Coalition Structures and Proportional Share Analysis for TU Cooperative Games

Selten in 1972 developed a new theory called �equal share analysis�. This theory was expressed in the form of three hypothesis about the outcome of characteristic function games with restricted coalition structures. First hypothesis says that there will be a strong tendency to form an �exhaustive coalition structure� which means that in the final coalition structure there are no coalitions who have incentives to merge. The second hypothesis asserts that the payoff vector will have a strong tendency to be in the equal division core, which is a new game theoretic concept introduced by Selten. A payoff vector is in the equal division core, if no coalition can divide its value equally among its members and in this way give more to each of the members than they receive in the payoff vector. After giving an order of strength within the players, the third hypothesis says that within a coalition a stronger player will not receive less than a weaker player. The result of the play is a configuration, that consists of a coalition structure which is a partition of N into permissible coalitions and a payoff vector. A configuration that satisfies the three hypothesis above is called regular configuration. Selten showed that a regular configuration exists for any game. In this paper we extend Selten�s result assuming that every player has an initial weight ai > 0 i = 1, . . . , n. We introduce the concept of the proportional division core which depends on the vector a and is very related to the equal division core. We say that a payoff vector is in the proportional division core if no coalition can divide it�s value proportionally to a among it�s members and in this way give more to all its members than they receive in the payoff vector divided by ai. On the other hand we introduce a new order of strength with respect to a within the players, and we consider that a coalition structure and a payoff vector is a proportional regular configuration if the coalition structure is exhaustive, the payoff vector is in the proportional division core and it preserves the order of strength with respect to a between the players in the same coalition. The main result in the paper is that for any cooperative TU game and for any vector a > 0 we can always guarantee the existence of a proportional regular configuration.