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

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.