Braess' paradox in a generalised traffic network

Zverovich, Vadim; Avineri, Erel · 2014 · Crossref

DOI: 10.1002/atr.1269

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Summary

This paper addresses the Braess paradox, a counterintuitive phenomenon in traffic networks where adding a new link increases overall congestion rather than reducing it. While previous research, notably by Pas and Principio, focused on symmetrical network configurations where specific links share identical volume-delay functions, such symmetry is rare in real-world infrastructure. Motivated by Valiant and Roughgarden’s finding that the global behavior of large random networks resembles Braess’s original four-node example, the authors aim to generalize the analysis to arbitrary, asymmetrical networks. The study seeks to determine the precise conditions under which the paradox occurs when link parameters—free flow travel times and delay coefficients—differ across the network. The authors model the traffic network as a game where users non-cooperatively minimize individual travel costs, leading to a Wardrop/Nash equilibrium. They introduce a generalized version of Braess’s network, replacing each link with a path of arbitrary length and assigning arbitrary linear volume-delay functions ($\alpha_{ij} + \beta_{ij}f_{ij}$) to each segment. By accounting for fixed external flows, the authors reduce the generalized network to a four-node configuration with five links, each having distinct linear travel time functions. The study derives explicit formulas for equilibrium travel times in both the network with the additional link ($N^+$) and without it ($N$). It defines specific "Braess numbers" and parameters based on the network’s delay coefficients and free flow times to characterize these equilibria. The main findings provide necessary and sufficient conditions for the occurrence of Braess’s paradox in this generalised setting. The authors establish a "Mega-Theorem" identifying four specific equilibrium scenarios where the paradox can arise, followed by four theorems that define the exact intervals of total traffic flow ($Q$) for which the paradox occurs in each scenario. These conditions depend on the positivity of specific Braess numbers and the relationship between total flow and network parameters. The paper also demonstrates that their general framework successfully reproduces and extends known results for symmetrical networks, including the conditions derived by Pas and Principio. Additionally, the authors introduce the concept of a "pseudo-paradox," showing that in asymmetrical configurations, this variant occurs for any total flow. The significance of this work lies in its removal of the restrictive symmetry assumptions prevalent in earlier literature, thereby providing a more realistic tool for transport planners. By establishing exact conditions for the paradox in arbitrary networks, the study highlights that Braess’s paradox is not an anomaly but a widespread risk in general transportation systems. This implies that infrastructure changes, such as adding new roads, must be rigorously analyzed using these generalized conditions to avoid unintended increases in congestion. The results reinforce the fundamental importance of Braess’s original configuration as a model for understanding equilibrium flows in complex, real-world networks.

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• traffic density