Cooperative games with diversity constraints

A cooperative game with diversity constraints is given by a cooperative game, a coalition structure which partitions the set of players into communities, and a vector of integers specifying, for each community, the minimal number of its members that a coalition must possess to be considered as diverse. We provide axioms for a value on the class of such co- operative games with diversity constraints. Some combinations of axioms characterize two values inspired by the Shapley value (Shapley, 1953) and the Owen value (Owen, 1977) for games with a coalition structure. More specifically, the Diversity Owen value is character- ized as the Owen value of the diversity-restricted game with a coalition structure, where the diversity-restricted game assigns a null worth to a coalition if it does not meet the diversity requirements or its original worth otherwise. Similarly, the Diversity Shapley value is char- acterized as the Shapley value of the diversity-restricted game (without coalition structure). Some of our axiomatic characterizations can be adapted to the class of simple games by replacing the Additivity axiom by the Transfer axiom (Dubey, 1975).