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Characterizations of Pareto-Nash Equilibria for Multiobjective Potential Population Games

Author

Listed:
  • Guanghui Yang

    (School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China)

  • Chanchan Li

    (School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China)

  • Jinxiu Pi

    (School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China)

  • Chun Wang

    (School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China)

  • Wenjun Wu

    (School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China)

  • Hui Yang

    (School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China)

Abstract

This paper studies the characterizations of (weakly) Pareto-Nash equilibria for multiobjective population games with a vector-valued potential function called multiobjective potential population games, where agents synchronously maximize multiobjective functions with finite strategies via a partial order on the criteria-function set. In such games, multiobjective payoff functions are equal to the transpose of the Jacobi matrix of its potential function. For multiobjective potential population games, based on Kuhn-Tucker conditions of multiobjective optimization, a strongly (weakly) Kuhn-Tucker state is introduced for its vector-valued potential function and it is proven that each strongly (weakly) Kuhn-Tucker state is one (weakly) Pareto-Nash equilibrium. The converse is obtained for multiobjective potential population games with two strategies by utilizing Tucker’s Theorem of the alternative and Motzkin’s one of linear systems. Precisely, each (weakly) Pareto-Nash equilibrium is equivalent to a strongly (weakly) Kuhn-Tucker state for multiobjective potential population games with two strategies. These characterizations by a vector-valued approach are more comprehensive than an additive weighted method. Multiobjective potential population games are the extension of population potential games from a single objective to multiobjective cases. These novel results provide a theoretical basis for further computing (weakly) Pareto-Nash equilibria of multiobjective potential population games and their practical applications.

Suggested Citation

  • Guanghui Yang & Chanchan Li & Jinxiu Pi & Chun Wang & Wenjun Wu & Hui Yang, 2021. "Characterizations of Pareto-Nash Equilibria for Multiobjective Potential Population Games," Mathematics, MDPI, vol. 9(1), pages 1-13, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:1:p:99-:d:474901
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    References listed on IDEAS

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    1. Sandholm, William H., 2009. "Large population potential games," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1710-1725, July.
    2. Cheung, Man-Wah & Lahkar, Ratul, 2018. "Nonatomic potential games: the continuous strategy case," Games and Economic Behavior, Elsevier, vol. 108(C), pages 341-362.
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    4. Christensen, Finn, 2017. "A necessary and sufficient condition for a unique maximum with an application to potential games," Economics Letters, Elsevier, vol. 161(C), pages 120-123.
    5. Sandholm, William H., 2001. "Potential Games with Continuous Player Sets," Journal of Economic Theory, Elsevier, vol. 97(1), pages 81-108, March.
    6. Ratul Lahkar, 2017. "Large Population Aggregative Potential Games," Dynamic Games and Applications, Springer, vol. 7(3), pages 443-467, September.
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