Viability analysis of the first-order mean field games

TL;DR

This paper investigates how solutions to deterministic mean field games depend on initial player distributions, establishing conditions under which a value multifunction accurately represents expected rewards.

Contribution

It provides a sufficient viability condition for the value multifunction in mean field games and derives its infinitesimal variant, advancing theoretical understanding.

Findings

Established viability condition for value multifunction

Derived infinitesimal viability condition

Provided theoretical framework for solution dependence

Abstract

The paper is concerned with the dependence of the solution of the deterministic mean field game on the initial distribution of players. The main object of study is the mapping which assigns to the initial time and the initial distribution of players the set of expected rewards of the representative player corresponding to solutions of mean field game. This mapping can be regarded as a value multifunction. We obtain the sufficient condition for a multifunction to be a value multifunction. It states that if a multifunction is viable with respect to the dynamics generated by the original mean field game, then it is a value multifunction. Furthermore, the infinitesimal variant of this condition is derived.