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Colini-Baldeschi, R., Cominetti, R., Mertikopoulos, P., & Scarsini, M. (2020). When Is Selfish Routing Bad? The Price of Anarchy in Light and Heavy Traffic. Oper. Res., 68(2), 411–434.
Abstract: This paper examines the behavior of the price of anarchy as a function of the traffic inflow in nonatomic congestion games with multiple origin/destination (O/D) pairs. Empirical studies in real-world networks show that the price of anarchy is close to 1 in both light and heavy traffic, thus raising the following question: can these observations be justified theoretically? We first show that this is not always the case: the price of anarchy may remain a positive distance away from 1 for all values of the traffic inflow, even in simple three-link networks with a single O/D pair and smooth, convex costs. On the other hand, for a large class of cost functions (including all polynomials) and inflow patterns, the price of anarchy does converge to 1 in both heavy and light traffic, irrespective of the network topology and the number of O/D pairs in the network. We also examine the rate of convergence of the price of anarchy, and we show that it follows a power law whose degree can be computed explicitly when the network's cost functions are polynomials.
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Cominetti, R., Scarsini, M., Schroder, M., & Stier-Moses, N. (2022). Approximation and Convergence of Large Atomic Congestion Games. Math. Oper. Res., Early Access.
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 War drop 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.
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