Games with fuzzy authorization structure: A Shapley value

A cooperative game consists of a set of players and a characteristic function which determines the maximal gain or minimal cost that every subset of players can achieve when they decide to cooperate, regardless of the actions that the other players take. It is often assumed that the players are free to participate in any coalition, but in some situations there are dependency relationships among the players that restrict their capacity to cooperate within some coalitions. Those relationships must be taken into account if we want to distribute the profits fairly. In this respect, several models have been proposed in literature. In all of them dependency relationships are considered to be complete, in the sense that either a player is allowed to fully cooperate within a coalition or they cannot cooperate at all. Nevertheless, in some situations it is possible to consider another option: that a player has a degree of freedom to cooperate within a coalition. A model for those situations is presented.