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Submodular Mean Field Games. Existence and Approximation of Solutions

Author

Listed:
  • Dianetti, Jodi

    (Center for Mathematical Economics, Bielefeld University)

  • Ferrari, Giorgio

    (Center for Mathematical Economics, Bielefeld University)

  • Fischer, Markus

    (Center for Mathematical Economics, Bielefeld University)

  • Nendel, Max

    (Center for Mathematical Economics, Bielefeld University)

Abstract

We study mean field games with scalar Itô-type dynamics and costs that are submodular with respect to a suitable order relation on the state and measure space. The submodularity assumption has a number of interesting consequences. Firstly, it allows us to prove existence of solutions via an application of Tarski's fixed point theorem, covering cases with discontinuous dependence on the measure variable. Secondly, it ensures that the set of solutions enjoys a lattice structure: in particular, there exist a minimal and a maximal solution. Thirdly, it guarantees that those two solutions can be obtained through a simple learning procedure based on the iterations of the best-response-map. The mean field game is first defined over ordinary stochastic controls, then extended to relaxed controls. Our approach allows also to treat a class of submodular mean field games with common noise in which the representative player at equilibrium interacts with the (conditional) mean of its state's distribution.

Suggested Citation

  • Dianetti, Jodi & Ferrari, Giorgio & Fischer, Markus & Nendel, Max, 2019. "Submodular Mean Field Games. Existence and Approximation of Solutions," Center for Mathematical Economics Working Papers 621, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:621
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    File URL: https://pub.uni-bielefeld.de/download/2936699/2936700
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    References listed on IDEAS

    as
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    2. A. Bensoussan & K. C. J. Sung & S. C. P. Yam & S. P. Yung, 2016. "Linear-Quadratic Mean Field Games," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 496-529, May.
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    8. Nendel, Max, 2019. "A Note on Stochastic Dominance and Compactness," Center for Mathematical Economics Working Papers 623, Center for Mathematical Economics, Bielefeld University.
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    Cited by:

    1. Shutian Liu & Yuhan Zhao & Quanyan Zhu, 2022. "Herd Behaviors in Epidemics: A Dynamics-Coupled Evolutionary Games Approach," Dynamic Games and Applications, Springer, vol. 12(1), pages 183-213, March.
    2. Nendel, Max, 2019. "A Note on Stochastic Dominance and Compactness," Center for Mathematical Economics Working Papers 623, Center for Mathematical Economics, Bielefeld University.

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    More about this item

    Keywords

    Mean field games; submodular cost function; complete lattice; first order stochastic dominance; Tarski's fixed point theorem.;
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