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Mean-Field Games Under Model Uncertainty

Zongxia Liang, Zhou Zhou, Yaqi Zhuang, Bin Zou

TL;DR

The paper studies discrete-time, finite-state mean-field games under model uncertainty, where agents face ambiguity in state transitions and the population flow becomes stochastic. It introduces an extended strategy space with state- and distribution-dependent relaxed controls and provides a dynamic-programming characterization of robust equilibria, along with a rigorous analysis of how finite-N equilibria relate to mean-field equilibria. The authors prove that a mean-field equilibrium yields an $\varepsilon$-Nash equilibrium for sufficiently large $N$ and that limits of $N$-agent equilibria converge to MF equilibria, while also establishing existence of finite-N equilibria and presenting a solvable two-state example with closed-form solutions. The framework highlights time-consistency and adaptive strategies under uncertainty and distinguishes itself from endogenous-uncertainty models by treating transition probabilities as exogenous but uncertain, leading to robust, distribution-dependent decision rules with practical implications for large-population decision-making under ambiguity.

Abstract

We study discrete-time, finite-state mean-field games (MFGs) under model uncertainty, where agents face ambiguity about the state transition probabilities. Each agent maximizes its expected payoff against the worst-case transitions within an uncertainty set. Unlike in classical MFGs, model uncertainty renders the population distribution flow stochastic. This leads us to consider strategies that depend on both individual states and the realized distribution of the population. Our main results establish the asymptotic relationship between $N$-agent games and MFGs: every MFG equilibrium constitutes an $\varepsilon$-Nash equilibrium for sufficiently large populations, and conversely, limits of $N$-agent equilibria are MFG equilibria. We also prove the existence of equilibria for finite-agent games and construct a solvable mean-field example with closed-form solutions.

Mean-Field Games Under Model Uncertainty

TL;DR

The paper studies discrete-time, finite-state mean-field games under model uncertainty, where agents face ambiguity in state transitions and the population flow becomes stochastic. It introduces an extended strategy space with state- and distribution-dependent relaxed controls and provides a dynamic-programming characterization of robust equilibria, along with a rigorous analysis of how finite-N equilibria relate to mean-field equilibria. The authors prove that a mean-field equilibrium yields an -Nash equilibrium for sufficiently large and that limits of -agent equilibria converge to MF equilibria, while also establishing existence of finite-N equilibria and presenting a solvable two-state example with closed-form solutions. The framework highlights time-consistency and adaptive strategies under uncertainty and distinguishes itself from endogenous-uncertainty models by treating transition probabilities as exogenous but uncertain, leading to robust, distribution-dependent decision rules with practical implications for large-population decision-making under ambiguity.

Abstract

We study discrete-time, finite-state mean-field games (MFGs) under model uncertainty, where agents face ambiguity about the state transition probabilities. Each agent maximizes its expected payoff against the worst-case transitions within an uncertainty set. Unlike in classical MFGs, model uncertainty renders the population distribution flow stochastic. This leads us to consider strategies that depend on both individual states and the realized distribution of the population. Our main results establish the asymptotic relationship between -agent games and MFGs: every MFG equilibrium constitutes an -Nash equilibrium for sufficiently large populations, and conversely, limits of -agent equilibria are MFG equilibria. We also prove the existence of equilibria for finite-agent games and construct a solvable mean-field example with closed-form solutions.
Paper Structure (25 sections, 18 theorems, 182 equations)

This paper contains 25 sections, 18 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Standard Framework and Notation
  4. Summary of Main Results and Contributions
  5. MFG and $N$-Agent Game under Model Uncertainty
  6. MFG Setup
  7. Characterization of MFG Equilibrium
  8. $N$-Agent Game
  9. Main Results
  10. Examples
  11. A Solvable MFG
  12. A Comparison Example
  13. Proofs of Main Results
  14. Preliminaries
  15. Technical Lemmas for Main Results
  16. ...and 10 more sections

Key Result

Proposition 2.1

For all admissible strategies $\pi^i, \pi \in \Pi^{\mathbb{T}}$ and all $(x,v) \in \Delta$, the robust utility function $V^{\pi^i, \pi}$ in eq:V_inf satisfies the DPP in eq:V. Moreover, $V^{\pi^i, \pi}$ in eq:V_inf is the unique bounded function on $\Pi^{\mathbb{T}} \times \Pi^{\mathbb{T}} \times \D

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1: Finite horizon extension
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.2
  • Remark 2.3
  • Definition 2.3
  • ...and 37 more