Contextual Bayesian optimization of congestion pricing with day-to-day dynamics

Congestion pricing is a common approach to alleviate urban traffic congestion. The design of second-best congestion pricing schemes is typically formulated as non-linear programming and bi-level optimization problems, where the lower-level problem involves either a static or dynamic network equilibrium model. The complexity of these bi-level toll optimization problems increases considerably when incorporating day-to-day dynamic models of travel behavior and dynamic models of network congestion. These models are often operationalized using simulation, and consequently, the toll design problem is a computationally challenging simulation-based optimization problem where the evaluation of a single candidate pricing scheme involves simulating the day-to-day model until convergence. In order to circumvent this issue, we propose a contextual Bayesian optimization (BO) framework, where the BO scheme is embedded within the day-to-day dynamic model by using temporal contextual information. The framework implicitly incorporates the relationship between the objective function across days and uses past days' observations (function evaluations) as weak priors when constructing the Gaussian process underlying the BO algorithm for the current day's toll optimization problem, resulting in gains in computational efficiency. The contextual BO approach is applied to the design of distance-based pricing schemes for the morning commute problem. We demonstrate numerically that the scheme converges to the system optimum, and moreover, utilizes a significantly smaller number of simulation evaluations (ten-fold reduction) than the standard approach wherein each function evaluation involves simulating the day-to-day model until convergence. From a policy perspective, we find that the distance-based schemes yield significant welfare gains relative to area-based schemes and show that the design of the distance-based tariff scheme can significantly affect distributional impacts. A suitably designed two-part tariff structure can partially offset the relatively large welfare losses of travelers with longer commute distances while maintaining overall welfare. The proposed contextual BO scheme is also extended to incorporate context specific demand and supply information, which can be of value to policy-makers when evaluating optimal toll design schemes under a wide range of scenarios in a computational tractable manner.