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A Characterization of Nash Equilibrium for the Games with Random Payoffs

Author

Listed:
  • Vikas Vikram Singh

    (Indian Institute of Technology Delhi)

  • Abdel Lisser

    (Université Paris Sud)

Abstract

We consider a two-player random bimatrix game where each player is interested in the payoffs which can be obtained with certain confidence. The payoff function of each player is defined using a chance constraint. We consider the case where the entries of the random payoff matrix of each player jointly follow a multivariate elliptically symmetric distribution. We show an equivalence between the Nash equilibrium problem and the global maximization of a certain mathematical program. The case where the entries of the payoff matrices are independent normal/Cauchy random variables is also considered. The case of independent normally distributed random payoffs can be viewed as a special case of a multivariate elliptically symmetric distributed random payoffs. As for Cauchy distribution, we show that the Nash equilibrium problem is equivalent to the global maximization of a certain quadratic program. Our theoretical results are illustrated by considering randomly generated instances of the game.

Suggested Citation

  • Vikas Vikram Singh & Abdel Lisser, 2018. "A Characterization of Nash Equilibrium for the Games with Random Payoffs," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 998-1013, September.
    • Handle: RePEc:spr:joptap:v:178:y:2018:i:3:d:10.1007_s10957-018-1343-0
    DOI: 10.1007/s10957-018-1343-0
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    References listed on IDEAS

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    Cited by:

    1. Filippo Fabiani & Barbara Franci, 2023. "On Distributionally Robust Generalized Nash Games Defined over the Wasserstein Ball," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 298-309, October.
    2. Singh, Vikas Vikram & Lisser, Abdel & Arora, Monika, 2021. "An equivalent mathematical program for games with random constraints," Statistics & Probability Letters, Elsevier, vol. 174(C).
    3. Hoang Nam Nguyen & Abdel Lisser & Vikas Vikram Singh, 2022. "Random Games Under Elliptically Distributed Dependent Joint Chance Constraints," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 249-264, October.
    4. Singh, Vikas Vikram & Lisser, Abdel, 2019. "A second-order cone programming formulation for two player zero-sum games with chance constraints," European Journal of Operational Research, Elsevier, vol. 275(3), pages 839-845.
    5. Rossana Riccardi & Giorgia Oggioni & Elisabetta Allevi & Abdel Lisser, 2023. "Complementarity formulation of games with random payoffs," Computational Management Science, Springer, vol. 20(1), pages 1-32, December.
    6. Shen Peng & Navnit Yadav & Abdel Lisser & Vikas Vikram Singh, 2021. "Chance-constrained games with mixture distributions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(1), pages 71-97, August.

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