finally, we check flexibility. Let σ = {C1 , . . . , Cm } ∈  be such that for each t = 1, . . . , k , Ct ∈
. Let
∈ D¯ be such that for each t = 1, . . . , m and each i ∈ Ct , top(N ,
i ) = Ct . By the definition of single-lapping rule, ϕ(
) = σ and σ ∈ R ϕ , which proves flexibility. The proof of necessity is parallel to the proof of Theorem 1. Let ϕ : D¯ →  satisfy strategy-proofness, individual rationality, non-bossiness, and flexibility. By Lemma 5, ϕ satisfies top-coalition. By Lemma 6, F ϕ satisfies the single-lapping property. Then, the same arguments of the proof of Theorem 1 apply to show that ϕ is a single-lapping rule. Acknowledgments I want to express my gratitude for the hospitality to the W.A. Wallis Institute of Political Economy at the University of Rochester where this research was initiated. I thank seminar audiences at Rochester, MEDS–Northwestern, Caltech, Carlos III, Málaga, and specially Dolors Berga, Matt Jackson, and William Thomson for their useful comments and suggestions. I am deeply indebted to an anonymous referee for extensive and thoughtful comments that allow me to shorten the proofs and to clarify the exposition. Financial support from the Consejería de Innovación, Ciencia y Empresa de la Junta de Andalucía (Proyecto de Excelencia SEJ-552), the Ministerio de Educación y Ciencia (Proyecto SEJ2005-04805 and Programa Ramón y Cajal 2006), the Fondo Social Europeo, and the Fundación Ramón Areces is gratefully acknowledged

Appendix: Omitted Proofs Lemma 7 If ϕ satisfies strategy-proofness, individual rationality, non-bossiness, and flexibility, then F ϕ satisfies Condition (a) of the single-lapping property. For each C ⊆ N , define the rule ϕ C as in the proof of Lemma 2. Note that ϕ’s strategy-proofness, individual rationality, non-bossiness, and flexibility imply that for each C ⊆ N , ϕ satisfies strategy-proofness, individual rationality, and non-bossiness. Moreover, ϕ’s top-coalition implies ϕ C ’s top-coalition and flexibility.

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C. Rodríguez-Álvarez

Claim 1 For each C ⊆ N with #C ≤ 3, F ϕ satisfies the single-lapping property. C

Proof The result holds trivially for each C with #C ≤ 2. Thus, let C = {i, j, k}. C Assume to the contrary that F ϕ does not satisfy Condition (a). Then, there are  ϕ  S, S ∈ F such that #(S ∩ S ) ≥ 2. We have two cases. Case (1 .i) F ϕ = {{i}, { j}, {k}, {i, j}, {i, j, k}}. ¯ ∈ D∗ be such that {i, j, k} ¯ k {i, k} ¯ k { j, k} ¯ k {k}. Let the rule ϕ¯ {i, j} : Let
k k ¯ ). By D¯ {i, j} →  be such that for each
{i, j} ∈ D¯ {i, j} , ϕ¯ {i, j} (
{i, j} ) ≡ ϕ C (
{i, j} ,
k ϕ C ’s strategy-proofness, ϕ¯ {i, j} satisfies strategy-proofness. By ϕ C ’s top-coalition, C

R ϕ¯

{i, j}

= {({i}, { j}, {k}), ({i, j}, {k}), {i, j, k}}. {i, j}

are unreBy Remark 1, agent i and agent j’s preferences over the partitions in R ϕ¯ stricted. Hence, ϕ¯ {i, j} satisfies strategy-proofness, its range contains three elements, and agents’ preferences over the elements of the range are unrestricted. Then, by the Gibbard–Satterthwaite Theorem, ϕ¯ {i, j} is dictatorial. Assume that i is the dictator for ∗ ϕ¯ {i, j} . Let
{i, j} ∈ D{i, j} be such that {i, j, k} i {i, j} i {i} and { j}  j {i, j}  j C {i, j, k}. Then, ϕ (
{i, j} ,
¯k ) = {i, j, k} , but { j}  j ϕ Cj (
{i, j} ,
¯k ), which violates individual rationality, a contradiction. Case (1 .ii) {{i}, { j}, {k}, {i, j}, { j, k}, {i, j, k}} ⊆ F ϕ . The line of argument of the previous case does not apply here. If agent k’s preferences are such that C is her best preferred coalition, then for some preferences of agents i and j, we can get ϕ C (
) = ({i}, { j, k}) . Agent i’s preferences over the partitions in the range of ϕ are no longer unrestricted because i is always indifferent between partitions [C] and ({i}, { j, k}). Thus, we cannot apply the Gibbard–Satterthwaite Theorem. Instead, we continue the line of argument of Claim 1 in the proof of Lemma 2. Take 4 ∈ D ∗ as defined in the proof of Claim 1 of Lemma 2. By the arguments employed
C C 4 ) = ({i}, { j}, {k}). there ϕ(
C 5 ∈ D ∗ be such that
5 =
4 , { j, k} 5 { j} 5 {i, j, k}
5 {i, j} , and Let
C C i i j j j 5 ,
4k ) = { j, k} . {i, j, k}
5k { j, k}
5k {i, k}
5k {k}. By top-coalition, ϕkC (
C\{k} 5 )
5 { j, k} . By individual rationality, because { j} 5 By strategy-proofness, ϕkC (
C k j
 5 = ({i}, { j, k}) . {i, j, k}, ϕ C
C 6 ∈ D ∗ be such that
6 5 6 6 6 Let
C C C\{ j} =
C\{ j} and {i, j, k}
j { j, k}
j {i, j}
j { j}. 6 Note that, by unanimity, ϕ C (
C\{i} ,
i3 ) = {i, j, k} . Hence, by strategy-proofness, C 6 6 C ϕi (
C )
i {i, j, k} . Then, ϕ (
C ) = {i, j, k} . 7 ∈ D ∗ be such that
7 6 7 4 Finally, let
C C C\{ j} =
C\{ j} and
j =
j . Then C

123


i7 :


7j :


7k :

{i, k} {i, j, k} {i} {i, j}

{i, j} {i, j, k} { j} { j, k}

{i, j, k} { j, k} {i, k} {k}

Strategy-proof coalition formation

451

4 and
7 consists of k’s preference. By Note that the only difference between
C C
7 7 C
6 C strategy-proofness, ϕ j
C
j ϕ j
C = {i, j, k}. By individual rationality, if
 7 ), then ϕ C (
7 ) = {i, j, k}. Hence, ϕ C
7 j ∈ ϕiC (
C C C = {i, j, k}. However, i

 7 4 ϕ C
4 , which violates strategy-proofness, a contradiction.   ϕkC
C k k C

Claim 2 . For each C ⊆ N , F ϕ satisfies the single-lapping property. C

Proof We prove first the result for coalitions formed by four agents. Again, the same arguments can be repeated iteratively to prove the result for arbitrary coalitions. Thus, X let X = {i, j, k, l}. Assume, to the contrary, that there are S, S  ∈ F ϕ such that C C #(S ∩ S  ) ≥ 2. Note that for each C, C  ⊆ N such that C ⊂ C  , F ϕ ⊂ F ϕ . Then, by Claim 1, S ∪ S  = X . We have two cases. Case (2 .i) Either there are S, S  ∈ F ϕ \X with #(S ∩ S  ) ≥ 2, or X ∈ F ϕ and there X X is S  ∈ F ϕ \X with #S  ≥ 2 and such that for each C ∈ F ϕ \{S  , X } with #C ≥ 2, C ∩ S  = ∅. The arguments applied in Case (i) of Claim 2 in the proof of Lemma 2 apply without any modification here. Just note that the reasoning works even if no agent belongs either to S\S  or to S  \S. X

Case (2 .ii) X ∈ F ϕ and for each C ∈ F ϕ \X with #C ≥ 2 there is C  ∈ F ϕ with #C  ≥ 2 and #(C ∩ C  ) = 1. X X Let C ∈ F ϕ \X with #C ≥ 2. Let j ∈ C and assume that there is C  ∈ F ϕ \{X, C} with j ∈ C  and #C  ≥ 2. Define C¯ ≡ C\{ j}. Take
4X ∈ D∗X as defined in Case (ii) of Claim 2 in the proof of Lemma 2. By the same arguments employed there, ϕ X (
4X ) = [X ]. Consider the agents i ∈ C¯ and k¯ ∈ C  described in the definition of profile
4X in ¯ Claim 2 of Lemma 2. Consider now the profile
5X ∈ D∗X , such that for each i  ∈ C, + 5 4 5 5 ¯ ˆ
i  =
i  , there is P j ∈ P such that N j (P j ) = X , k = max(N , P j ), for each k ∈ C  and each kˆ  ∈ / C  , kˆ P j kˆ  , and
5 =
+ (P 5 ), and for each k ∈ X \C, there is P 5 ∈ P X

X

j

j

j

X

k

5 such that j = max(N , Pk5 ), Nk+ (Pk5 ) = X , and
5k =
+ k (Pk ). Note that by unanim3 5 X 5 X ity, ϕ (
i ,
X \{i} ) = X . By strategy-proofness, ϕi (
X )
i5 X . Then, k¯ ∈ ϕiX (
∗X ). Consider now
X ∈ D∗X , such that for each k ∈ C  , there is Pk ∈ P with N + (Pk ) = C  and
k =
− (Pk ), and for each l ∈ X \C  ,
l =
l5 . By top-coalition, for each k ∈ C  , ϕkX (
X ) = C  . Let k  ∈ C  , by strategy-proofness, ϕkX (
X \{k  } ,
∗k  )
∗k  C  . By Claim 1 and Case (2 .i), either ϕkX (
X \{k  } ,
∗k  ) = X or ϕkX (
X \{k  } ,
∗k  ) = C  . Thus, C  ⊆ ϕkX (
X \{k  } ,
∗k  ). Repeating the argument with the remaining agents in C  , we obtain that for each k ∈ C  , C  ⊆ ϕk (
∗X ). Thus, k¯ ∈ ϕiX (
∗X ) and k¯ ∈ C  imply that C  ⊆ ϕiX (
∗X ). Finally, by Claim 1 and Case (2 .i), ϕ X (
∗X ) = X . Repeating the ¯ we obtain that ϕ X (
5 ) = X . argument as many times as necessary for each i ∈ C, X 6 ∗ 6 4 , while
6 5 =
Finally, let
X ∈ D X , be such that for each
C =
C X \C X \C . That is, we only change agent j’s preferences with respect to the previous profile. By strategy-proofness, ϕ Xj (
6X )
6 ϕ Xj (
5X ) = X . By individual rationality, for each ¯ if j ∈ ϕ X (
6 ) , then ϕ X (
6 ) = X . Hence, ϕ X (
6 ) = X . Clearly,
6 i ∈ C, i X i X X X

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only differs from
4X in the preferences of the agents who belong to X \C. Let k ∈ X \C. By strategy-proofness, we have that ϕkX (
6X \{k} ,
4k )
4k ϕkX (
6X ) = X . Then, j ∈ ϕkX (
6X \{k} ,
4k ). By individual rationality, there is i ∈ C¯ such that i ∈ ϕ Xj (
6X \{k} ,
4k ). Then {i, j, k} ∈ ϕkX (
6X \{k} ,
4k ). Thus, Claim 1 and Case (2 .i), ϕ X (
6X \{k} ,
4k ) = X . Repeating the argument as many times as necessary, we obtain ϕ X (
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