Variational analysis of Nash equilibria for a model of traffic flow

Authors:
Alberto Bressan, Chen Jie Liu, Wen Shen and Fang Yu

Journal:
Quart. Appl. Math. **70** (2012), 495-515

MSC (2010):
Primary 35L65; Secondary 35Q93, 49J21, 49N70, 49N90

DOI:
https://doi.org/10.1090/S0033-569X-2012-01304-9

Published electronically:
May 2, 2012

MathSciNet review:
2986132

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Abstract | References | Similar Articles | Additional Information

Abstract: The paper is concerned with Nash equilibrium solutions for the Lighthill-Whitham model of traffic flow, where each driver chooses his own departure time in order to minimize the sum of a departure cost and an arrival cost. Estimates are provided on how much the Nash solution may change, depending on the cost functions and on the flux function of the conservation law. It is shown that this equilibrium solution can also be determined as a global minimizer for a functional , measuring the maximum total cost among all drivers, in a given traffic pattern. The last section of the paper introduces two evolution models, describing how the traffic pattern can change, day after day. It is assumed that each driver adjusts his departure time based on previous experience, in order to lower his own cost. Numerical simulations are reported, indicating a possible instability of the Nash equilibrium.

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Additional Information

**Alberto Bressan**

Affiliation:
Department of Mathematics, Penn State University, University Park, Pennsylvania 16802

Email:
bressan@math.psu.edu

**Chen Jie Liu**

Affiliation:
Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China

Email:
cjliusjtu@gmail.com

**Wen Shen**

Affiliation:
Department of Mathematics, Penn State University, University Park, Pennsylvania 16802

Email:
shen{\textunderscore}w@psu.edu

**Fang Yu**

Affiliation:
Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China

Email:
yufang0820@sjtu.edu.cn

DOI:
https://doi.org/10.1090/S0033-569X-2012-01304-9

Received by editor(s):
November 16, 2011

Published electronically:
May 2, 2012

Dedicated:
Dedicated to Constantine Dafermos in the occasion of his 70th birthday

Article copyright:
© Copyright 2012
Brown University

The copyright for this article reverts to public domain 28 years after publication.