  The theory of first-order mean field type differential games examines the
systems of infinitely many identical agents interacting via some external media
under assumption that each agent is controlled by two players. We study the
approximations of the value function of the first-order mean field type
differential game using solutions of model finite-dimensional differential
games. The model game appears as a mean field type continuous time Markov game,
i.e., the game theoretical problem with the infinitely many agents and dynamics
of each agent determined by a controlled finite state nonlinear Markov chain.
Given a supersolution (resp. subsolution) of the Hamilton-Jacobi equation for
the model game, we construct a suboptimal strategy of the first (resp. second)
player and evaluate the approximation accuracy using the modulus of continuity
of the reward function and the distance between the original and model games.
This gives the approximations of the value function of the mean field type
differential game by values of the finite-dimensional differential games.
Furthermore, we present the way to build a finite-dimensional differential game
that approximates the original game with a given accuracy.
