IDEAS home Printed from https://ideas.repec.org/a/eee/transb/v195y2025ics0191261525000608.html
   My bibliography  Save this article

A game-theoretical model of road pricing with an endogenized user-equilibrium with multiple user classes

Author

Listed:
  • Malik, Gaurav
  • Proost, Stef
  • M․J․ Tampère, Chris

Abstract

This paper presents a game-theoretical model of road pricing. The model incorporates an endogenized demand and path-choice user-equilibrium with variable user demand and multiple user classes. Different to most of the literature, the proposed model allows to compute in a direct way the optimal tolls, rather than by trial and error of exogenous toll values and tackles the problem of inactive paths that can become active (and vice-versa). Additionally, games with multiple government in different settings can be solved. We proceed in four stages. Firstly, the user-equilibrium model is developed to predict the response of general users to toll instruments of the government(s). Modelling of multiple user classes allows to differentiate users who have different Value of Time and Willingness-To-Pay for their trips. Further, it allows such users to be targeted by different toll instruments. Secondly, a single-player optimization problem is formulated to find optimal toll values for a government acting as a Stackelberg leader over the users. Thirdly, to handle the non-uniqueness of user-equilibrium path flows, a heuristic-based post-processing method is presented that helps in identifying suitable access restrictions necessary to avoid the suboptimal user responses. Fourthly, the single-player optimization problem is used as a building block to develop a general game-theoretical framework that can be applied to different competition scenarios between different types of governments with each, possibly, tolling a different part of the network or the society. The model is, then, applied to four illustrative case-studies. The first case-study involves a single-player optimization problem and ends with a comparison of three solution methods. Mixed Integer Quadratic Program is shown to be the fastest as well as the most consistent. The second case-study involves a game-theoretical problem with two governments and two user classes, and four competition scenarios are elaborated. It is demonstrated how the central objective function can only be worsened by any type of competition between players, and that players have an incentive to take leadership to convert a Nash game to a Stackelberg game. The third case study specifically addresses the non-uniqueness of user-equilibrium path flows, and two different levels of access restrictions are assessed in the post-processing. Finally, the fourth case study shows an application of the single-player optimization problem to a real-world urban mobility problem.

Suggested Citation

  • Malik, Gaurav & Proost, Stef & M․J․ Tampère, Chris, 2025. "A game-theoretical model of road pricing with an endogenized user-equilibrium with multiple user classes," Transportation Research Part B: Methodological, Elsevier, vol. 195(C).
  • Handle: RePEc:eee:transb:v:195:y:2025:i:c:s0191261525000608
    DOI: 10.1016/j.trb.2025.103211
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0191261525000608
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.trb.2025.103211?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Bassolas, Aleix & Ramasco, José J. & Herranz, Ricardo & Cantú-Ros, Oliva G., 2019. "Mobile phone records to feed activity-based travel demand models: MATSim for studying a cordon toll policy in Barcelona," Transportation Research Part A: Policy and Practice, Elsevier, vol. 121(C), pages 56-74.
    2. Najmi, Ali & Waller, Travis & Rashidi, Taha H., 2023. "Equity in network design and pricing: A discretely-constrained MPEC problem," Transportation Research Part A: Policy and Practice, Elsevier, vol. 176(C).
    3. May, Anthony & Shepherd, Simon & Sumalee, Agachai, 2004. "4. Optimal Locations And Charges For Cordon Schemes," Research in Transportation Economics, Elsevier, vol. 9(1), pages 87-105, January.
    4. Nathan H. Gartner, 1980. "Optimal Traffic Assignment with Elastic Demands: A Review Part II. Algorithmic Approaches," Transportation Science, INFORMS, vol. 14(2), pages 192-208, May.
    5. (Jeff) Ban, Xuegang & Dessouky, Maged & Pang, Jong-Shi & Fan, Rong, 2019. "A general equilibrium model for transportation systems with e-hailing services and flow congestion," Transportation Research Part B: Methodological, Elsevier, vol. 129(C), pages 273-304.
    6. Adler, Nicole & Brudner, Amir & Proost, Stef, 2021. "A review of transport market modeling using game-theoretic principles," European Journal of Operational Research, Elsevier, vol. 291(3), pages 808-829.
    7. De Borger, B. & Dunkerley, F. & Proost, S., 2007. "Strategic investment and pricing decisions in a congested transport corridor," Journal of Urban Economics, Elsevier, vol. 62(2), pages 294-316, September.
    8. Vosough, Shaghayegh & de Palma, André & Lindsey, Robin, 2022. "Pricing vehicle emissions and congestion externalities using a dynamic traffic network simulator," Transportation Research Part A: Policy and Practice, Elsevier, vol. 161(C), pages 1-24.
    9. Zhang, H. M. & Ge, Y. E., 2004. "Modeling variable demand equilibrium under second-best road pricing," Transportation Research Part B: Methodological, Elsevier, vol. 38(8), pages 733-749, September.
    10. Stef Proost & Jonas Westin, 2017. "Race to the top in traffic calming," Papers in Regional Science, Wiley Blackwell, vol. 96(2), pages 401-422, June.
    11. Ming-Hua Lin & John Gunnar Carlsson & Dongdong Ge & Jianming Shi & Jung-Fa Tsai, 2013. "A Review of Piecewise Linearization Methods," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-8, November.
    12. Wang, Guangmin & Gao, Ziyou & Xu, Meng & Sun, Huijun, 2014. "Joint link-based credit charging and road capacity improvement in continuous network design problem," Transportation Research Part A: Policy and Practice, Elsevier, vol. 67(C), pages 1-14.
    13. Nicola Basilico & Stefano Coniglio & Nicola Gatti & Alberto Marchesi, 2020. "Bilevel programming methods for computing single-leader-multi-follower equilibria in normal-form and polymatrix games," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 8(1), pages 3-31, March.
    14. Shatanawi, Mohamad & Alatawneh, Anas & Mészáros, Ferenc, 2022. "Implications of static and dynamic road pricing strategies in the era of autonomous and shared autonomous vehicles using simulation-based dynamic traffic assignment: The case of Budapest," Research in Transportation Economics, Elsevier, vol. 95(C).
    15. De Borger, Bruno & Proost, Stef, 2021. "Road tolls, diverted traffic and local traffic calming measures: Who should be in charge?," Transportation Research Part B: Methodological, Elsevier, vol. 147(C), pages 92-115.
    16. Huo, Jinbiao & Liu, Zhiyuan & Chen, Jingxu & Cheng, Qixiu & Meng, Qiang, 2023. "Bayesian optimization for congestion pricing problems: A general framework and its instability," Transportation Research Part B: Methodological, Elsevier, vol. 169(C), pages 1-28.
    17. Stella C. Dafermos, 1973. "Toll Patterns for Multiclass-User Transportation Networks," Transportation Science, INFORMS, vol. 7(3), pages 211-223, August.
    18. Bar-Gera, Hillel, 2010. "Traffic assignment by paired alternative segments," Transportation Research Part B: Methodological, Elsevier, vol. 44(8-9), pages 1022-1046, September.
    19. Nyga, Andreas & Minnich, Aljoscha & Schlüter, Jan, 2020. "The effects of susceptibility, eco-friendliness and dependence on the Consumers’ Willingness to Pay for a door-to-door DRT system," Transportation Research Part A: Policy and Practice, Elsevier, vol. 132(C), pages 540-558.
    20. Abulibdeh, Ammar & Zaidan, Esmat, 2018. "Analysis of factors affecting willingness to pay for high-occupancy-toll lanes: Results from stated-preference survey of travelers," Journal of Transport Geography, Elsevier, vol. 66(C), pages 91-105.
    21. Nathan H. Gartner, 1980. "Optimal Traffic Assignment with Elastic Demands: A Review Part I. Analysis Framework," Transportation Science, INFORMS, vol. 14(2), pages 174-191, May.
    22. Proost, Stef & Sen, Ahksaya, 2006. "Urban transport pricing reform with two levels of government: A case study of Brussels," Transport Policy, Elsevier, vol. 13(2), pages 127-139, March.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Joakim Ekström & Leonid Engelson & Clas Rydergren, 2009. "Heuristic algorithms for a second-best congestion pricing problem," Netnomics, Springer, vol. 10(1), pages 85-102, April.
    2. Zhang, H. M. & Ge, Y. E., 2004. "Modeling variable demand equilibrium under second-best road pricing," Transportation Research Part B: Methodological, Elsevier, vol. 38(8), pages 733-749, September.
    3. Hong Gao & Kai Liu & Xinchao Peng & Cheng Li, 2020. "Optimal Location of Fast Charging Stations for Mixed Traffic of Electric Vehicles and Gasoline Vehicles Subject to Elastic Demands," Energies, MDPI, vol. 13(8), pages 1-16, April.
    4. Hörcher, Daniel & De Borger, Bruno & Graham, Daniel J., 2023. "Subsidised transport services in a fiscal federation: Why local governments may be against decentralised service provision," Economics of Transportation, Elsevier, vol. 34(C).
    5. Watling, D.P. & Shepherd, S.P. & Koh, A., 2015. "Cordon toll competition in a network of two cities: Formulation and sensitivity to traveller route and demand responses," Transportation Research Part B: Methodological, Elsevier, vol. 76(C), pages 93-116.
    6. Yang, Hai, 1997. "Sensitivity analysis for the elastic-demand network equilibrium problem with applications," Transportation Research Part B: Methodological, Elsevier, vol. 31(1), pages 55-70, February.
    7. Frédéric Babonneau & Jean-Philippe Vial, 2008. "An Efficient Method to Compute Traffic Assignment Problems with Elastic Demands," Transportation Science, INFORMS, vol. 42(2), pages 249-260, May.
    8. Toon Vandyck & Stef Proost, 2012. "Inefficiencies in regional commuting policy," Papers in Regional Science, Wiley Blackwell, vol. 91(3), pages 659-689, August.
    9. Zhang, Honggang & Liu, Zhiyuan & Wang, Jian & Wu, Yunchi, 2023. "A novel flow update policy in solving traffic assignment problems: Successive over relaxation iteration method," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 174(C).
    10. Aalami, Soheila & Kattan, Lina, 2022. "Proportionally fair flow markets for transportation networks," Transportation Research Part B: Methodological, Elsevier, vol. 157(C), pages 24-41.
    11. Feder, Christophe, 2018. "Decentralization and spillovers: A new role for transportation infrastructure," Economics of Transportation, Elsevier, vol. 13(C), pages 36-47.
    12. Geng, Lijun & Lu, Zhigang & He, Liangce & Zhang, Jiangfeng & Li, Xueping & Guo, Xiaoqiang, 2019. "Smart charging management system for electric vehicles in coupled transportation and power distribution systems," Energy, Elsevier, vol. 189(C).
    13. Vo, Khoa D. & Lam, William H.K. & Chen, Anthony & Shao, Hu, 2020. "A household optimum utility approach for modeling joint activity-travel choices in congested road networks," Transportation Research Part B: Methodological, Elsevier, vol. 134(C), pages 93-125.
    14. Esteve Codina & Lídia Montero, 2006. "Approximation of the steepest descent direction for the O-D matrix adjustment problem," Annals of Operations Research, Springer, vol. 144(1), pages 329-362, April.
    15. Penchina, Claude M., 2004. "Minimal-revenue congestion pricing: some more good-news and bad-news," Transportation Research Part B: Methodological, Elsevier, vol. 38(6), pages 559-570, July.
    16. Wang, Aihu & Tang, Yuanhua & Mohmand, Yasir Tariq & Xu, Pei, 2022. "Modifying link capacity to avoid Braess Paradox considering elastic demand," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 605(C).
    17. Vickerman, Roger, 2008. "Provision of public transport under conflicting regulatory regimes," Transportation Research Part A: Policy and Practice, Elsevier, vol. 42(9), pages 1176-1182, November.
    18. Ubbels, Barry & Verhoef, Erik T., 2008. "Governmental competition in road charging and capacity choice," Regional Science and Urban Economics, Elsevier, vol. 38(2), pages 174-190, March.
    19. Hörcher, Daniel & Tirachini, Alejandro, 2021. "A review of public transport economics," Economics of Transportation, Elsevier, vol. 25(C).
    20. Fung, Chau Man & Proost, Stef, 2017. "Can we decentralize transport taxes and infrastructure supply?," Economics of Transportation, Elsevier, vol. 9(C), pages 1-19.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:transb:v:195:y:2025:i:c:s0191261525000608. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/548/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.