On the Price of Anarchy for Flows over Time

Abstract

Dynamic network flows, or network flows over time, constitute an important model for real-world situations in which steady states are unusual, such as urban traffic and the internet. These applications immediately raise the issue of analyzing dynamic network flows from a game-theoretic perspective. In this paper, we study dynamic equilibria in the deterministic fluid queuing model in single-source, single-sink networks—arguably the most basic model for flows over time. In the last decade, we have witnessed significant developments in the theoretical understanding of the model. However, several fundamental questions remain open. One of the most prominent ones concerns the price of anarchy, measured as the worst-case ratio between the minimum time required to route a given amount of flow from the source to the sink and the time a dynamic equilibrium takes to perform the same task. Our main result states that, if we could reduce the inflow of the network in a dynamic equilibrium, then the price of anarchy is bounded by a factor, parameterized by the longest path length that converges to [Formula: see text], and this is tight. This significantly extends a result by Bhaskar et al. (SODA 2011). Furthermore, our methods allow us to determine that the price of anarchy in parallel-link and parallel-path networks is exactly 4/3. Finally, we argue that, if a certain, very natural, monotonicity conjecture holds, the price of anarchy in the general case is exactly [Formula: see text].

The Impact of Spillback on the Price of Anarchy for Flows Over Time

“…Are there any bounds in the case of c ≤ 1 e or is it possible to enforce c > 1 e through some Stackelberg-like strategy? Do the results of the recent work of Correa et al [7] also transfer to the spillback setting? On the more applied side of the research it would also be very interesting to algorithmically identify street segments (edges) which are especially vulnerable for spillback.…”

“…This new research includes the study of existence, uniqueness and the long-term behavior of Nash flows over time by Cominetti et al [4,5,6], the work by Macko et al [14] about the Braess paradox for flows over time as well as the extension to multi-terminal settings [21]. Of special interest to the paper at hand are the results by Bhaskar et al [1] and very recently by Correa et al [7] about the PoA for flows over time. Since it was already shown that the evacuation-PoA (maximizing the flow amount within some time horizon) is unbounded [12], they focus on the time-PoA (minimizing the completion time for a given flow amount) for which they establish an upper bound of e e−1 under some constraints on the capacities of the network.…”

“…inequality we used[7, Claim 12] and in the last equality we used|x | = r 0 . Now let c f i := min{c v (θ) : v ∈ V, θ ∈ (θ i−1 , θ i )}, i.e.…”

Bridging the user equilibrium and the system optimum in static traffic assignment: how the cooperation among drivers can solve the congestion problem in city networks

“…They identified four empirical rules that leads to an increase of the price of anarchy and numerical evidences show that the price of anarchy follows a power law decay for large demands. Most recent findings on the magnitude of the price of anarchy depending on the flow magnitude can be found in in Correa et al (2019) and in Colini-Baldeschi et al (2020). The former focuses on bound for the price of anarchy while the latter confirm that the price of anarchy follows a power law with respect to the magnitude of the flow and they stated that, given polynomial latency functions, it can be explicitly evaluated.…”

Computation of Dynamic Equilibria in Series-Parallel Networks

“…By reducing the capacities of the links, a leader can ensure that a fixed amount of flow in a dynamic equilibrium reaches the sink within a factor of e e−1 compared to the time it would take in a social optimum of the original network. Further progress on this price-of-anarchy was made by Correa, Cristi, and Oosterwijk (2019), who prove a bound of e e−1 under weaker assumptions, which get dispensable if a certain monotonicity conjecture holds. Cominetti, Correa, and Olver (2017) consider the long-term behavior of dynamic equilibria.…”