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On the Price of Anarchy in Packet Routing Games with FIFO

Daniel Schmand, Torben Schürenberg, Martin Strehler

TL;DR

This paper studies dynamic atomic packet routing games with FIFO queues on directed graphs, where each packet (player) aims to reach a fixed destination as quickly as possible. It derives the first non-trivial bounds for PoA in FIFO packet routing by proving a constant upper bound $\text{PoA} \le 2$ for the natural class of linear multigraphs under uniformly fastest route equilibria, and constructs instances showing $\text{PoS} \ge \frac{e}{e-1}$, which also implies a matching lower bound for PoA in related continuous-flow models under a monotonicity assumption. The methods connect discrete routing to flows over time, with extensions to edge-capacitated settings and graph subdivisions, and align with the monotonicity conjecture proved for these graph classes. Overall, the results reveal a tight gap between worst- and best-case equilibria in a natural FIFO setting, and provide insight into how network structure governs inefficiency in dynamic selfish routing in both discrete and continuous-time formulations.

Abstract

We investigate packet routing games in which network users selfishly route themselves through a network over discrete time, aiming to reach the destination as quickly as possible. Conflicts due to limited capacities are resolved by the first-in, first-out (FIFO) principle. Building upon the line of research on packet routing games initiated by Werth et al., we derive the first non-trivial bounds for packet routing games with FIFO. Specifically, we show that the price of anarchy is at most 2 for the important and well-motivated class of uniformly fastest route equilibria introduced by Scarsini et al. on any linear multigraph. We complement our results with a series of instances on linear multigraphs, where the price of stability converges to at least $\frac{e}{e-1}$. Furthermore, our instances provide a lower bound for the price of anarchy of continuous Nash flows over time on linear multigraphs which establishes the first lower bound of $\frac{e}{e-1}$ on a graph class where the monotonicity conjecture is proven by Correa et al.

On the Price of Anarchy in Packet Routing Games with FIFO

TL;DR

This paper studies dynamic atomic packet routing games with FIFO queues on directed graphs, where each packet (player) aims to reach a fixed destination as quickly as possible. It derives the first non-trivial bounds for PoA in FIFO packet routing by proving a constant upper bound for the natural class of linear multigraphs under uniformly fastest route equilibria, and constructs instances showing , which also implies a matching lower bound for PoA in related continuous-flow models under a monotonicity assumption. The methods connect discrete routing to flows over time, with extensions to edge-capacitated settings and graph subdivisions, and align with the monotonicity conjecture proved for these graph classes. Overall, the results reveal a tight gap between worst- and best-case equilibria in a natural FIFO setting, and provide insight into how network structure governs inefficiency in dynamic selfish routing in both discrete and continuous-time formulations.

Abstract

We investigate packet routing games in which network users selfishly route themselves through a network over discrete time, aiming to reach the destination as quickly as possible. Conflicts due to limited capacities are resolved by the first-in, first-out (FIFO) principle. Building upon the line of research on packet routing games initiated by Werth et al., we derive the first non-trivial bounds for packet routing games with FIFO. Specifically, we show that the price of anarchy is at most 2 for the important and well-motivated class of uniformly fastest route equilibria introduced by Scarsini et al. on any linear multigraph. We complement our results with a series of instances on linear multigraphs, where the price of stability converges to at least . Furthermore, our instances provide a lower bound for the price of anarchy of continuous Nash flows over time on linear multigraphs which establishes the first lower bound of on a graph class where the monotonicity conjecture is proven by Correa et al.

Paper Structure

This paper contains 27 sections, 18 theorems, 18 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Informal Model Description
  4. Related Literature
  5. Our Contribution
  6. Preliminaries
  7. Network Loading:
  8. Social Objective:
  9. Example:
  10. Equilibria:
  11. Price of Anarchy and Price of Stability:
  12. Understanding an Optimal State $S^*$:
  13. Upper Bound on the Price of Anarchy
  14. Lower Bound on the Price of Stability
  15. Constructing Bad Instances
  16. ...and 12 more sections

Key Result

Lemma 1

For every packet routing game $\Gamma$ on a linear multigraph and $S\in \mathcal{S}_{\operatorname{eq}}$ it holds that $C(S)= C_n(S)$.

Figures (3)

  • Figure 1: An example to illustrate the notation. The left figure depicts the positions of players at $t=1$, the right figure for $t=2$.
  • Figure 2: Visualization of graphs in $\mathcal{G}_{k,\ell}$. For a given $\ell$, the graph $G$ consists of all layers left of the red curved line and the last node before the line serves as destination node $d$.
  • Figure 3: The transit time of edge $e_4^1$ is $4$. All other edges have a transit time of $1$. Network, in which there is a state for $9$ players, where player 9 is strictly earlier at node $v_1$ than player 8, while simultaneously all players arrive as early as possible at the destination $d$.

Theorems & Definitions (32)

  • Lemma 1
  • proof
  • Proposition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Proposition 2
  • proof
  • Lemma 5
  • Lemma 6
  • ...and 22 more