The Price of Anarchy for Instantaneous Dynamic Equilibria

TL;DR

This work analyzes the efficiency of instantaneous dynamic equilibria (IDE) in deterministic queueing flows, contrasting them with full-information dynamic equilibria. It delivers the first quantitative upper bounds for IDE in single-sink networks: IDE flows terminate within $\mathcal{O}(U\tau)$ time and both makespan and total travel-time PoA are $\mathcal{O}(U\tau)$. Complementing this, the authors construct complex networks yielding a lower bound of $\Omega(U\log \tau)$ on both PoA measures, illustrating a fundamental efficiency gap due to short-sighted, real-time routing. The results illuminate the trade-off between information availability and efficiency in dynamic traffic models, with implications for transport policy and real-time routing systems. Overall, the paper advances understanding of IDE performance and establishes baseline bounds that guide future refinements and model extensions.

Abstract

We consider flows over time within the deterministic queueing model 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 have been studied since the eighties, the efficiency of the solution concept is not well understood. We study the price of anarchy for this model and show an upper bound of order $\mathcal{O}(U\cdot τ)$ for single-sink instances, where $U$ denotes the total inflow volume and $τ$ the sum of edge travel times. We complement this upper bound with a family of quite complex instances proving a lower bound of order $Ω(U\cdot\logτ)$.

Figures (15)

• Figure 1: An edge $e=vw$. As the inflow rate at node $v$ exceeds the edge's capacity, a queue forms at its tail.
• Figure 2: The evolution of an IDE flow over the time horizon $[0,4]$. At time $\theta =4$ all remaining particles are on one of the edges leading directly towards the sink $t$. Thus, the last particles will arrive at the sink three time steps later, i.e. at time $\theta=7$. In contrast the optimal makespan would be $\theta=6$, which can be achieved by only sending flow along the direct edges $s_1t$ and $s_2t$ (note, that the maximal inflow rate into sink $t$ is $0$ up to time $2$, then $1$ up to time $3$ and finally $2$ after that -- thus time $\theta=6$ is the earliest point at which the total flow volume of $7$ may have reached the sink). The price of anarchy with respect to the makespan for this example is therefore $7/6$.
• Figure 3: The total flow volume $F^\Delta$ present in the network at different times for the optimal solution (left) and the IDE solution (right) described in \ref{['fig:ide']}. The total shaded area corresponds to the sum of travel times in each of the flows (cf. \ref{['lemma:SumOfTraveltimesAltDef']}), which is $22$ and $25$, respectively. Thus, the price of anarchy with respect to total travel times is $25/22$ in this instance.
• Figure 4: The logical structure of \ref{['sec:UpperBounds']}. Implication arrows are used to indicate which statements are used to prove which other statements. A superscript (V) is used to highlight statements for which their proofs directly use the fact that in the Vickrey bottleneck model queues operate at capacity (i.e. \ref{['eq:FeasibleFlow-QueueOpAtCap']}). Additionally, the proof of the lemma with grey background (from GHS18) depends on the way particles in an IDE predict the future travel times (and, thus, indirectly also depends on the way congestion is modelled). Therefore, these are the main places where one would possibly have to adjust the proofs if one wants to transfer our results to a different flow model.
• Figure 5: An example network with the nodes placed according to their pessimistic distance $\tilde{d}_{v}$. The bold edges form the path $P_{\max}$ in this network. \ref{['lemma:TerminationAcyclicNetworksV2:claim:LBforPessFlow']} states that as long as there is remaining flow volume to the left of any level $k$ this flow volume crosses this level at most $\tau(P_{\max})-k$ time steps later at a rate of at least $1$.

Theorems & Definitions (50)

• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• Remark 2.6
• Definition 2.7
• Definition 2.8