Selection of equilibria in a linear quadratic mean-field game

In this paper, we address an instance of uniquely solvable mean-field game with a common noise whose corresponding counterpart without common noise has several equilibria. We study the selection problem for this mean-field game without common noise via three approaches. A common approach is to select, amongst all the equilibria, those yielding the minimal cost for the representative player. Another one is to select equilibria that are included in the support of the zero noise limit of the mean-field game with common noise. A last one is to select equilibria supported by the limit of the mean-field component of the corresponding $N$-player game as the number of players goes to infinity. The contribution of this paper is to show that, for the class under study, the last two approaches select the same equilibria, but the first approach selects another one.