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Robust mean-field games under entropy-based uncertainty

François Delarue, Pierre Lavigne

Abstract

In this article, we introduce a new class of entropy-penalized robust mean field game problems in which the representative agent is opposed to Nature. The agent's objective is formulated as a min-max stochastic control problem, in which Nature distorts the reference probability measure at an entropic cost. As a consequence, the distribution of the continuum of agents represented by the player is given by the effective measure induced by Nature. Existence of a mean-field game equilibrium is established via a Schauder fixed point argument. To ensure uniqueness, we introduce a joint flat anti-monotonicity and displacement monotonicity condition, extending the classical Lasry-Lions monotonicity framework. Finally, we present two classes of N -player games for which the mean-field game limit yields $ε$-Nash equilibria.

Robust mean-field games under entropy-based uncertainty

Abstract

In this article, we introduce a new class of entropy-penalized robust mean field game problems in which the representative agent is opposed to Nature. The agent's objective is formulated as a min-max stochastic control problem, in which Nature distorts the reference probability measure at an entropic cost. As a consequence, the distribution of the continuum of agents represented by the player is given by the effective measure induced by Nature. Existence of a mean-field game equilibrium is established via a Schauder fixed point argument. To ensure uniqueness, we introduce a joint flat anti-monotonicity and displacement monotonicity condition, extending the classical Lasry-Lions monotonicity framework. Finally, we present two classes of N -player games for which the mean-field game limit yields -Nash equilibria.
Paper Structure (34 sections, 21 theorems, 161 equations)

This paper contains 34 sections, 21 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem.
  3. FBSDE formulation
  4. Literature.
  5. Contributions.
  6. Organization of the article.
  7. Notations and definitions
  8. Spaces of random variables and processes.
  9. Spaces of measures.
  10. Duality.
  11. Miscellaneous.
  12. Robust control within a fixed environment
  13. Set-up
  14. Solvability of the robust control problem
  15. Stability Estimates
  16. ...and 19 more sections

Key Result

Theorem 2

Let $\mu \in {\mathcal{M}}_{2-r}({\mathbb R}^n)$. Then, there exists a unique saddle point $({\psi},{q}) \in \mathcal{A} \times \mathcal{Q}$ to Problem pb:optim-mfg, i.e. Moreover, if a pair $(\psi,q) \in \mathcal{A} \times \mathcal{Q}$ is a solution to the problem pb:optim-mfg, then the tuples $(\psi,p,k,X)$, obtained by solving in ${\mathscr A}$ the two decoupled equations in optim:condition-pr

Theorems & Definitions (45)

  • Remark 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • Definition 7
  • ...and 35 more