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Trial-and-Error Procedure for Implementing a Road-Pricing Scheme

This paper deals with a road-pricing scheme that aims at alleviating congestion phenomena or air and noise pollution in some areas of a transportation network in such a way that the volumes of traffic flow on entry links to these areas do not exceed their respective predetermined thresholds by levying appropriate tolls at these links. This paper begins to show that the road-pricing scheme is equivalent to a problem that determines optimal Lagrangian multipliers for a user equilibrium traffic assignment problem with link capacity constraints. It then proceeds to devise a novel trial-and-error procedure requiring observed traffic flows at the entry links only, to identify a solution for the road-pricing scheme when link travel time functions, origin–destination demand functions, and users’ value of travel time are unknown. The procedure is as follows. A trial on a set of given tolls is conducted, and then the resultant link flows are observed. According to these observed traffic flows, a new set of tolls for the next trial is adjusted by executing a simple projection operation. The trial-and-error procedure is, in fact, a variation of a gradient projection method for dual formulation of the traffic assignment problem, and its convergence can be guaranteed under mild conditions. Accordingly, a conjecture for the convergence of trial-and-error implementation of the congestion pricing proposed by economists is rigorously proved. Furthermore, the iterative procedure presented in this paper in practice can facilitate the estimation of such tolls by land transport authorities.