A Dynamic Game Model of Collective Choice: Stochastic Dynamics and Closed Loop Solutions

We consider within the framework of the Mean Field Games theory a dynamic discrete choice model with social interactions, where a large number of agents/players are choosing between two alternatives while influenced by the group’s behavior. We introduce the “Min-LQG” optimal control problem, a modified Linear Quadratic Gaussian (LQG) optimal control problem that includes a minimum term in its final cost to capture the discrete choice phenomenon. We give an explicit solution of the Min-LQG problem and show that at each instant, the dynamic discrete choice model can be interpreted as a static discrete choice model where the cost of choosing one of the alternative includes an additional term that increases with the risk of being driven to the other alternative by the Wiener process. Finally, the mean field equations are given. The fixed point problem will be studied in a next version of this article. Index Terms Mean Field Games, Stochastic Optimal Control, Discrete Choice Models.