Subordinated hedonic games

Abstract

Hedonic games are simple models of coalition formation whose main solution concept is that of core-partition. Several conditions guaranteeing the existence of core-partitions have been proposed so far. In this paper, we explore hedonic games where a reduced family of coalitions determines the development of the game. We allow each coalition to select a subset of it so as to act as its set of representatives (a distribution). Then, we introduce the notion of subordination of a hedonic game to a given distribution. Subordination roughly states that any player chosen as a representative for a coalition has to be comfortable with this decision. With subordination we have a tool, within hedonic games, to compare how a "convenient" agreement reached by the sets of representatives of dierent groups of a society is "valued" by the rest of the society. In our approach, a "convenient" agreement is a core-partition, so this paper is devoted to relate the core of a hedonic game with the core of a hedonic game played by the sets of representatives. Thus we have to tackle the existence problem of core-partitions in a reduced game where the only coalitions that matter are those prescribed by the distribution as a set of representatives. We also study how a distribution determines the whole set of core-partitions of a hedonic game. As an interesting example, we introduce the notion of hedonic partitioning game, which resembles partitioning games studied in the case where a utility, transferable or not, is present. The existence result obtained in this new class of games is later used to provide a non constructive proof of the existence of a stable matching in the marriage model.