Nash Equilibrium in Fuzzy Random Bi-Matrix Games

Abstract

Most of the existing works on games under uncertainty consider only one type of uncertainty: fuzzy, random, rough, etc. However, in real-world games, it often happens that randomness and fuzziness simultaneously affect the interaction between players and their payoffs. Investigating fuzzy random games is challenging as it is difficult to express the preferences of players in the presence of two different types of uncertainty. This paper presents a new approach to games involving randomness and fuzziness. Specifically, we consider bi-matrix games where the payoffs are fuzzy random variables. Using probability and possibility measures, we formulate related fuzzy chance-constrained games. Then, we introduce Nash equilibrium for these games. Next, we establish sufficient conditions for the existence of this equilibrium. Further, the problem of its computing is formulated as a nonlinear complementarity problem. Finally, examples of market competition games and pollution management are given to illustrate the application potential of the proposed approach. The novelty and advantage of this work are that it grants the players the freedom to choose the probability and possibility confidence/satisfaction levels at which they want Nash equilibrium to be, and equilibrium computation is simpler compared to existing approaches to fuzzy random bi-matrix games.