Generic Properties of First Order Mean Field Games

TL;DR

This paper investigates the generic properties of deterministic first-order mean field games, establishing conditions for existence, uniqueness, and stability of solutions, and providing examples of both single and multiple solutions.

Contribution

It proves that for most mean field games, the best reply map is single-valued, ensuring the existence of strong solutions, and explores stability and non-existence scenarios.

Findings

Most mean field games have a single, well-defined solution.

Open sets of games exhibit either stable or unstable unique solutions.

Some games with terminal constraints have no solutions at all.

Abstract

We consider a class of deterministic mean field games, where the state associated with each player evolves according to an ODE which is linear w.r.t. the control. Existence, uniqueness, and stability of solutions are studied from the point of view of generic theory. Within a suitable topological space of dynamics and cost functionals, we prove that, for nearly all mean field games(in the Baire category sense) the best reply map is single valued for a.e. player. As a consequence, the mean field game admits a strong (not randomized) solution. Examples are given of open sets of games admitting a single solution, and other open sets admitting multiple solutions. Further examples show the existence of an open set of MFG having a unique solution which is asymptotically stable w.r.t. the best reply map, and another open set of MFG having a unique solution which is unstable. We conclude with an example of a MFG with terminal constraints which does not have any solution, not even in the mild sense with randomized strategies.