We consider a discrete-time nonatomic routing game with variable demand and uncertain costs. Given a routing network with single origin and destination, the cost function of each edge depends on some uncertain persistent state parameter. At every period, a random traffc demand is routed through the network according to a Bayes-Wardrop equilibrium. The realized costs are publicly observed and the Bayesian belief about the state parameter is updated. We say that there is strong learning when beliefs converge to the truth and weak learning when the equilibrium flow converges to the complete-information flow. We characterize the networks for which learning occurs. We prove that these networks have a series-parallel structure and provide a counterexample to prove that the condition is necessary.
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