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Optimal Control and Potential Games in the Mean Field

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arxiv 2408.00733 v2 pith:O7DWZ4AR submitted 2024-08-01 math.OC

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keywords fieldmeancontrolcontrolsequilibriagamegamesgeneral
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We study a mean field optimal control problem with general non-Markovian dynamics, including 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 necessarily potential, and the Nash equilibria derived as the minimizers of the control problem are closely connected to McKean-Vlasov equations of Langevin type. To illustrate the general theory, we present several examples, including a mean field game of controls with interactions through a price variable, and mean field Cucker-Smale Flocking and Kuramoto models. We also establish the invariance property of the value function, a key ingredient used in our proofs.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On Mean-field Singular Stochastic Control Problems

    math.OC 2026-07 accept novelty 6.0 of 10

    Under convexity, potential MFG equilibria solve mean-field singular control; for the mean-field monotone follower with strategic complementarities the optimum is a free boundary uniquely solving a nonlinear integral equation.

  2. Stationary Heterogeneous-Agent Models in Continuous Time

    econ.TH 2025-10 conditional novelty 6.0 of 10

    In a continuous-time heterogeneous-agent model, small government deficits can produce two stationary equilibria and hence two price levels, while surpluses give a unique equilibrium.

  3. A particle system approach towards the global well-posedness of master equations for potential mean field games of control

    math.OC 2024-12 conditional novelty 6.0 of 10

    Under an extended displacement convexity condition, classical solutions to the HJB and master equations for generalized mean field control and potential mean field games of controls exist globally, including in the de...

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