# Nash equilibria for a model of traffic flow with several groups of drivers

• Volume: 18, Issue: 4, page 969-986
• ISSN: 1292-8119

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## Abstract

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Traffic flow is modeled by a conservation law describing the density of cars. It is assumed that each driver chooses his own departure time in order to minimize the sum of a departure and an arrival cost. There are N groups of drivers, The i-th group consists of κi drivers, sharing the same departure and arrival costs ϕi(t),ψi(t). For any given population sizes κ1,...,κn, we prove the existence of a Nash equilibrium solution, where no driver can lower his own total cost by choosing a different departure time. The possible non-uniqueness, and a characterization of this Nash equilibrium solution, are also discussed.

## How to cite

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Bressan, Alberto, and Han, Ke. "Nash equilibria for a model of traffic flow with several groups of drivers." ESAIM: Control, Optimisation and Calculus of Variations 18.4 (2012): 969-986. <http://eudml.org/doc/272910>.

@article{Bressan2012,
abstract = {Traffic flow is modeled by a conservation law describing the density of cars. It is assumed that each driver chooses his own departure time in order to minimize the sum of a departure and an arrival cost. There are N groups of drivers, The i-th group consists of κi drivers, sharing the same departure and arrival costs ϕi(t),ψi(t). For any given population sizes κ1,...,κn, we prove the existence of a Nash equilibrium solution, where no driver can lower his own total cost by choosing a different departure time. The possible non-uniqueness, and a characterization of this Nash equilibrium solution, are also discussed.},
author = {Bressan, Alberto, Han, Ke},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {scalar conservation law; Hamilton-Jacobi equation; Nash equilibrium; non-uniqueness},
language = {eng},
number = {4},
pages = {969-986},
publisher = {EDP-Sciences},
title = {Nash equilibria for a model of traffic flow with several groups of drivers},
url = {http://eudml.org/doc/272910},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Bressan, Alberto
AU - Han, Ke
TI - Nash equilibria for a model of traffic flow with several groups of drivers
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2012
PB - EDP-Sciences
VL - 18
IS - 4
SP - 969
EP - 986
AB - Traffic flow is modeled by a conservation law describing the density of cars. It is assumed that each driver chooses his own departure time in order to minimize the sum of a departure and an arrival cost. There are N groups of drivers, The i-th group consists of κi drivers, sharing the same departure and arrival costs ϕi(t),ψi(t). For any given population sizes κ1,...,κn, we prove the existence of a Nash equilibrium solution, where no driver can lower his own total cost by choosing a different departure time. The possible non-uniqueness, and a characterization of this Nash equilibrium solution, are also discussed.
LA - eng
KW - scalar conservation law; Hamilton-Jacobi equation; Nash equilibrium; non-uniqueness
UR - http://eudml.org/doc/272910
ER -

## References

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1. [1] J.P. Aubin and A. Cellina, Differential inclusions. Set-Valued Maps and Viability Theory. Springer-Verlag, Berlin (1984). Zbl0538.34007MR755330
2. [2] A. Bressan and K. Han, Optima and equilibria for a model of traffic flow. SIAM J. Math. Anal.43 (2011) 2384–2417. Zbl1236.90024MR2861667
3. [3] A. Cellina, Approximation of set valued functions and fixed point theorems. Ann. Mat. Pura Appl.82 (1969) 17–24. Zbl0187.07701MR263046
4. [4] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory. Springer-Verlag, New York (1998). Zbl1047.49500MR1488695
5. [5] T.L. Friesz, Dynamic Optimization and Differential Games, Springer, New York (2010). Zbl1207.91004
6. [6] T.L. Friesz, T. Kim, C. Kwon and M.A. Rigdon, Approximate network loading and dual-time-scale dynamic user equilibrium. Transp. Res. Part B (2010).
7. [7] A. Fügenschuh, M. Herty and A. Martin, Combinatorial and continuous models for the optimization of traffic flows on networks. SIAM J. Optim.16 (2006) 1155–1176. Zbl1131.90015MR2219137
8. [8] M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models. AIMS Series on Applied Mathematics, Springfield, Mo. (2006). Zbl1136.90012MR2328174
9. [9] M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks. J. Optim. Theory Appl.126 (2005) 589–616. Zbl1079.49024MR2164806
10. [10] M. Herty, C. Kirchner and A. Klar, Instantaneous control for traffic flow. Math. Methods Appl. Sci.30 (2007) 153–169. Zbl1117.90030MR2285119
11. [11] L.C. Evans, Partial Differential Equations, 2nd edition. American Mathematical Society, Providence, RI (2010). Zbl0902.35002
12. [12] P.D. Lax, Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math.10 (1957) 537–566. Zbl0081.08803MR93653
13. [13] M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A 229 (1955) 317–345. Zbl0064.20906MR72606
14. [14] P.I. Richards, Shock waves on the highway. Oper. Res.4 (1956), 42–51. MR75522
15. [15] J. Smoller, Shock waves and reaction-diffusion equations, 2nd edition. Springer-Verlag, New York (1994). Zbl0508.35002MR1301779

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