Mean Field Games with Branching

TL;DR

This paper introduces a branching mechanism into mean field games to model populations with varying sizes over time, providing new theoretical insights and numerical examples for such dynamic systems.

Contribution

It extends classical mean field game theory by incorporating population branching, offering existence results and approximation of Nash equilibria in variable-population models.

Findings

Existence of solutions for the branching mean field game model.

Approximate Nash equilibria in large populations with branching.

Numerical illustration using a linear-quadratic model.

Abstract

Mean field games are concerned with the limit of large-population stochastic differential games where the agents interact through their empirical distribution. In the classical setting, the number of players is large but fixed throughout the game. However, in various applications, such as population dynamics or economic growth, the number of players can vary across time which may lead to different Nash equilibria. For this reason, we introduce a branching mechanism in the population of agents and obtain a variation on the mean field game problem. As a first step, we study a simple model using a PDE approach to illustrate the main differences with the classical setting. We prove existence of a solution and show that it provides an approximate Nash-equilibrium for large population games. We also present a numerical example for a linear--quadratic model. Then we study the problem in a general setting by a probabilistic approach. It is based upon the relaxed formulation of stochastic control problems which allows us to obtain a general existence result.