A class of consistent share functions for games in coalition structure

A cooperative game with transferable utility {or simply a TU-game{ describes a situation in which players can obtain certain payo®s by cooperation. A value function for these games is a function which assigns to every such a game a distribution of the payo®s over the players in the game. An alternative type of solutions are share functions which assign to every player in a TU-game its share in the payo®s to be distributed. In this paper we consider cooperative games in which the players are organized into an a priori coalition structure being a ̄nite partition of the set of players. We introduce a general method for de ̄ning a class of share functions for such games in coalition structure using a multiplication property that states that the share of player i in the total payo® is equal to the share of player i in some internal game within i's a priori coalition, multiplied by the share of this coalition in an external game between the a priori given coalitions. We show that these coalition structure share functions satisfy certain consistency properties. We provide axiomatizations of this class of coalition structure share functions using these consistency and multiplication properties. JEL classi ̄cation number: C71