Fuzzy Harsanyi solution for a kind of fuzzy coalition games

Abstract

The fuzzy linear game in Ref. [32] is a class of fuzzy cooperative games with fuzzy coalition (fuzzy coalition game for short), which admits the difference of interactions for different fuzzy coalitions. The fuzzy Shapley value for fuzzy linear game has been studied. As Shapley value is just one point in the Harsanyi solution, it is interesting to study the explicit form of Harsanyi solution in the fuzzy coalition game. This paper defines the fuzzy Harsanyi solution which distributes the Harsanyi dividends such that the dividend shares of players in each fuzzy coalition are proportional to the corresponding players' participation index. When the fuzzy coalition payoff is average distributed between players in the fuzzy coalition, the fuzzy Harsanyi solution also coincides with the fuzzy Shapley value just as their relationship in a crisp cooperative game. We provide two axiomatic characterizations for the fuzzy Harsanyi solution: one uses axioms, and the other uses distribution matrices. The fuzzy Harsanyi solution is a unique value on the fuzzy linear game that satisfies fuzzy efficiency, fuzzy null player and fuzzy additivity property. Meanwhile, based on probability distribution, fuzzy Harsanyi solution is seen as an expected marginal contribution of each player in the fuzzy coalition, which provides more choices for players in the fuzzy coalition.