Study Of Bi-matrix Game With Fuzzy Pay-offs

This thesis studies bi-matrix games in which the elements of the pay-off matrices are characterized as fuzzy variables. Expected equilibrium solution with fuzzy expected value operator is defined which is a new kind of Nash equilibrium solution. The existence of expected equilibrium solutions is proved and then the method of solving the games is discuss in two different ways according to the cases that the fuzzy-offs are independent of each other or not.In the case of non-independence, the sequence of the method is: construct a set-valued mapping satisfying the conditions in the definition; make the equilibrium solution correspondent with Kakutani fixed point; search the Kakutani fixed point in the mix-strategy product space. In searching for the solutions, objective functions with fuzzy variables are simulated by Neural Network (NN) based on fuzzy simulation and fixed points are obtained by Particle Swarm Optimization (PSO) algorithm.In the case of independence, expected equilibrium solutions are obtained by solving quadratic programming with constraints. Due to the feature that the expected equilibrium solutions are correspondent with the solutions of quadratic programming, the solutions are obtained through quadratic programming with constraints by Particle Swarm Optimization algorithm (PSO) by searching the mix-strategy product space. In the process of solving quadratic programming, the problem with constraints is changed into the problem without constraints after substituting the constraints into the objective function by using penalty function.In both cases, some numerical examples are given to illustrate the effectiveness of the algorithm.