Generalized Shapely Value for Games with a Coalition Structure

In cooperative games with transferable utility, there is usually no restriction on the possible coalitions that can materialize. A significant departure from this tradition, occurs in Moulin [1995], where the concept of admissible coalitions arise. In this paper we consider cooperative games with admissible coalitions, requiring that both the grand coalition, as well as, the null coalition are always admissible. We call such games, ‘games with a coalition structure’. We define the concept of a core for such games and introduce a generalization of the notion of Shapley value. We define this generalized Shapley value to be the unique value satisfying the Dummy Player Property, Anonymity and Linearity. All these properties have been adapted from the standard context to our framework in such a way, that the existence of a unique value satisfying these properties is guaranteed. We subsequently consider specific coalition structures and obtain closed form solutions for the generalized Shapley value in each case.