Convergence of Large Atomic Congestion Games

We study the convergence of sequences of atomic unsplittable congestion games with an increasing number of players. We consider two situations. In the first setting, each player has a weight that tends to zero, in which case the mixed equilibria of the finite games converge to the set of Wardrop equilibria of the corresponding nonatomic limit game. In the second case, players have unit weights, but participate in the game with a probability that tends to zero. In this case, the mixed equilibria converge to the set of Wardrop equilibria of another nonatomic game with suitably defined costs, which can be seen as a Poisson game in the sense of Myerson (1998). In both settings we show that the price of anarchy of the sequence of games converges to the price of anarchy of the nonatomic limit. Beyond the case of congestion games, we establish a general result on the convergence of large games with random players towards a Poisson game.