k-Coalitional Cooperative Games

In most previous models of coalition structure generation, it is assumed that agents may partition themselves into any coalition structure. In practice, however, there may be physical and organizational constraints that limit the number of co-existing coalitions. In this paper, we introduce k-coalitional games: a type of partition function game especially designed to model such situations. We propose an extension of the Shapley value for these games, and study its axiomatic and computational properties. In particular, we show that, under some conditions, it can be computed in polynomial time given two existing representations of coalitional games with externalities. Finally, we use k-coalitional games to analyse the relative importance of geographical locations in the game of Diplomacy.