Synthesis of Strategic Games with Multiple Nash Equilibria - A Global Optimization Approach

This paper presents an extension of the resuts obtained in previous work by the author concerning the application of global optimization techniques to the design of finite strategic games with mixed strategies. In that publication the Fuzzy ASA global optimization method was applied to many examples of synthesis of strategic games with one previously specified Nash equilibrium, evidencing its ability in finding payoff functions whose respective games present those equilibria, possibly among others. That is to say, it was shown it is possible to establish in advance a Nash equilibrium for a generic finite state strategic game and to compute payoff functions that will make it feasible to reach the chosen equilibrium, allowing players to converge to the desired profile, considering that it is an equilibrium of the game as well. Going beyond this state of affairs, the present article shows that it is possible to "impose" multiple Nash equilibria to finite strategic games by following the same reasoning as before, but with a slight change: using the same fundamental theorem of Richard D. McKelvey, modifying the original prescribed objective function and globally minimizing it. The proposed method, in principle, is able to find payoff functions that result in games featuring an arbitrary number of Nash equiibria, paving the way to a substantial number of potential applications.