We consider an instance of non uniquely solvable mean field games without common noise, whose counter-part with common noise is uniquely solvable. Because uniqueness fails in this particular example, the question of selection amongst the available equilibria arises naturally. 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, the one(s) 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. We show that while the last two approaches lead to the same selection, they both disagree with the first approach.
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