Computing Optimal Tolls in Routing Games without Knowing the Latency Functions

We consider the following question: in a nonatomic routing game, can the tolls that induce the minimum latency flow be computed without knowing the latency functions? Since the latency functions are unknown, we assume we are given an oracle that has access to the underlying routing game. A query to the oracle consists of tolls on edges, and the response is the resulting equilibrium flow. We show that in this model, it is impossible to obtain optimal tolls. However, if we augment the oracle so that it returns the total latency of the equilibrium flow induced by the tolls in addition to the flow itself, then the required tolls can be computed with a polynomial number of queries.