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On Time-Inconsistency in Mean Field Games

Erhan Bayraktar, Zhenhua Wang

Abstract

We investigate an infinite-horizon time-inconsistent mean-field game (MFG) in a discrete time setting. We first present a classic equilibrium for the MFG and its associated existence result. This classic equilibrium aligns with the conventional equilibrium concept studied in MFG literature when the context is time-consistent. Then we demonstrate that while this equilibrium produces an approximate optimal strategy when applied to the related $N$-agent games, it does so solely in a precommitment sense. Therefore, it cannot function as a genuinely approximate equilibrium strategy from the perspective of a sophisticated agent within the $N$-agent game. To address this limitation, we propose a new consistent equilibrium concept in both the MFG and the $N$-agent game. We show that a consistent equilibrium in the MFG can indeed function as an approximate consistent equilibrium in the $N$-agent game. Additionally, we analyze the convergence of consistent equilibria for $N$-agent games toward a consistent MFG equilibrium as $N$ tends to infinity.

On Time-Inconsistency in Mean Field Games

Abstract

We investigate an infinite-horizon time-inconsistent mean-field game (MFG) in a discrete time setting. We first present a classic equilibrium for the MFG and its associated existence result. This classic equilibrium aligns with the conventional equilibrium concept studied in MFG literature when the context is time-consistent. Then we demonstrate that while this equilibrium produces an approximate optimal strategy when applied to the related -agent games, it does so solely in a precommitment sense. Therefore, it cannot function as a genuinely approximate equilibrium strategy from the perspective of a sophisticated agent within the -agent game. To address this limitation, we propose a new consistent equilibrium concept in both the MFG and the -agent game. We show that a consistent equilibrium in the MFG can indeed function as an approximate consistent equilibrium in the -agent game. Additionally, we analyze the convergence of consistent equilibria for -agent games toward a consistent MFG equilibrium as tends to infinity.
Paper Structure (17 sections, 16 theorems, 178 equations)

This paper contains 17 sections, 16 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Mean field games under the classic equilibrium notion
  3. Mean field games with time-inconsistency
  4. Proof for Theorem \ref{['thm.exist.nonstatoinary']}
  5. Precommitment equilibrium in the $N$-agent game
  6. $N$-agent games with time-inconsistency
  7. Equilibria in Definition \ref{['def.nonstationary.relax']} lead to precommitment approximate Nash equilibria for N-agent games
  8. Consistent Equilibria
  9. The concept of consistent MFG equilibria
  10. Revisiting the $N$-agent game
  11. An example analysis for the existence of continuous consistent MFG equilibria
  12. Proofs for Theorem \ref{['thm.MFG.Neps']} and Theorem \ref{['thm.Nequi.converge']}
  13. A formula of $J^{N,1,( \bm{{\tilde{\pi}}}^N_{-1}, \varpi)\otimes_1{\bm{{\tilde{\pi}}}}^N}(x, \nu)$
  14. Lemmata
  15. Proof of Theorem \ref{['thm.MFG.Neps']}
  16. ...and 2 more sections

Key Result

Proposition 2.1

Suppose Assumption assum.bound holds. Fix $\nu\in {\mathcal{P}}([d])$. Consider $(\pi,\mu)\in \Pi\times \Lambda$ with $\mu_0=\nu$. Then $(\pi,\mu)\in \Pi\times \Lambda$ is an equilibrium in Definition def.nonstationary.relax for the MFG with initial population distribution $\nu$if and only if $\pi$

Theorems & Definitions (47)

  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.1
  • proof
  • Theorem 2.1
  • Lemma 2.1
  • ...and 37 more