A Simplified Expression of Share Functions for Cooperative Games with Fuzzy Coalitions

Abstract

In this paper, we discuss the notion of Share functions for cooperative games with fuzzy coalitions or simply fuzzy cooperative games. We obtain the Share functions for some special classes of fuzzy games, namely the fuzzy games in proportional value form and the fuzzy games in Choquet integral form. The Shapley Share and Banzhaf Share functions for these classes are derived.

Solidarity Value and Solidarity Share Functions for TU Fuzzy Games

“…In this section we compile the definitions and results necessary for the development of the present study from [2,6,8,9,11,12,15,16,18]. We start with the notion of solidarity values and share functions in crisp games.…”

“…TU fuzzy games derived from their crisp counterparts are found in the literature; see, for example, [9][10][11][12][13][14][15]. The Shapley share function for TU fuzzy games is studied in [16]. The relevance of the solidarity value and the corresponding share function for TU fuzzy games can be realized in situations where players with partial participations are marginally unproductive but being the part of the cooperative endeavor may be rewarded with some nonzero payoffs.…”

“…These two classes are continuous with respect to the standard metric and also monotonic when the associated crisp game is monotonic. Moreover they build on the idea of nonadditive interactions among the players; for more details we refer to [15][16][17].…”

α-consensus Value of Cooperative Game with Intuitionistic Fuzzy Payment

“…As we all known, due to the influence of economic, political and other social factors, in some scenarios, there are some uncertainties in the game process. So fuzzy cooperative games have been extensively studied [11][12][13][14]. Fuzzy sets can represent the uncertain information of cooperative games, but they cannot reflect the hesitation degrees of the players.…”