Generic uniqueness of equilibrium in large crowding games

A crowding game is a noncooperative game in which the payoff of each player depends only on the player's action and the size of the set of players choosing that particular action: The larger the set, the smaller the payoff. Finite, n-player crowding games often have multiple equilibria. However, a large crowding game generically has just one equilibrium, and the equilibrium payoffs in such a game are always unique. Moreover, the sets of equilibria of the m-replicas of a finite crowding game generically converge to a singleton as m tends to infinity. This singleton consists of the unique equilibrium of the `limit' large crowding game. This equilibrium genetically has the following graph-theoretic property: The bipartite graph, in which each player in the original, finite crowding game is joined with all best-response actions for (copies of) that player, does not contain cycles.