Strategy-Proof Coalition Formation

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. However, reasonable strategy-proof rules exist if appropriate restrictions are imposed on agents’ preferences. In coalition formation problems, such domain For further references, see the recent works by Banerjee, Konishi, and Sönmez (2001), Barberà and Gerber (2003), Bogomolnaina and Jackson (2002), and Pápai (2004). 1 C en tr o d e E st u d io s A n d al u ce s restrictions arise naturally. On the one hand, while coalition formation rules select a partition for each preference profile, 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 class of problems consists of situations in which there are no complementarities among the members of a coalition. That is, the preferences of an agent i regarding the convenience of an agent j joining the coalition i belongs to, do not depend on the coalition to which i is assigned. 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, Sonnenschein, and Zhou (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 when agents’ preferences are additively representable or separable has not been addressed in the literature. Besides strategy-proofness, we would like our rules to satisfy four additional properties. Our rules should be individually rational, Pareto efficient, non-bossy, and flexible. Individual rationality is a participation constraint. It means that no agent should ever be worse-off than she would be if staying alone. Pareto efficiency requires that the whole society of agents should not prefer a partition formed by feasible coalitions rather than forming the partition that the rules recommends. Non-bossiness is a collusion-proof requirement. It says that if a change in an agent’s preferences does not affect the coalition to which this agent is assigned, then the remaining agents are also unaffected by this change of preferences. Flexibility is implied by Pareto efficiency. It says that every partition formed by a collection of feasible coalitions belongs to the range of the rule. Hence, flexibility implies that feasible disjoint coalitions are mutually compatible. We provide two characterizations of a family of rules, the family of single-lapping rules, that fulfill the previous axioms in minimally rich domains of preferences (as the domain of additively representable preferences). 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. The 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. 2 C en tr o d e E st u d io s A n d al u ce s single-lapping property was firstly introduced by Pápai (2004), who shows that it is a necessary and sufficient condition for the existence of a unique core-stable partition of the society for every profile of agents’ preferences. In fact, single-lapping rules always select the unique core-stable partition of the society, in the sense that no feasible coalition of agents unanimously prefer joining each other rather than staying at the coalition they are assigned to. Hence, our results provide further evidence on the relation between the noncooperative game theory concept of strategy-proofness and the cooperative game theory concept of the existence of a unique core-stable partition. Before proceeding with the formal analysis, we review the most related literature. This paper is is closely related to the already mentioned Pápai (2004). Pápai’s main focus is on finding necessary and sufficient conditions on the set of feasible coalitions for t uniqueness of core-stable partitions. Additionally, this author shows that, given an initial set of coalitions that satisfy the single-lapping property, its associated single-lapping rule is the unique rule that satisfies strategy-proofness, individual rationality, and Pareto efficiency when agents’ preferences over coalitions 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 implied directly by strategy-proofness and the remaining axioms. Moreover, we show that the results also hold in more restricted domains of preferences over coalitions. The manipulability of coalition formation rules has also been studied by Alcalde and Revilla (2004), Cechlárová and Romero-Medina (2001), Sönmez (1999), and Takamiya (2003). However, these works focus on different domains of preferences that are not consistent with additively representable or separable preferences. More specifically, Alcalde and Revilla (2004), and Cechlárová and Romero-Medina (2001) assume that agents’ preferences over coalitions are based on the best or the worst group of agents in each coalition. In these environments, they prove the existence of strategy-proof rules that always select core-stable partitions. Finally, Sönmez (1999) proposes a general model of allocation of indivisible goods which includes our coalition formation model as a special case. This author focuses on problems for which there always exist core-stable partitions. Under some assumptions on agents’ preferences, Sönmez (1999) shows that there exist strategy-proof, individually rational, and Pareto efficient rules only if the set of core-stable partitions is always essentially single-valued. Takamiya (2003) proves that the converse result also holds under additional assumptions on preferences –such as strict preferences and no 3 C en tr o d e E st u d io s A n d al u ce s consumption externalities– that are fulfilled in coalition formation problems. The remainder of the paper is organized as follows. In Section 2, we present the model and basic notation. In Section 3, we present different domains of preferences over coalitions and the notion of minimally rich domain. In Section 4, we introduce the main axioms while in Section 5 we present single-lapping rules and provide the characterization results. In Section 6, we prove Theorem 2. We include the proofs of some intermediate results and supplemental material in the Appendices.