Equilibrium in a dynamic model of congestion with large and small users

Individual users of transportation facilities often control a significant share of total traffic. Examples include airlines, rail and maritime freight shippers, urban goods delivery companies, taxi companies, and ride-sharing companies. These users have an incentive to internalize the congestion delays their own vehicles impose on each other by adjusting the timing of their trips. We investigate simultaneous trip-timing decisions by large users and small users in a dynamic model of congestion. Unlike previous work, we allow for heterogeneity of trip-timing preferences and for the presence of small users such as individual commuters and fringe airlines. We derive the optimal fleet departure schedule for a large user as a best-response to the aggregate departure rate of other users. We show that when the vehicles in a large user’s fleet have sufficiently diverse preferred arrival times, there may exist a pure-strategy Nash-equilibrium (PSNE) in which the large user schedules vehicles when there is a queue. This highlights that the problem of non-existence of a PSNE identified in Silva et al. (2016) for the case of symmetric large users hinges on the assumption that all vehicles in each fleet have the same preferred arrival time. We also develop some examples to identify under what conditions a PSNE exists, and when the large user departs. The examples illustrate how self-internalization of congestion by a large user can affect the nature of equilibrium and the travel costs that it and other users incur.