Cooperative Games under Augmenting Systems

The goal of this paper is to develop a theoretical framework in order to analyze cooperative games in which only certain coalitions are allowed to form. We will axiomatize the structure of such allowable coalitions using the theory of antimatroids, a notion developed to combinatorially abstract sets. There have been previous models developed to confront the problem of unallowable coalitions. Games restricted by a communication graph were introduced by Myerson (1977) and Owen (1986). Another type of combinatorial structure introduced by Gilles, Owen and van den Brink (1992) is equivalent to a subclass of antimatroids. We introduce a new combinatorial structure called augmenting system which is a generalization of the antimatroid structure and the system of connected subgraphs of a graph. The main result of the paper is a direct formula of Shapley’s and Banzhaf’s values for games under augmenting systems restrictions.