The core-partition of a hedonic game

Abstract

A pure hedonic game describes the situation where player's utility depends only on the identity of the members of the group he belongs to. The paper provides a necessary and sufficient condition for the existence of core-partition in hedonic games. The condition is based on a new concept of balancedness, called pivotal balancedness. Pivotal balancedness involves especially the notion pivotal distribution that associates to each coalition a subgroup of players in the coalition. Then, we proceed to a review of several sufficient conditions for core-partition existence showing how the results can be unified through suitably chosen pivotal distributions.

Subordinated Hedonic Games

“…They are a straightforward extension of hedonic games as stated by Banerjee et al [6] and Bogomolnaia and Jackson [7] where there is no restriction on the family of admissible coalitions. For this enlarged class of hedonic games we state two balancedness conditions extending those of ordinal balancedness of Bogomolnaia and Jackson [7] and pivotal balancedness of Iehlé [8]. Following those authors, we show that both conditions are sufficient to guarantee the existence of core partitions, while pivotal balancedness is also a necessary condition.…”

“…In order to deal with the reduced game played by the coalitions of representatives we present some results about a hedonic game with a restricted family of admissible coalitions. They are simple extensions of the main existence results proved by Iehlé [8]. Pápai [9] has already used the idea of restrict the family of feasible coalitions to get uniqueness results about core partitions in hedonic games, while the idea of subordination (Section 3) seems to be new.…”

“…There are several sufficient conditions for the existence of core partitions, being those of top-coalition properties [6], consecutiveness, and ordinal balancedness [7] among the most well known. Iehlé [8] gives a complete answer to the existence problem in a general setting by introducing the notion of pivotal balancedness. To define pivotal balancedness first it is associated one of its subsets to each coalition .…”

“…To define pivotal balancedness first it is associated one of its subsets to each coalition . Iehlé [8] calls this family of subsets a distribution. Iehlé's condition states that a hedonic game has nonempty core if and only if there is a distribution such that the game is pivotally balanced with respect to this distribution (see Section 2).…”

Status-Seeking in Hedonic Games with Heterogeneous Players

“…Banerjee et al (2001), Bogomolnaia and Jackson (2002), and Ihlé (2007), among others, introduce various notions of stability and provide sufficient conditions for the existence of stable partitions in hedonic games. In this literature, however, identity is summarized in the index of each agent and authors do not discuss heterogeneous types of agents.…”