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Mean Field Games Models—A Brief Survey

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  • Diogo Gomes
  • João Saúde

Abstract

The mean-field framework was developed to study systems with an infinite number of rational agents in competition, which arise naturally in many applications. The systematic study of these problems was started, in the mathematical community by Lasry and Lions, and independently around the same time in the engineering community by P. Caines, Minyi Huang, and Roland Malhamé. Since these seminal contributions, the research in mean-field games has grown exponentially, and in this paper we present a brief survey of mean-field models as well as recent results and techniques. In the first part of this paper, we study reduced mean-field games, that is, mean-field games, which are written as a system of a Hamilton–Jacobi equation and a transport or Fokker–Planck equation. We start by the derivation of the models and by describing some of the existence results available in the literature. Then we discuss the uniqueness of a solution and propose a definition of relaxed solution for mean-field games that allows to establish uniqueness under minimal regularity hypothesis. A special class of mean-field games that we discuss in some detail is equivalent to the Euler–Lagrange equation of suitable functionals. We present in detail various additional examples, including extensions to population dynamics models. This section ends with a brief overview of the random variables point of view as well as some applications to extended mean-field games models. These extended models arise in problems where the costs incurred by the agents depend not only on the distribution of the other agents, but also on their actions. The second part of the paper concerns mean-field games in master form. These mean-field games can be modeled as a partial differential equation in an infinite dimensional space. We discuss both deterministic models as well as problems where the agents are correlated. We end the paper with a mean-field model for price impact. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Diogo Gomes & João Saúde, 2014. "Mean Field Games Models—A Brief Survey," Dynamic Games and Applications, Springer, vol. 4(2), pages 110-154, June.
    • Handle: RePEc:spr:dyngam:v:4:y:2014:i:2:p:110-154
    DOI: 10.1007/s13235-013-0099-2
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    References listed on IDEAS

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    1. A. Bensoussan & K. Sung & S. Yam, 2013. "Linear–Quadratic Time-Inconsistent Mean Field Games," Dynamic Games and Applications, Springer, vol. 3(4), pages 537-552, December.
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    4. Aim'e Lachapelle & Jean-Michel Lasry & Charles-Albert Lehalle & Pierre-Louis Lions, 2013. "Efficiency of the Price Formation Process in Presence of High Frequency Participants: a Mean Field Game analysis," Papers 1305.6323, arXiv.org, revised Aug 2015.
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    11. Alberto Bressan & Khai T. Nguyen, 2023. "Generic Properties of First-Order Mean Field Games," Dynamic Games and Applications, Springer, vol. 13(3), pages 750-782, September.
    12. Marcel Nutz, 2016. "A Mean Field Game of Optimal Stopping," Papers 1605.09112, arXiv.org, revised Nov 2017.
    13. Paolo Dai Pra & Elena Sartori & Marco Tolotti, 2019. "Climb on the Bandwagon: Consensus and Periodicity in a Lifetime Utility Model with Strategic Interactions," Dynamic Games and Applications, Springer, vol. 9(4), pages 1061-1075, December.
    14. Luo, Peng & Tangpi, Ludovic, 2024. "Laplace principle for large population games with control interaction," Stochastic Processes and their Applications, Elsevier, vol. 171(C).
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    21. V. N. Kolokoltsov & O. A. Malafeyev, 2018. "Corruption and botnet defense: a mean field game approach," International Journal of Game Theory, Springer;Game Theory Society, vol. 47(3), pages 977-999, September.
    22. Stefanny Ramirez & Dario Bauso, 2023. "Dynamic Games with Strategic Complements and Large Number of Players," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 1-21, April.
    23. V. N. Kolokoltsov & O. A. Malafeyev, 2017. "Mean-Field-Game Model of Corruption," Dynamic Games and Applications, Springer, vol. 7(1), pages 34-47, March.
    24. Naci Saldi & Tamer Bas¸ ar & Maxim Raginsky, 2020. "Approximate Markov-Nash Equilibria for Discrete-Time Risk-Sensitive Mean-Field Games," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1596-1620, November.
    25. Hoang, Lê Nguyên & Soumis, François & Zaccour, Georges, 2019. "The return function: A new computable perspective on Bayesian–Nash equilibria," European Journal of Operational Research, Elsevier, vol. 279(2), pages 471-485.

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