The Impact of Spillback on the Price of Anarchy for Flows Over Time

TL;DR

This paper investigates how spillback effects influence the price of anarchy in dynamic traffic flows, revealing unbounded PoA in general but bounded PoA under certain network constraints, and showing spillback can sometimes accelerate equilibrium formation.

Contribution

It extends the understanding of spillback effects on the price of anarchy in flows over time, providing bounds and quantifying spillback's impact on equilibrium quality.

Findings

PoA is unbounded with spillback even in networks with unit capacities.

PoA can be arbitrarily large, with the Braess ratio also unbounded.

Spillback can sometimes speed up the convergence to equilibrium.

Abstract

Flows over time enable a mathematical modeling of traffic that changes as time progresses. In order to evaluate these dynamic flows from a game theoretical perspective we consider the price of anarchy (PoA). In this paper we study the impact of spillback effects on the PoA, which turn out to be substantial. It is known that, in general, the PoA is unbounded in the spillback setting. We extend this by showing that it is still unbounded even when considering networks with unit edge capacities and that the Braess ratio can be arbitrarily large. In contrast to that, we show that on a fixed network the PoA as a function of the flow amount is bounded by a constant and also upper bound the PoA for the set of networks where the outflow capacities satisfy certain constraints depending on the quickest flow. This upper bound only depends on the worst spillback factor of the Nash flows over time of the given network. It therefore provides a way to quantify the impact of spillback to the quality of the dynamic equilibria. In addition, we show the surprising fact that the introduction of spillback behavior can actually speed up dynamic equilibria in some networks.