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On Approximate Nash Equilibria in Mean Field Games

Mao Fabrice Djete, Nizar Touzi

Abstract

In the context of large population symmetric games, approximate Nash equilibria are introduced through equilibrium solutions of the corresponding mean field game in the sense that the individual gain from optimal unilateral deviation under such strategies converges to zero in the large population size asymptotic. We show that these strategies satisfy an $Ł^\infty$ notion of approximate Nash equilibrium which guarantees that the individual gain from optimal unilateral deviation is small uniformly among players and uniformly on their initial characteristics. We establish these results in the context of static models and in the dynamic continuous time setting, and we cover situations where the agents' criteria depend on the conditional law of the controlled state process.

On Approximate Nash Equilibria in Mean Field Games

Abstract

In the context of large population symmetric games, approximate Nash equilibria are introduced through equilibrium solutions of the corresponding mean field game in the sense that the individual gain from optimal unilateral deviation under such strategies converges to zero in the large population size asymptotic. We show that these strategies satisfy an notion of approximate Nash equilibrium which guarantees that the individual gain from optimal unilateral deviation is small uniformly among players and uniformly on their initial characteristics. We establish these results in the context of static models and in the dynamic continuous time setting, and we cover situations where the agents' criteria depend on the conditional law of the controlled state process.
Paper Structure (10 sections, 5 theorems, 120 equations)

This paper contains 10 sections, 5 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. One-period approximate Nash equilibria
  3. Finite population Nash equilibrium
  4. The mean field game
  5. Approximate Nash Equilibrium: main results
  6. Continuous time approximate Nash equilibria
  7. The continuous time mean field game
  8. Continuous time approximate Nash equilibria
  9. Existence of one-period MFG solutions
  10. Existence of MFG solution in continuous time

Key Result

Proposition 2.5

Let Assumption ${\rm H}_p$ hold for some $p >1$, and let $\eta\in\mathbb{L}^q$ and $\mu_0\in{\cal P}_q(\mathbb{S})$ for some $q>p$. Then there exists a solution of the MFG $\widehat{\mu} \in {\cal P}_p (\mathbb{X} \times A)$ under either one of the following conditions: (MFG1) There exists an optima (MFG2) Or, $F^e_{x_0}(\mu,a)=\widehat{F}^e_{x_0}\left(\mu^X, \langle f^1, \mu \rangle,\cdots, \lang

Theorems & Definitions (16)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Remark 2.8
  • proof : Proof of \ref{['thm:approx_nash_normal']}
  • Definition 3.1
  • ...and 6 more