Characterization of the Equilibrium Strategy of Fuzzy Bimatrix Games Based onL-RFuzzy Variables

Cun-Lin Li

doi:10.1155/2012/824790

Abstract

This paper deals with bimatrix games in uncertainty environment based on several types of ordering, which Maeda proposed. But Maeda’s models was just made based on symmetrical triangle fuzzy variable. In this paper, we generalized Maeda’s model to the non-symmetrical environment. In other words, we investigated the fuzzy bimatrix games based on nonsymmetricalL-Rfuzzy variables. Then the pseudoinverse of a nonconstant monotone function was given and the concept of crisp parametric bimatrix games was introduced. At last, the existence condition of Nash equilibrium strategies of the fuzzy bimatrix games is proposed and (weak) Pareto equilibrium of the fuzzy bimatrix games was obtained through the Nash equilibrium of the crisp parametric bimatrix.

Highlights

Nash presented noncooperative game theory 1, 2 based on the assumption that each player has a well-defined quantitative utility function over a set of the player’s strategy, each player attempts to optimize his own expected payoffs, and each is assumed to know the extensive game completely
This paper developed a new theoretical approach to deal with the bimatrix games with fuzzy payoffs
By investigating crisp parametric bimatrix games, we presented a method to figure out the Nashe quilibrium strategy and weak Pareto equilibrium strategy of fuzzy bimatrix games

Summary

Introduction

Nash presented noncooperative game theory 1, 2 based on the assumption that each player has a well-defined quantitative utility function over a set of the player’s strategy, each player attempts to optimize his own expected payoffs, and each is assumed to know the extensive game completely. Since the expected payoffs of the player are fuzzy variable, we should define new concepts of equilibrium strategy to investigate their properties. Takashi 12, 13 presented three kinds of equilibrium strategies of fuzzy matrix games based on especial symmetric triangular fuzzy variable and investigated the existence condition of these equilibrium strategies. This paper was going to generalize Maeda’s model and investigate all types of equilibrium strategies based on more general asymmetric L-R fuzzy variables. We introduce the crisp parametric matrix games and its equilibrium strategies in order to find the Pareto equilibrium strategies of fuzzy bimatrix games. The relation between the Nash equilibrium strategies of crisp bimatrix parametric games and Pareto equilibrium strategies of fuzzy bimatrix games is established

Preliminaries

Conclusion