Approximation and Convergence of Large Atomic Congestion Games

Abstract

A. 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 (1998b). 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.

A convergence analysis of the price of anarchy in atomic congestion games

“…They confirm the intuition that atomic congestion games can be thought of as non-atomic congestion games when d max is tiny, the number of users is huge, and T is moderate, i.e., neither too small nor too large. Our convergence result for the mixed PoA actually generalizes those of Cominetti et al [11] to the case that T → ∞. This is a non-trivial generalization, since it does not require the existence of the limit non-atomic congestion game, which is a premise in the analysis of Cominetti et al [11].…”

“…The results of Cominetti et al [11] are inspiring and seminal. They confirm the intuition that atomic congestion games can be thought of as non-atomic congestion games when d max is tiny, the number of users is huge, and T is moderate, i.e., neither too small nor too large.…”

“…Conditions implying the convergence of mixed Nash equilibria in atomic congestion games to pure Nash equilibria in non-atomic congestion games have also been studied in, e.g., [11,19,21,22,26], and others. Among these papers, Cominetti et al [11] is the closest to our work.…”

“…Conditions implying the convergence of mixed Nash equilibria in atomic congestion games to pure Nash equilibria in non-atomic congestion games have also been studied in, e.g., [11,19,21,22,26], and others. Among these papers, Cominetti et al [11] is the closest to our work. They showed that mixed Nash equilibria of an atomic congestion game with strictly increasing cost functions converge in distribution to pure Nash equilibria of a limit non-atomic congestion game, when the total demand T converges to a constant T 0 ∈ (0, ∞), the maximum individual demand d max converges to 0, and the number of users converges to ∞.…”

Information Design in Large Games

“…Convergence of finite players potential games to nonatomic potential games has been studied by Sandholm (2001). The relation between Nash equilibria and Wardrop equilibria in congestion games has been studied by Haurie and Marcotte (1985) in the atomic splittable case and by Cominetti et al (2021) in the atomic nonsplittable case. Similar results appear in the literature about mean-field games (see, e.g., Fischer, 2018, 2021;Lacker, 2020).…”