Value function for first order mean field games

The first order mean field games are considered within the framework of minimax solutions. It's known that the solution to mean field game is a pair consisting of continuous function of position and function of time with values in the set of probabilities. We show the nonuniqueness of the minimax solution. Thus, we consider th multivalued mapping which assigns initial time and probability the set of continuous functions of state variable. The value of each function at the point is assumed to be equal to the expected outcome of the player which starts at the given point. We present the conditions on the multifunction those analogs of dynamic programming principle. The existence theorem for the multifunction satisfying dynamic programming is also proved.