Control theory approach to continuous-time finite state mean field games

TL;DR

This paper models finite state mean field games as control problems with constraints, analyzing how solutions depend on initial distributions and characterizing the maximal solution set through viability theory.

Contribution

It introduces the concept of a value multifunction for finite state mean field games and characterizes the maximal multifunction using viability theory and control reformulation.

Findings

Defined the value multifunction for mean field games.

Provided sufficient conditions for a multifunction to be a value multifunction.

Characterized the maximal multifunction via backward attainability sets.

Abstract

In the paper, we use the equivalent formulation of a finite state mean field game as a control problem with mixed constraints to study the dependence of solutions to finite state mean field game on an initial distribution of players. We introduce the concept of value multifunction of the mean field game that is a mapping assigning to an initial time and an initial distribution a set of expected outcomes of the representative player corresponding to solutions of the mean field game. Using the control reformulation of the finite state mean field game, we give the sufficient condition on a given multifunction to be a value multifunction in the terms of the viability theory. The maximal multifunction (i.e. the mapping assigning to an initial time and distribution the whole set of values corresponding to solutions of the mean field game) is characterized via the backward attainability set for the certain control systems.