Toll Design for Routing Games With Stochastic Demands

We consider the toll design problem for non-atomic routing games with stochastic demands. Different from standard toll design problems where the demand for each origin-destination pair is a fixed number, the stochastic demand for each origin-destination pair is a random variable following a specified and known distribution. Our goal is to obtain a toll design for the traffic network to minimize a global cost function, that can, e.g., be the price of anarchy. We formulate the problem into a mathematical program with equilibrium constraints. We then develop a two-loop stochastic algorithm to solve this optimization. Our algorithm is guaranteed to converge to a solution under some assumptions. A numerical example is presented to illustrate the efficacy of our method.