We consider within the framework of the Mean Field Games theory a dynamic collective choice model, where a large number of agents are choosing among a set of alternatives while influenced by the group's behavior. Our model can be interpreted as a stylized version of opinion crystallization in an election for example. We introduce the "Min-LQG" optimal control problem, a modified Linear Quadratic Gaussian optimal control problem that includes a minimum term in its final cost to capture the discrete choice phenomenon. We consider two cases: deterministic dynamics with random initial conditions and stochastic dynamics. In the first case, the agents make their choices prior start moving, while in the second case, the Wiener processes make the players indecisive. We construct a one to one map between the epsilon Nash equilibria and the fixed points of a finite dimensional operator. These fixed points are the potential distributions of the choices between the alternatives. Finally, we establish the existence of at least one epsilon Nash equilibrium.
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