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

In the paper, we study the dependence of solutions of the continuous-time finite state mean field game on initial distribution of players. Our approach relies on 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 reformulation of the finite state mean field game as a control problem with mixed constraints, 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 an initial 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 system.