[Paper Review] Mean-Field Games Under Model Uncertainty

Zongxia Liang, Zhou Zhou+2 more|arXiv (Cornell University)|Jan 18, 2026

Game Theory and Applications0 citations

TL;DR

This paper studies discrete-time, finite-state mean-field games with model uncertainty, where agents face worst-case transition probabilities and strategies depend on both individual state and realized population distribution. It establishes the link between finite-agent equilibria and mean-field equilibria under uncertainty and provides a solvable example.

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.

Motivation & Objective

• Extend mean-field game theory to discrete-time, finite-state settings under model uncertainty about state transitions.
• Allow strategies to depend on both an agent's state and the realized population distribution.
• Establish the asymptotic relationship between N-agent equilibria and mean-field equilibria under worst-case transitions.
• Prove existence of equilibria for finite-agent games and illustrate with a solvable example.

Proposed method

• Introduce robust optimization into the MFG framework with an uncertainty set for transition kernels.
• Define a distributionally robust objective via a dynamic programming principle (DPP) that encodes worst-case transitions.
• Extend strategy space to state- and distribution-dependent (and relaxed) controls.
• Establish a DPP-based characterization (Bellman operator) to compute the robust value function.
• Prove convergence results: any MFG equilibrium is an ε-Nash equilibrium for large N, and finite-agent equilibria converge to MFG equilibria.
• Demonstrate existence of finite-agent equilibria via Kakutani’s fixed point theorem and provide a two-state closed-form MFG example.

Experimental results

Research questions

• RQ1 What is the form and implications of robust optimization in discrete-time mean-field games under model uncertainty?
• RQ2 How does model uncertainty affect the evolution of the population distribution and the structure of optimal strategies?
• RQ3 What are the relationships between finite-agent Nash equilibria and mean-field equilibria when transitions are uncertain?
• RQ4 Can equilibria exist for finite-agent games and can a solvable example be constructed with closed-form solutions?

Key findings

• Mean-field equilibria under model uncertainty yield ε-Nash equilibrilibria for sufficiently large populations.
• Limits of N-agent equilibria converge to mean-field equilibria under uncertainty.
• Equilibria for finite-agent games exist, established via Kakutani’s fixed-point theorem.
• A solvable two-state mean-field game with closed-form equilibrium strategy, worst-case kernel, and value function is constructed.
• Distribution-dependent (state and distribution) strategies are essential because the realized population distribution becomes stochastic under uncertainty.