Dynamic equilibrium at a congestible facility under market power

Erik T. Verhoef, Hugo E. Silva*

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review


This paper studies equilibrium and optimum at a congested facility when firms have market power; e.g., when a few airlines jointly use a congested airport. Unlike most of the previous literature, we characterize the equilibrium in terms of timing of arrivals in a continuous-time congestion model when firms simultaneously schedule services. Using the Henderson-Chu dynamic model of flow congestion in a multiple-firm setting, we find that a stable and unique Nash equilibrium in pure strategies always exists. Importantly, it also exists in cases where it fails to exist under bottleneck congestion (notably when the value of schedule late exceeds the value of travel delays). We find that symmetric firms schedule arrivals inefficiently, and strongly concentrated around the desired arrival time so that the peak is shorter and delays are higher than socially optimal. We show that when firms are asymmetric in terms of output, all firms schedule vehicles in the peak center, around the desired arrival time, with arrival windows increasing with firm size such that a smaller firm's window is always fully contained in a larger firm's window and only the largest firm operates in the early and late shoulders. Furthermore, for any pair of asymmetric firms, the larger firm has a higher instantaneous arrival rate at any moment where both firms schedule arrivals. Our results also show that even though self-internalization can be substantial, there is scope for decentralizing the first-best outcome through time-varying tolls.

Original languageEnglish
Pages (from-to)174-192
Number of pages19
JournalTransportation Research Part B: Methodological
Publication statusPublished - 1 Nov 2017


  • Congestion pricing
  • Dynamic congestion
  • Internalization
  • Market power

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