ABSTRACT
We discuss several sets of cooperative games in which the Shapley value assigns zero payoffs to all players. Each set spans the kernel of the Shapley value and leads to a different characterization of games with identical Shapley values. The problem of identifying all games that generate a given vector of Shapley values became known as the “inverse problem” in the literature. The special games we identify reflect a clear balance of power among players and coalitions and deliver intuitive axiomatizations of the Shapley value. We explain how each basis of the kernel of the Shapley value can be augmented to construct a basis of the space of all games. Finally, we also survey alternative bases for the kernel of the Shapley value from the literature and explore other interesting games that belong to the kernel of the Shapley value.
