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A Bayesian optimization approach to find Nash equilibria

Author

Listed:
  • Victor Picheny

    (Université de Toulouse, INRA)

  • Mickael Binois

    (The University of Chicago Booth School of Business)

  • Abderrahmane Habbal

    (Université Côte d’Azur)

Abstract

Game theory finds nowadays a broad range of applications in engineering and machine learning. However, in a derivative-free, expensive black-box context, very few algorithmic solutions are available to find game equilibria. Here, we propose a novel Gaussian-process based approach for solving games in this context. We follow a classical Bayesian optimization framework, with sequential sampling decisions based on acquisition functions. Two strategies are proposed, based either on the probability of achieving equilibrium or on the stepwise uncertainty reduction paradigm. Practical and numerical aspects are discussed in order to enhance the scalability and reduce computation time. Our approach is evaluated on several synthetic game problems with varying number of players and decision space dimensions. We show that equilibria can be found reliably for a fraction of the cost (in terms of black-box evaluations) compared to classical, derivative-based algorithms. The method is available in the R package GPGame available on CRAN at https://cran.r-project.org/package=GPGame .

Suggested Citation

  • Victor Picheny & Mickael Binois & Abderrahmane Habbal, 2019. "A Bayesian optimization approach to find Nash equilibria," Journal of Global Optimization, Springer, vol. 73(1), pages 171-192, January.
    • Handle: RePEc:spr:jglopt:v:73:y:2019:i:1:d:10.1007_s10898-018-0688-0
    DOI: 10.1007/s10898-018-0688-0
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    References listed on IDEAS

    as
    1. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    2. Matthew Plumlee, 2014. "Fast Prediction of Deterministic Functions Using Sparse Grid Experimental Designs," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(508), pages 1581-1591, December.
    3. Roustant, Olivier & Ginsbourger, David & Deville, Yves, 2012. "DiceKriging, DiceOptim: Two R Packages for the Analysis of Computer Experiments by Kriging-Based Metamodeling and Optimization," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 51(i01).
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