Dynamic Atomic Congestion Games with Seasonal Flows

Abstract

We propose a model of discrete time dynamic congestion games with atomic players and a single source-destination pair. The latencies of edges are composed by free-flow transit times and possible queuing time due to capacity constraints. We give a precise description of the dynamics induced by the individual strategies of players and of the corresponding costs, either when the traffic is controlled by a planner, or when players act selfishly. In parallel networks, optimal and equilibrium behavior eventually coincides, but the selfish behavior of the first players has consequences that cannot be undone and are paid by all future generations. In more general topologies, our main contributions are three-fold.First, we show that equilibria are usually not unique. In particular, we prove that there exists a sequence of networks such that the price of anarchy is equal to n − 1, where n is the number of vertices, and the price of stability is equal to 1.Second, we illustrate a new dynamic version of Braess's paradox: the presence of initial queues in a network may decrease the long-run costs in equilibrium. This paradox may arise even in networks for which no Braess's paradox was previously known.Third, we propose an extension to model seasonalities by assuming that departure flows fluctuate periodically over time. We introduce a measure that captures the queues induced by periodicity of inflows. This measure is the increase in costs compared to uniform departures for optimal and equilibrium flows in parallel networks.

Atomic Dynamic Flow Games: Adaptive vs. Nonadaptive Agents

“…As in other models of atomic dynamic network flows, complicated and sometimes unpredictable chain effects form a great obstacle in our analysis. For example, a Braess-like paradox that resembles the one in Scarsini et al (2018) (but with a different flavor) still exists in our model (see Example 4…”

“…For each booth, at most one car can pass through it each time and begin to travel along the corresponding lane with a uniform speed (meaning the transit time of road e can be viewed as a constant t e ). In this paper, based on this atomic DQ rule, we propose a model that is very similar to the one in Scarsini et al (2018) but has a crucial difference on the tie-breaking rules.…”

“…Note that the above characterization and properties are satisfied by all NEs of game Γ N without any additional constraints on agent behaviors or network topologies, whereas the literature usually can establish the properties for only some special NEs or NEs on special networks (Harks et al 2018, Scarsini et al 2018. In particular, while the existence of a strong NE (which must be an NE by definition) is known in the literature of atomic dynamic flow games (e.g., Werth et al 2014),…”

“…Subsequent studies are extensive since the last two decades, encompass various models to investigate equilibrium behaviors of selfish agents, and adopt a wide variety of methodologies from mathematical programming, optimal control, variational inequalities, algorithmic game theory, and simulations (see Peeta and Ziliaskopoulos 2001, Koch and Skutella 2009, Cominetti et al 2017, and the references therein). Under dynamic queuing, little is known about general equilibrium properties, until recent exciting progress on deriving equilibrium existence, uniqueness, characterizations and constructions (Meunier and Wagner 2010, Koch and Skutella 2011, Cominetti et al 2015, Scarsini et al 2018. We discuss the study of equilibria for two major subbranches of dynamic flow games, atomic models and nonatomic models, in the following two subsections, respectively.…”

“…Second, a latency function imposes the same delay upon all agents who travel along the road segment at any time, even if their travel periods along the segment do not overlap, which is unreasonable, as for example travel in peak hours takes more time than in off-peak hours. A well-recognized method to overcome the above weaknesses is to apply the deterministic queuing (DQ) rule (Vickrey 1969, Hendrickson and Kocur 1981, Koch and Skutella 2011, Cominetti et al 2015, Scarsini et al 2018).…”