Core in a simple coalition formation game

Abstract

We analyze the core of a class of coalition formation game in which every player's payoff depends only on the members of her coalition. We first consider anonymous games and additively separable games. Neither of these strong properties guarantee the existence of a core allocation, even if additional strong properties are imposed. We then introduce two top-coalition properties each of which guarantee the existence. We show that these properties are independent of the Scarf-balancedness condition. Finally we give several economic applications.

Group activity selection problem with approval preferences

“…The model considered in this paper is closely related to that of anonymous hedonic games (Banerjee et al 2001), where, just as in our setting, players have to split into groups and each player has preferences over possible group sizes. The main difference between anonymous hedonic games and our problem is that, in our setting, the agents' preferences depend not only on the group size, but also on the activity that has been allocated to their group; thus, our model can be seen as a generalization of anonymous hedonic games.…”

“…The main difference between anonymous hedonic games and our problem is that, in our setting, the agents' preferences depend not only on the group size, but also on the activity that has been allocated to their group; thus, our model can be seen as a generalization of anonymous hedonic games. We remark that we can represent our problem as a general (i.e., nonanonymous) hedonic game (Banerjee et al 2001;Bogomolnaia and Jackson 2002), by creating a dummy agent for each activity and endowing it with suitable preferences (see Sect. 2.2 for details).…”

“…Recall that a hedonic game (Banerjee et al 2001;Bogomolnaia and Jackson 2002) is given by a set of agents N , and, for each agent i ∈ N , a weak order ≥ i over all coalitions (i.e., subsets of N ) that include i (the associated strict preference and indifference relations are denoted by > i and = i , respectively). That is, in a hedonic game each agent has preferences over coalitions that he can be a part of.…”

Strategy-proof coalition formation

“…However, reasonable strategy-proof rules exist if appropriate restrictions are imposed on agents' preferences. In coalition formation problems, such domain restrictions 1 For further references, see the recent works by Banerjee, Konishi, and Sönmez [3], Barberà and Gerber [4], Bogomolnaia and Jackson [6], and Pápai [15]. 1 arise naturally.…”

A Note on the Impossibility of a Satisfactory Concept of Stability for Coalition Formation Games

“…In general, coalition structures in the core of the coalition formation game would be great candidates for a solution (though not necessarily the only ones). But since the core is empty unless one imposes strong assumptions on the domain of games (see Banerjee, Konishi, andSönmez, 2001, andBogomolnaia andJackson, 2002), it has rather limited predictive power. For our example, the only possibilities left by the preceding axioms were to either identify three stable coalition structures or none.…”