A fuzzy value for the fuzzy TU-game based on the relative group marginal contribution of coalition with restricted size

Abstract

Considering the group marginal contribution of coalitions (with size no larger than k), we study the cooperative game (transferable utility game (TU-game)) with a fuzzy payoff (i.e., fuzzy TU-game). In order to reflect the impact of the coalition's size on the group contribution, the fuzzy relative group's marginal contribution (i.e., fuzzy RGMC) is introduced, and a new fuzzy value (i.e., the k-lateral per-capita value) is then proposed for the fuzzy TU-game based on the fuzzy RGMC. The k-lateral per-capita value is an extended form of the fuzzy Shapley value, where the coalition is restricted in size and the group contribution is considered. Similar to the fuzzy Shapley value, the k-lateral per-capita value satisfies the common axioms in the fuzzy TU-game, such as efficiency, symmetry and additivity, but not the null player property. In addition, the potential function of the fuzzy TU-game is extended, and the k-lateral per-capita value is characterized by a potential function. The proposed new value (i.e., k-lateral per-capita value) can be used as a solution for the fuzzy TU-game but it also considers the RGMC of a coalition with restricted size.