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  Quarterly of Applied Mathematics
Quarterly of Applied Mathematics
  
Online ISSN 1552-4485; Print ISSN 0033-569X
 

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
Published electronically: May 2, 2012
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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 $ \Phi $, 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: http://dx.doi.org/10.1090/S0033-569X-2012-01304-9
PII: S 0033-569X(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.



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