Policy Optimization for Linear-Quadratic Zero-Sum Mean-Field Type Games

TL;DR

This paper studies zero-sum mean-field type games with linear-quadratic structure, deriving explicit Nash equilibria and proposing policy gradient methods for both model-based and sample-based optimization, demonstrating convergence through numerical experiments.

Contribution

It provides explicit Nash equilibrium strategies for linear-quadratic zero-sum mean-field games and introduces policy gradient algorithms for their optimization.

Findings

Explicit Nash equilibrium strategies derived.

Policy gradient methods successfully optimize strategies.

Numerical experiments confirm convergence of algorithms.

Abstract

In this paper, zero-sum mean-field type games (ZSMFTG) with linear dynamics and quadratic utility are studied under infinite-horizon discounted utility function. ZSMFTG are a class of games in which two decision makers whose utilities sum to zero, compete to influence a large population of agents. In particular, the case in which the transition and utility functions depend on the state, the action of the controllers, and the mean of the state and the actions, is investigated. The game is analyzed and explicit expressions for the Nash equilibrium strategies are derived. Moreover, two policy optimization methods that rely on policy gradient are proposed for both model-based and sample-based frameworks. In the first case, the gradients are computed exactly using the model whereas they are estimated using Monte-Carlo simulations in the second case. Numerical experiments show the convergence of the two players' controls as well as the utility function when the two algorithms are used in different scenarios.