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Advances in Dynamic Traffic Assgmnt: TAC

A New Relationship Between Wardrop’s User Equilibrium and Nash Equilibrium

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Abstract

Traditionally traffic assignment problems (TAPs) have been formulated as non-cooperative games (NCGs), and Wardrop’s user equilibrium (UE) of a TAP as Nash equilibrium (NE) of the corresponding NCG. In this study, in contrast, we propose a mapping from finite NCGs (F-NCGs) to asymmetric TAPs (A-TAPs) on a two-way or urban network, in which the travel cost of one link depends on flows on other links. With the help of three route-based formulations of UE, we show that NE of an F-NCG is equivalent to UE of the corresponding A-TAP. Based on the new relationship, we extend Nash’s fixed point formulation of NE to prove the existence of static UE. This study provides a more complete picture of the relationships between NCGs and TAPs. In the future we will be interested in studying Nash’s fixed point formulation for dynamic UE.

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Notes

  1. 1 Such a relationship has motivated studies on the price of anarchy caused by selfish route choices in both transportation and computer networks (e.g. Roughgarden 2005). Arguably, however, the price of anarchy can be defined in lieu of such a relationship, since it is based on the concepts of UE and system optimal, both of which are well-defined in TAPs.

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Acknowledgments

The author would like to thank Dr. Fan Yang for pointing out possible relationship between the FIFO dynamical system and replicator dynamics through private communications in 2005. The views and results are the author’s alone.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, California Institute for Telecommunications and Information Technology, Institute of Transportation Studies, University of California, Irvine, CA, 92697-3600, USA

    Wen-Long Jin

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Correspondence to Wen-Long Jin.

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Jin, WL. Advances in Dynamic Traffic Assgmnt: TAC. Netw Spat Econ 15, 617–634 (2015). https://doi.org/10.1007/s11067-014-9250-x

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