TY - JOUR

T1 - Approximation and Convergence of Large Atomic Congestion Games

AU - Cominetti, Roberto

AU - Scarsini, Marco

AU - Schröder, Marc

AU - Stier-Moses, Nicolás

N1 - Funding Information:
Funding: R. Cominetti gratefully acknowledges the support of Proyecto Anillo [Grant ANID/PIA/ ACT192094]. M. Scarsini’s work was partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilitàe le loro Applicazioni, Istituto Nazionale di Alta Matematica [Grant 2020 “Random walks on random games”] and the Progetti di Rilevante Interesse Nazionale 2017 [Grant “Algorithms, Games, and Digital Markets”]. This research also received partial support from the European Cooperation in Science and Technology [Action European Network for Game Theory].
Publisher Copyright:
Copyright: © 2022 INFORMS.

PY - 2023/5

Y1 - 2023/5

N2 - 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 in which 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 toward 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 toward Poisson games.

AB - 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 in which 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 toward 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 toward Poisson games.

KW - Poisson games

KW - Wardrop equilibrium

KW - nonatomic congestion games

KW - price of anarchy

KW - price of stability

KW - symmetric equilibrium

KW - total variation

KW - unsplittable atomic congestion games

UR - http://www.scopus.com/inward/record.url?scp=85161019026&partnerID=8YFLogxK

U2 - 10.1287/moor.2022.1281

DO - 10.1287/moor.2022.1281

M3 - Article

AN - SCOPUS:85161019026

SN - 0364-765X

VL - 48

SP - 784

EP - 811

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

IS - 2

ER -