Average Unfairness in Routing Games

TL;DR

This paper introduces average unfairness as a new fairness measure in routing games, compares it with existing measures, and analyzes its impact on system efficiency and fairness tradeoffs.

Contribution

It defines average unfairness, compares it theoretically with loaded and UE unfairness, and studies the constrained optimization problem to improve fairness-efficiency tradeoffs.

Findings

Worst-case unfairness measures coincide and depend on latency function steepness.

Average unfairness is always less than or equal to loaded unfairness.

Optimal flows with average unfairness constraints achieve lower total latency in parallel-link networks.

Abstract

We propose average unfairness as a new measure of fairness in routing games, defined as the ratio between the average latency and the minimum latency experienced by users. This measure is a natural complement to two existing unfairness notions: loaded unfairness, which compares maximum and minimum latencies of routes with positive flow, and user equilibrium (UE) unfairness, which compares maximum latency with the latency of a Nash equilibrium. We show that the worst-case values of all three unfairness measures coincide and are characterized by a steepness parameter intrinsic to the latency function class. We show that average unfairness is always no greater than loaded unfairness, and the two measures are equal only when the flow is fully fair. Besides that, we offer a complete comparison of the three unfairness measures, which, to the best of our knowledge, is the first theoretical analysis in this direction. Finally, we study the constrained system optimum (CSO) problem, where one seeks to minimize total latency subject to an upper bound on unfairness. We prove that, for the same tolerance level, the optimal flow under an average unfairness constraint achieves lower total latency than any flow satisfying a loaded unfairness constraint. We show that such improvement is always strict in parallel-link networks and establish sufficient conditions for general networks. We further illustrate the latter with numerical examples. Our results provide theoretical guarantees and valuable insights for evaluating fairness-efficiency tradeoffs in network routing.