A Differential Evolution Algorithm Based on Nikaido-Isoda Function for Solving Nash Equilibrium in Nonlinear Continuous Games

Abstract

A differential evolution algorithm for solving Nash equilibrium in nonlinear continuous games is presented in this paper, called NIDE (Nikaido-Isoda differential evolution). At each generation, parent and child strategy profiles are compared one by one pairwisely, adapting Nikaido-Isoda function as fitness function. In practice, the NE of nonlinear game model with cubic cost function and quadratic demand function is solved, and this method could also be applied to non-concave payoff functions. Moreover, the NIDE is compared with the existing Nash Domination Evolutionary Multiplayer Optimization (NDEMO), the result showed that NIDE was significantly better than NDEMO with less iterations and shorter running time. These numerical examples suggested that the NIDE method is potentially useful.

A Nikaido Isoda-Based Hybrid Genetic Algorithm and Relaxation Method for Finding Nash Equilibrium

“…where 0 ≤ α s ≤ 1. Assigning x s to the NI function (2), the optimum response function y max (x s ) is obtained from solving the optimization problem (7). y max (x s ) is then input into (8) and results in x s+1 , which is successively used in the NI function (2).…”

“…To extend the relaxation method to the case of the non-differentiable NI function, the methodologies that can solve the optimization problems with non-differentiable objective functions are used. It was discussed that analytical methods are not able to handle complicated or non-differentiable payoff functions while mathematical programming approaches may not reach the NE point, due to not being reliable due to their dependence on initial searching points [7]. Meta-heuristic algorithms are more effective in dealing with non-differentiable, non-linear, complex payoff functions from their fundamental procedures [7].…”

“…It was discussed that analytical methods are not able to handle complicated or non-differentiable payoff functions while mathematical programming approaches may not reach the NE point, due to not being reliable due to their dependence on initial searching points [7]. Meta-heuristic algorithms are more effective in dealing with non-differentiable, non-linear, complex payoff functions from their fundamental procedures [7]. Accordingly, the evolutionary computation was proposed for finding the NE points [8].…”

“…Accordingly, the evolutionary computation was proposed for finding the NE points [8]. The so-called Nash dominance concept was defined and has been directly used or modified in later research [7,9,10]. The concept was implemented in Non-dominated Sorting Genetic Algorithm II (NSGAII) [11] with changing from the Pareto dominance to the defined Nash dominance.…”

“…Genetic Algorithm (GA) and DE were again utilized as optimization tools. Instead of checking the Nash Equilibrium condition based on the Nash dominance concept, the NI function was employed for the purpose [7]. However, the new actions were still created by the evolutionary algorithm DE.…”