Linear fuzzy game with coalition interaction and its coincident solutions

Abstract

In this study, we consider the solution concepts for fuzzy coalition games (i.e., cooperative games with fuzzy coalition) under a certain participation level. In general, cooperative games with fuzzy coalition are based on the assumption that all the fuzzy coalition values for different fuzzy coalitions must be represented by the same formula, which may omit the coalition interaction under different participation ratios for players. Considering these conditions, we propose the coincident fuzziness form for games with fuzzy coalition, which are represented by a mapping from the characteristic function of the crisp game to that of the fuzzy coalition game. The proposed fuzzy coalition games admit the differences in coalition interactions for different fuzzy coalitions, where the coalition interactions are represented by the fuzzy coalition values in different ways (or formulas). For a fixed fuzzy coalition, the maximum fuzzy coalition game is proven to be the Choquet integral form on the condition that the associate crisp game is convex. In order to seek appropriate solutions for the proposed games based on a certain participation level, the Fuzzy-Shapley axioms are defined, and the explicit Shapley value is represented by the Shapley value of the associated crisp games. Moreover, the fuzzy core of this proposed fuzzy coalition game is the stable solution set for the given fuzzy coalition, which is also denoted by the crisp cores. Furthermore, we study the relationship between the Fuzzy-Shapley value and the fuzzy core. By adding restrictions on the fuzzy core, we propose a strong Fuzzy-core, which is a more stable solution than the fuzzy core. In addition, the fuzzy core is equivalent to the strong Fuzzy-core under some condition.