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Optimal control and potential games in the mean field

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  • Höfer, Felix
  • Soner, H. Mete

Abstract

We study a mean-field optimal control problem with general non-Markovian dynamics that allow both common noise and jumps. We show that its minimizers are Nash equilibria of an associated mean field game of controls. These types of games are potential, and the resulting Nash equilibria are closely linked to McKean-Vlasov equations of Langevin type. Furthermore, as a byproduct of our main result, we obtain the existence of strong common noise equilibria for potential games. To illustrate the general theory, we present several examples, including a mean field game of controls with interactions through a price variable, as well as mean-field Cucker-Smale Flocking and Kuramoto models. We also establish the invariance property of the value function, which is a key ingredient used in our proofs.

Suggested Citation

  • Höfer, Felix & Soner, H. Mete, 2026. "Optimal control and potential games in the mean field," Stochastic Processes and their Applications, Elsevier, vol. 199(C).
    • Handle: RePEc:eee:spapps:v:199:y:2026:i:c:s0304414926001031
    DOI: 10.1016/j.spa.2026.104971
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    References listed on IDEAS

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    1. René Carmona & Christy V. Graves, 2020. "Jet Lag Recovery: Synchronization of Circadian Oscillators as a Mean Field Game," Dynamic Games and Applications, Springer, vol. 10(1), pages 79-99, March.
    2. Lacker, Daniel, 2015. "Mean field games via controlled martingale problems: Existence of Markovian equilibria," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2856-2894.
    3. Mao Fabrice Djete & Dylan Possamaï & Xiaolu Tan, 2022. "McKean–Vlasov Optimal Control: Limit Theory and Equivalence Between Different Formulations," Mathematics of Operations Research, INFORMS, vol. 47(4), pages 2891-2930, November.
    4. Clémence Alasseur & Imen Ben Taher & Anis Matoussi, 2020. "An Extended Mean Field Game for Storage in Smart Grids," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 644-670, February.
    5. Samuel Daudin, 2022. "Optimal Control of Diffusion Processes with Terminal Constraint in Law," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 1-41, October.
    6. Mátyás Barczy & Zenghu Li & Gyula Pap, 2015. "Yamada-Watanabe Results for Stochastic Differential Equations with Jumps," International Journal of Stochastic Analysis, Hindawi, vol. 2015, pages 1-23, January.
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