Learning marginal-cost pricing via a trial-and-error procedure with day-to-day flow dynamics

This paper investigates the convergence of the trial‐and‐error procedure to achieve the system optimum by incorporating the day‐to‐day evolution of traffic flows. The path flows are assumed to follow an ‘excess travel cost dynamics’ and evolve from disequilibrium states to the equilibrium day by day. This implies that the observed link flow pattern during the trial‐and‐error procedure is in disequilibrium. By making certain assumptions on the flow evolution dynamics, we prove that the trial‐and‐error procedure is capable of learning the system optimum link tolls without requiring explicit knowledge of the demand functions and flow evolution mechanism. A methodology is developed for updating the toll charges and choosing the inter‐trial periods to ensure convergence of the iterative approach towards the system optimum. Numerical examples are given in support of the theoretical findings.