Lattice approximations of the first-order mean field type differential games

TL;DR

This paper develops a method to approximate the value function of first-order mean field type differential games using finite-dimensional models, enabling practical computation and analysis of complex infinite-agent systems.

Contribution

It introduces a novel approach to approximate mean field differential games via finite-dimensional Markov chain models, with explicit error estimates.

Findings

Constructed suboptimal strategies from supersolutions and subsolutions.

Provided bounds on approximation accuracy based on the modulus of continuity.

Outlined a method to build finite-dimensional games with desired accuracy.

Abstract

The theory of first-order mean field type differential games examines the systems of infinitely many identical agents interacting via some external media under assumption that each agent is controlled by two players. We study the approximations of the value function of the first-order mean field type differential game using solutions of model finite-dimensional differential games. The model game appears as a mean field type continuous time Markov game, i.e., the game theoretical problem with the infinitely many agents and dynamics of each agent determined by a controlled finite state nonlinear Markov chain. Given a supersolution (resp. subsolution) of the Hamilton-Jacobi equation for the model game, we construct a suboptimal strategy of the first (resp. second) player and evaluate the approximation accuracy using the modulus of continuity of the reward function and the distance between the original and model games. This gives the approximations of the value function of the mean field type differential game by values of the finite-dimensional differential games. Furthermore, we present the way to build a finite-dimensional differential game that approximates the original game with a given accuracy.