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Fuzzy Weighted Pareto–Nash Equilibria of Multi-Objective Bi-Matrix Games with Fuzzy Payoffs and Their Applications

Author

Listed:
  • Wen Li

    (School of Science, Wuhan University of Science and Technology, Wuhan 430065, China)

  • Deyi Li

    (School of Science, Wuhan University of Science and Technology, Wuhan 430065, China
    These authors contributed equally to this work.)

  • Yuqiang Feng

    (School of Science, Wuhan University of Science and Technology, Wuhan 430065, China
    These authors contributed equally to this work.)

  • Du Zou

    (School of Science, Wuhan University of Science and Technology, Wuhan 430065, China
    These authors contributed equally to this work.)

Abstract

Based on our previous research, this paper further discusses the multi-objective bi-matrix game with fuzzy payoffs (MBGFP), which is a special case of the fuzzy constrained multi-objective game with fuzzy payoffs. We first prove that any bi-matrix game with interval payoffs (BGIP) has at least one Pareto–Nash equilibrium. Then, with the help of BGIP, we obtain the necessary and sufficient conditions for the existence of fuzzy Pareto–Nash equilibrium of MBGFP. Secondly, based on the bilinear programming method for calculating Nash equilibrium in crisp bi-matrix games, we established a bilinear programming method with parameters for calculating fuzzy Pareto–Nash equilibrium. By considering the importance of each objective to the players, MBGFP is transformed into a bi-matrix game with fuzzy payoffs (BGFP). Furthermore, we obtained the necessary and sufficient conditions for the existence of fuzzy weighted Pareto–Nash equilibrium and its calculation method. Finally, a practical example is used to illustrate the effectiveness of our proposed calculation method.

Suggested Citation

  • Wen Li & Deyi Li & Yuqiang Feng & Du Zou, 2023. "Fuzzy Weighted Pareto–Nash Equilibria of Multi-Objective Bi-Matrix Games with Fuzzy Payoffs and Their Applications," Mathematics, MDPI, vol. 11(20), pages 1-18, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4266-:d:1258676
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    References listed on IDEAS

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    1. Li, Deng-Feng, 2012. "A fast approach to compute fuzzy values of matrix games with payoffs of triangular fuzzy numbers," European Journal of Operational Research, Elsevier, vol. 223(2), pages 421-429.
    2. R. E. Bellman & L. A. Zadeh, 1970. "Decision-Making in a Fuzzy Environment," Management Science, INFORMS, vol. 17(4), pages 141-164, December.
    3. Li, Deng-Feng, 2011. "Linear programming approach to solve interval-valued matrix games," Omega, Elsevier, vol. 39(6), pages 655-666, December.
    4. Chandra, S. & Aggarwal, A., 2015. "On solving matrix games with pay-offs of triangular fuzzy numbers: Certain observations and generalizations," European Journal of Operational Research, Elsevier, vol. 246(2), pages 575-581.
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