The Price of Anarchy for Instantaneous Dynamic Equilibria

Abstract

We consider flows over time within the deterministic queueing model of Vickrey and study the solution concept of instantaneous dynamic equilibrium (IDE), in which flow particles select at every decision point a currently shortest path. The length of such a path is measured by the physical travel time plus the time spent in queues. Although IDE has been studied since the eighties, the worst-case efficiency of the solution concept with respect to the travel time is not well understood. We study the price of anarchy for this model under the makespan and total travel time objective and show the first known upper bound for both measures of order [Formula: see text] for single-sink instances, where U denotes the total inflow volume and τ denotes the sum of edge travel times. We complement this upper bound with a family of quite complex instances proving a lower bound of order [Formula: see text]. Funding: The research of the authors was funded by the Deutsche Forschungsgemeinschaft [Grants HA 8041/1-1 and HA 8041/4-1].

Convergence of a Packet Routing Model to Flows over Time

“…It asks that for every node in the network that an agent uses in their path, 1 Other equilibria notions distinct from Nash equilibria have been considered in the literature, in which agents make decisions without full information of the overall traffic situation. We refer to Graf, Harks and Sering [GHS20] and references therein for a discussion of instantaneous dynamic equilibria (see also [GH23a,GH23b]), where agents make decisions as they traverse the network based on current queues; and to Graf, Harks, Kollias and Merkl [GHKM22] for a very interesting approach to a much more general information framework where users use predictions of future congestion patterns. not just the sink, the agent's departure time from that node is at most δ later than its earliest possible arrival time (considering all possible routes to the node).…”

Convergence of Approximate and Packet Routing Equilibria to Nash Flows Over Time

“…1 Other equilibria notions distinct from Nash equilibria have been considered in the literature, in which agents make decisions without full information of the overall traffic situation. We refer to Graf, Harks and Sering [GHS20] and references therein for a discussion of instantaneous dynamic equilibria (see also [GH23a], [GH23b]), where agents make decisions as they traverse the network based on current queues; and to Graf, Harks, Kollias and Merkl [GHKM22] for a very interesting approach to a much more general information framework where users use predictions of future congestion patterns.…”