Dynamic Coalition Formation in the Apex Game

Abstract

This paper studies stability of coalition structures in the apex game by the use of Bloch's (1996) model of dynamic coalition formation. In our model, players' payoffs are given by the coalition values and a cost for proposal is newly introduced. We study stable coalition structures by the subgame perfect equilibrium and we show that in the apex game stable coalition structures depend on the cost for making a proposal and who the first proposer is. As opposed to the static analysis by Hart and Kurz (1983), it turns out that the apex may form a two-person coalition with a minor player. Owen (1977) defined the coalition value, that is, a generalized Shapley value (Shapley, 1953) with a priori coalition structure. Hart and Kurz (1983) studied stability of coalition structures using the coalition value. Their models were presented as strategic form games, and the strong Nash equilibrium (Au­ mann, 1967) was used to study stable coalition structures. In particular, they studied stable coalition structures in the apex game, and showed that if the number of the players is greater than or equal to five, the coalition of all minor players is a unique stable coalition. On the other hand, a dynamic process of coalition formation was investigated in Bloch (1996). In this paper, we apply Bloch's dynamic coalition forma­ tion model to the apex game after slightly modifying the model. Players' payoffs are given by the coalition value and costs for making proposals are newly introduced. We use the subgame perfect equilibrium to study stable coalition structures.