A simplified expression of the Shapley function for fuzzy game

In this paper, a simplified expression of the Shapley function for games with fuzzy coalition is proposed, which can be regarded as the generalization of Shapley functions defined in some particular games with fuzzy coalition. The simplified expression of the Shapley function is compared with two definitions established by Butnariu, Tsurumi et al. A conclusion is drawn that the simplified expression of the Shapley function is equivalent to Butnariu's definition when characteristic function is a game with proportional values, and is equivalent to Tsurumi's definition when characteristic function is a game with Choquet integral forms. Furthermore, from an angle of interaction between two participation levels, the properties of the two games defined by Butnariu and Tsurumi are respectively studied.

[1]  Michel GRABISCH,et al.  The Interaction and Möbius Representations of Fuzzy Measures on Finite Spaces, -Additive Measures: A Survey , 2022 .

[2]  L. Shapley.  A Value for n-person Games , 1988 .

[3]  Najmeh Mahjouri,et al.  Optimal Inter-Basin Water Allocation Using Crisp and Fuzzy Shapley Games , 2010 .

[4]  Dan Butnariu,et al.  Shapley mappings and the cumulative value for n-person games with fuzzy coalitions , 2008, Eur. J. Oper. Res..

[5]  M. Sakawa,et al.  A lexicographical solution concept in an n -person cooperative fuzzy game , 1994 .

[6]  Ivan Kojadinovic,et al.  Modeling interaction phenomena using fuzzy measures: on the notions of interaction and independence , 2003, Fuzzy Sets Syst..

[7]  Milan Mares,et al.  Review: "Models in Cooperative Game Theory, Crisp, Fuzzy and Multi-Choice Games" by Rodica Branzei, Stef Tijs and Dinko Dimitrov , 2006, Kybernetika.

[8]  Shigeo Muto,et al.  On cores and stable sets for fuzzy games , 2004, Fuzzy Sets Syst..

[9]  G. Owen.  Multilinear Extensions of Games , 1972 .

[10]  M. Sugeno,et al.  Fuzzy Measures and Integrals: Theory and Applications , 2000 .

[11]  Masahiro Inuiguchi,et al.  A Shapley function on a class of cooperative fuzzy games , 2001, Eur. J. Oper. Res..

[12]  Qiang Zhang,et al.  The measure of interaction among players in games with fuzzy coalitions , 2008, Fuzzy Sets Syst..

[13]  Michel Grabisch,et al.  K-order Additive Discrete Fuzzy Measures and Their Representation , 1997, Fuzzy Sets Syst..

[14]  Ivan Kojadinovic,et al.  An axiomatic approach to the measurement of the amount of interaction among criteria or players , 2005, Fuzzy Sets Syst..

[15]  D. Butnariu.  Stability and Shapley value for an n-persons fuzzy game , 1980 .

[16]  Michel Grabisch,et al.  An axiomatic approach to the concept of interaction among players in cooperative games , 1999, Int. J. Game Theory.

[17]  J. Aubin.  Mathematical methods of game and economic theory , 1979 .