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

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  • Napat Harnpornchai

    (Center of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Chiang Mai 50200, Thailand)

  • Wiriyaporn Wonggattaleekam

    (Department of Agricultural Economy and Development, Faculty of Agriculture, Chiang Mai University, Chiang Mai 50200, Thailand)

Abstract

Nash Equilibrium (NE) plays a crucial role in game theory. The relaxation method in conjunction with the Nikaido–Isoda (NI) function, namely the NI-based relaxation method, has been widely applied to the determination of NE. Genetic Algorithm (GA) with adaptive penalty is introduced and incorporated in the original NI-based relaxation method. The GA enhances the capability in the optimization step for computing the optimum response function. The optimization of the non-convex and non-concave NI function is made possible by GA. The proposed method thus combines the advantageous feature of the GA in its optimization capability and that of the relaxation method in its implementation simplicity together. The applicability of the method is shown through the illustrative examples, including the generalized Nash Equilibrium problem with nonlinear payoff functions and coupled constraints, the game with multiple strategic variables for individual players, and the non-differentiable payoff functions. All test example results suggest the appropriate crossover and mutation rate to be 0.05 and 0.002 for use in GA. These numbers are closed to the recommended values by DeJong. The proposed method shows its capability of finding correct NEs in all test examples.

Suggested Citation

  • Napat Harnpornchai & Wiriyaporn Wonggattaleekam, 2021. "A Nikaido Isoda-Based Hybrid Genetic Algorithm and Relaxation Method for Finding Nash Equilibrium," Mathematics, MDPI, vol. 10(1), pages 1-17, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2021:i:1:p:81-:d:711850
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    References listed on IDEAS

    as
    1. Feng He & Wei Zhang & Guoqiang Zhang, 2016. "A Differential Evolution Algorithm Based on Nikaido-Isoda Function for Solving Nash Equilibrium in Nonlinear Continuous Games," PLOS ONE, Public Library of Science, vol. 11(9), pages 1-17, September.
    2. A. Heusinger & C. Kanzow, 2009. "Relaxation Methods for Generalized Nash Equilibrium Problems with Inexact Line Search," Journal of Optimization Theory and Applications, Springer, vol. 143(1), pages 159-183, October.
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