Approximation and Convergence of Large Atomic Congestion Games

Roberto Cominetti (), Marco Scarsini, Marc Schröder () and Nicolás Stier-Moses ()
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Roberto Cominetti: Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Peñalolén 7941169, Región Metropolitana, Chile
Marc Schröder: School of Business and Economics, Maastricht University, 6211 LK Maastricht, Netherlands
Nicolás Stier-Moses: Core Data Science, Meta, Menlo Park, California 94025

Mathematics of Operations Research, 2023, vol. 48, issue 2, 784-811

Abstract: 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.

Keywords: Primary: 91A14; secondary: 91A07; 91A43; 60B10; unsplittable atomic congestion games; nonatomic congestion games; Wardrop equilibrium; Poisson games; symmetric equilibrium; price of anarchy; price of stability; total variation (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:48:y:2023:i:2:p:784-811

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