  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 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.
