Linear-Quadratic Zero-Sum Mean-Field Type Games: Optimality Conditions and Policy Optimization

TL;DR

This paper analyzes zero-sum mean-field type games with linear-quadratic structure, deriving optimality conditions and proposing policy gradient methods for both model-based and sample-based settings, supported by numerical experiments.

Contribution

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

Findings

Explicit Nash equilibrium strategies derived.

Policy gradient methods successfully optimize controls.

Numerical experiments confirm convergence of utility and controls.

Abstract

In this paper, zero-sum mean-field type games (ZSMFTG) with linear dynamics and quadratic cost 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 indistinguishable 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 optimality conditions of the game are analysed for both open-loop and closed-loop controls, 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 model-based case, the gradients are computed exactly using the model, whereas they are estimated using Monte-Carlo simulations in the sample-based case. Numerical experiments are conducted to show the convergence of the utility function as well as the two players' controls.