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

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  • 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
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    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. 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.
    6. Stella C. Dafermos, 1973. "Toll Patterns for Multiclass-User Transportation Networks," Transportation Science, INFORMS, vol. 7(3), pages 211-223, August.
    7. Bar-Gera, Hillel, 2010. "Traffic assignment by paired alternative segments," Transportation Research Part B: Methodological, Elsevier, vol. 44(8-9), pages 1022-1046, September.
    8. 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).
    9. 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.
    10. 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.
    11. 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).
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. (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.
    17. 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.
    18. 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.
    19. 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.
    20. 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.
    21. 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.
    22. 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.
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