Bayesian Congestion Game with Traffic Manager: Binary Signal Case

We study the problem of designing an efficient signaling policy for a traffic manager (TM) when parts of a network experience unpredictable congestion and delays due to external factors, such as accidents or construction that blocks some lanes. To this end, we consider a simple network with two parallel routes, one of which suffers from more unpredictable congestion than the other. In order to understand drivers' behavior, we consider two scenarios - (i) no information from TM and (ii) a binary signal from TM - and assume that a fraction of drivers, called informed drivers, can make use of the TM signal in the latter case and make decisions based on updated information in order to minimize their expected delay. We show that an optimal signaling policy for TM is a threshold policy: the TM warns the drivers of high congestion if and only if the observed congestion on the route with more unpredictable delays exceeds some threshold. Under a mild condition, this optimal signaling policy is guaranteed to reduce the expected overall network delay. Moreover, we examine how the optimal threshold policy behaves when the congestion delay is either very sensitive or insensitive to traffic and provide a lower bound on the optimal threshold.