Average Unfairness in Routing Games
Pan-Yang Su, Arwa Alanqary, Bryce L. Ferguson, Manxi Wu, Alexandre M. Bayen, Shankar Sastry
TL;DR
The paper introduces average unfairness $U^{A}$ as a population-average fairness metric for routing games and establishes its formal relation to existing worst-case measures $U^{L}$ and $U^{UE}$. It proves that the worst-case values of these measures coincide, equal to the latency-class steepness bound $\gamma(\mathcal{L})$, and that $U^{A} \le U^{L}$ with equality only for fully fair flows. It then analyzes the constrained system optimum (CSO) under unfairness constraints, showing that for the same tolerance, the average-unfairness-constrained optimum achieves strictly lower total latency than the loaded-unfairness counterpart in parallel-link networks and often in general networks, supported by theoretical conditions and numerical experiments. The results provide theoretical guarantees and practical insights into fairness-efficiency tradeoffs in routing, with implications for designing fairness-aware routing policies and algorithms.
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.