Equilibrium in Functional Stochastic Games with Mean-Field Interaction

TL;DR

This paper develops a semi-explicit method to find Nash equilibria in mean-field stochastic games with linear-quadratic costs, demonstrating convergence from finite-player to mean-field equilibria and applying it to diverse examples.

Contribution

It introduces a novel operator-resolvent approach to derive equilibria and proves convergence and stability results for mean-field and finite-player games.

Findings

Explicit semi-closed form for Nash equilibria

Convergence of finite-player game equilibria to mean-field equilibrium

Application to various complex stochastic game models

Abstract

We consider a general class of finite-player stochastic games with mean-field interaction, in which the linear-quadratic cost functional includes linear operators acting on controls in $L^2$. We propose a novel approach for deriving the Nash equilibrium of the game semi-explicitly in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations of the second kind and deriving their solution in semi-explicit form. Furthermore, by proving stability results for the system of stochastic Fredholm equations, we derive the convergence of the equilibrium of the $N$-player game to the corresponding mean-field equilibrium. As a by-product, we also derive an $\varepsilon$-Nash equilibrium for the mean-field game, which is valuable in this setting as we show that the conditions for existence of an equilibrium in the mean-field limit are less restrictive than in the finite-player game. Finally, we apply our general framework to solve various examples, such as stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact.