Unique Stability in Simple Coalition Formation Games

In this paper we study the existence and uniqueness of stable coalition structures, when the value of a coalition to a member depends solely on the other members of the coalition. We search for collections of admissible coalitions that induce uniquely stable coalition structures, that is, ensure that there is a unique stable coalition structure at every preference profile when only admissible coalitions may form. A collection of coalitions satisfies the single-lapping property if (a) no two coalitions have more than one member in common, and (b) in a cycle formed by coalitions with a non-empty intersection all the coalitions have the same member in common. We prove that a collection of coalitions induces a unique stable coalition structure if and only if it satisfies the single-lapping property. We also provide a graph representation of collections of coalitions that satisfy the single-lapping property. This alternative representation is used to explore implications of our characterization result for matching problems such as the marriage and roommate problems. We also discuss the implications for the existence of strategyproof rules of coalition formation.