On Nash Equilibria in a Finite Game for Fuzzy Sets of Strategies

Abstract

The present paper investigates a finite game with fuzzy sets of player strategies. It is proven that Nash equilibria constitute a type-2 fuzzy set defined on the universal set of strategy profiles. Furthermore, the corresponding type-2 membership function is provided. This paper demonstrates that the Nash equilibria type-2 fuzzy set of the game can be decomposed based on the secondary membership grades into a finite collection of crisp sets. Each of these crisp sets represents the Nash equilibria set of the corresponding game with crisp sets of player strategies. A characteristic feature of the proposed decomposition approach is its independence from the chosen method for calculating the Nash equilibria in crisp subgames. Some properties of game equilibria T2FSs are studied. These sets correspond to specific partitions or cuts of the original fuzzy sets of player strategies. An illustrative example is also included for clarity.