  We consider the question of whether, and in what sense, Wardrop equilibria
provide a good approximation for Nash equilibria in atomic unsplittable
congestion games with a large number of small players. We examine two different
definitions of small players. In the first setting, we consider games where
each player's weight is small. We prove that when the number of players goes to
infinity and their weights to zero, the random flows in all (mixed) Nash
equilibria for the finite games converge in distribution to the set of Wardrop
equilibria of the corresponding nonatomic limit game. In the second setting, we
consider an increasing number of players with a unit weight that participate in
the game with a decreasingly small probability. In this case, the Nash
equilibrium flows converge in total variation towards Poisson random variables
whose expected values are Wardrop equilibria of a different nonatomic game with
suitably-defined costs. The latter can be viewed as symmetric equilibria in a
Poisson game in the sense of Myerson, establishing a plausible connection
between the Wardrop model for routing games and the stochastic fluctuations
observed in real traffic. In both settings we provide explicit approximation
bounds, and we study the convergence of the price of anarchy. Beyond the case
of congestion games, we prove a general result on the convergence of large
games with random players towards Poisson games.
