Asymptotic Nash Equilibria of Finite-State Ergodic Markovian Mean Field Games

TL;DR

The paper addresses asymptotic Nash equilibria in finite-state ergodic MFGs with Markovian controls, showing that both time-dependent MFG-derived and master-equation-derived profiles yield a $\varepsilon$-Nash equilibrium in the $n$-player game with $\varepsilon$ scaling as $C/\sqrt{n}$. It establishes propagation of chaos for the empirical measures and derives a comprehensive large deviations theory for these measures and their invariant laws. It also provides explicit analytical results for a quadratic stationary ergodic MFG example and develops a deep Galerkin method to numerically solve the ergodic master equation, enabling direct comparison of strategies and costs under the two approaches. The work highlights that master-equation based strategies can concentrate the empirical distribution more tightly near the mean-field limit and yield lower realized costs, with explicit rate-function formulas in dimension two. Overall, the results deepen the connection between finite-player Nash equilibria and mean-field predictions in finite-state ergodic settings and offer practical computational tools for strategy design in large populations.

Abstract

Mean field games (MFGs) model equilibria in games with a continuum of weakly interacting players as limiting systems of symmetric $n$-player games. We consider the finite-state, infinite-horizon problem with ergodic cost. Assuming Markovian strategies, we first prove that any solution to the MFG system gives rise to a $(C/\sqrt{n})$-Nash equilibrium in the $n$-player game. We follow this result by proving the same is true for the strategy profile derived from the master equation. We conclude the main theoretical portion of the paper by establishing a large deviation principle for empirical measures associated with the asymptotic Nash equilibria. Then, we contrast the asymptotic Nash equilibria using an example. We solve the MFG system directly and numerically solve the ergodic master equation by adapting the deep Galerkin method of Sirignano and Spiliopoulos. We use these results to derive the strategies of the asymptotic Nash equilibria and compare them. Finally, we derive an explicit form for the rate functions in dimension two.

Figures (6)

• Figure 1: We compare $\Gamma_0$ and $\bar{\Gamma}$ by the difference in rates $\gamma^*_y(x,\Delta_x U_0(\cdot,\eta)) - \gamma^*_y(x,\Delta_x \bar{u})$ in the case $d=2$.
• Figure 2: We compare $\Gamma_0$ and $\bar{\Gamma}$ by the difference in rates $\gamma^*_y(x,\Delta_x U_0(\cdot,\eta)) - \gamma^*_y(x,\Delta_x \bar{u})$ in the case $d=3$.
• Figure 3: Due to the elevated rate for players of $\Gamma_0$ near the boundary of ${\mathcal{P}}([2])$, we see a greater concentration of mass toward the center of the simplex, relative to $\bar{\Gamma}$.
• Figure 4: The left graph shows the convergence of each realized cost to $\varrho$ the optimal cost in the mean field limit. The right graph highlights the fact that $\varrho^{n, U_0} < \varrho^{n, \bar{u}}$.
• Figure 5: The difference in realized value $\varrho^{n, \bar{u}} - \varrho^{n, U_0}$.

Theorems & Definitions (33)

• Definition 3.1
• Remark 3.1
• Definition 3.2
• Remark 3.2
• Definition 3.3: Ergodic MFE
• Proposition 3.1
• Proposition 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.5