### Abstract

The paper is concerned with the Lighthill-Whitham model of traffic flow, where the density of cars is described by a scalar conservation law. A cost functional is introduced, depending on the departure and arrival times of each driver. Under natural assumptions, we prove the existence of a unique globally optimal solution, minimizing the total cost to all drivers. This solution contains no shocks and can be explicitly described. We also prove the existence of a Nash equilibrium solution, where no driver can lower his individual cost by changing his own departure time. A characterization of the Nash solution is provided, establishing its uniqueness. Some explicit examples are worked out, comparing the costs of the optimal and the equilibrium solutions. The analysis also yields a strategy for optimal toll pricing.

Original language | English (US) |
---|---|

Pages (from-to) | 2384-2417 |

Number of pages | 34 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 43 |

Issue number | 5 |

DOIs | |

State | Published - Nov 21 2011 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Mathematical Analysis*,

*43*(5), 2384-2417. https://doi.org/10.1137/110825145

}

*SIAM Journal on Mathematical Analysis*, vol. 43, no. 5, pp. 2384-2417. https://doi.org/10.1137/110825145

**Optima and equilibria for a model of traffic flow.** / Bressan, Alberto; Han, Ke.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Optima and equilibria for a model of traffic flow

AU - Bressan, Alberto

AU - Han, Ke

PY - 2011/11/21

Y1 - 2011/11/21

N2 - The paper is concerned with the Lighthill-Whitham model of traffic flow, where the density of cars is described by a scalar conservation law. A cost functional is introduced, depending on the departure and arrival times of each driver. Under natural assumptions, we prove the existence of a unique globally optimal solution, minimizing the total cost to all drivers. This solution contains no shocks and can be explicitly described. We also prove the existence of a Nash equilibrium solution, where no driver can lower his individual cost by changing his own departure time. A characterization of the Nash solution is provided, establishing its uniqueness. Some explicit examples are worked out, comparing the costs of the optimal and the equilibrium solutions. The analysis also yields a strategy for optimal toll pricing.

AB - The paper is concerned with the Lighthill-Whitham model of traffic flow, where the density of cars is described by a scalar conservation law. A cost functional is introduced, depending on the departure and arrival times of each driver. Under natural assumptions, we prove the existence of a unique globally optimal solution, minimizing the total cost to all drivers. This solution contains no shocks and can be explicitly described. We also prove the existence of a Nash equilibrium solution, where no driver can lower his individual cost by changing his own departure time. A characterization of the Nash solution is provided, establishing its uniqueness. Some explicit examples are worked out, comparing the costs of the optimal and the equilibrium solutions. The analysis also yields a strategy for optimal toll pricing.

UR - http://www.scopus.com/inward/record.url?scp=81255169947&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=81255169947&partnerID=8YFLogxK

U2 - 10.1137/110825145

DO - 10.1137/110825145

M3 - Article

AN - SCOPUS:81255169947

VL - 43

SP - 2384

EP - 2417

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 5

ER -