On the Existence and Uniqueness of Equilibrium in the Bottleneck Model with Atomic Users

Abstract

This paper investigates the existence and uniqueness of equilibrium in the Vickrey bottleneck model when each user controls a positive fraction of total traffic. Users simultaneously choose departure schedules for their vehicle fleets. Each user internalizes the congestion cost that each of its vehicles imposes on other vehicles in its fleet. We establish three results. First, a pure strategy Nash equilibrium (PSNE) may not exist. Second, if a PSNE does exist, identical users may incur appreciably different equilibrium costs. Finally, a multiplicity of PSNE can exist in which no queuing occurs but departures begin earlier or later than in the system optimum. The order in which users depart can be suboptimal as well. Nevertheless, by internalizing self-imposed congestion costs individual users can realize much, and possibly all, of the potential cost savings from either centralized traffic control or time-varying congestion tolls

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

“…Consequently, a candidate PSNE departure schedule may exist in which a large user cannot gain by rescheduling any single vehicle in its fleet, yet it can gain by rescheduling a positive measure of vehicles. These difficulties underlie the non-existence of a PSNE in Silva et al (2017). We then show in Section 4 that if vehicles in the large user's fleet have sufficiently diverse trip-timing preferences, a PSNE may exist in which some -or even all -of the large user's vehicles do queue.…”

“…In this paper we extend Silva et al (2017) by investigating the existence and nature of PSNE in the bottleneck model for a wider range of market structures and under more general assumptions about trip-timing preferences. Unlike both Verhoef and Silva (2017) and Silva et al (2017), we allow for the presence of small users as well as large users.…”

“…For example, at Hartsfield-Jackson Atlanta International (ATL) American Airlines has a 3% market share while Delta's is 73% (Bureau of Transportation Statistics, 2017). 7 See Silva et al (2017) for a brief review.…”

“…Our paper follows up on recent work by Verhoef and Silva (2017) and Silva et al (2017) who focus on determining under what conditions a Pure Strategy Nash Equilibrium (PSNE) in departure-time decisions exists. These two studies employ different deterministic congestion models that are best suited to describe road traffic congestion.…”

“…The second paper to adopt Nash equilibrium, by Silva et al (2017), uses the Vickrey (1969) bottleneck model in which congestion takes the form of queuing behind a bottleneck with a fixed flow capacity. Silva et al consider two large users controlling identical vehicles with linear trip-timing preferences.…”