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.

翻译：本文提出平均不公平性作为路由博弈中一种新的公平性度量，其定义为用户平均延迟与最小延迟之比。该度量是现有两种不公平性概念的自然补充：负载不公平性（比较具有正流量的路径的最大与最小延迟）和用户均衡不公平性（比较最大延迟与纳什均衡延迟）。我们证明所有三种不公平性度量的最坏情况值相互重合，并由延迟函数类固有的陡度参数所刻画。我们表明平均不公平性始终不大于负载不公平性，且仅当流量完全公平时两者相等。此外，我们首次对这三种不公平性度量进行了完整的比较分析，据我们所知，这是该方向上的首个理论分析。最后，我们研究了约束系统最优问题，即在给定不公平性上界约束下最小化总延迟。我们证明，在相同容忍度下，平均不公平性约束下的最优流量总能获得比满足负载不公平性约束的任何流量更低的总延迟。我们证明在平行链路网络中这种改进总是严格成立的，并为一般网络建立了充分条件。我们进一步通过数值算例对后者进行了说明。本研究结果为评估网络路由中公平性与效率的权衡提供了理论保证和重要见解。