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Dynamic Atomic Congestion Games with Seasonal Flows

61 Pages Posted: 13 Jun 2013 Last revised: 19 Jun 2016

Marco Scarsini

LUISS, Dipartimento di Economia e Finanza

Marc Schröder

Maastricht University

Tristan Tomala

HEC Paris - Economics & Decision Sciences

Date Written: June 17, 2016


We propose a model of discrete time dynamic congestion games with atomic players and a single source-destination pair. The latencies of edges are composed by free-flow transit times and possible queuing time due to capacity constraints. We give a precise description of the dynamics induced by the individual strategies of players and of the corresponding costs, either when the traffic is controlled by a planner, or when players act selfishly. In parallel networks, optimal and equilibrium behavior eventually coincides, but the selfish behavior of the first players has consequences that cannot be undone and are paid by all future generations. In more general topologies, our main contributions are three-fold.

First, we show that equilibria are usually not unique. In particular, we prove that there exists a sequence of networks such that the price of anarchy is equal to n-1, where n is the number of vertices, and the price of stability is equal to 1.

Second, we illustrate a new dynamic version of Braess's paradox: the presence of initial queues in a network may decrease the long-run costs in equilibrium. This paradox may arise even in networks for which no Braess's paradox was previously known.

Third, we propose an extension to model seasonalities by assuming that departure flows fluctuate periodically over time. We introduce a measure that captures the queues induced by periodicity of inflows. This measure is the increase in costs compared to uniform departures for optimal and equilibrium flows in parallel networks.

Keywords: Network games, dynamic flows, price of seasonality, price of anarchy, max-flow min-cut

JEL Classification: C72, C73

Suggested Citation

Scarsini, Marco and Schröder, Marc and Tomala, Tristan, Dynamic Atomic Congestion Games with Seasonal Flows (June 17, 2016). HEC Paris Research Paper No. ECO/SCD-2013-1016. Available at SSRN: or

Marco Scarsini

LUISS, Dipartimento di Economia e Finanza ( email )

Viale Romania 32
Rome, RM 00197

Marc Schröder

Maastricht University ( email )

P.O. Box 616
Maastricht, 6200MD

Tristan Tomala (Contact Author)

HEC Paris - Economics & Decision Sciences ( email )

1 rue de la Liberation
Paris, 78351

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