An Axiomatic Approach to Fairness

We present a set of five axioms for fairness measures in resource allocation. A family of fairness measures satisfying the axioms is constructed. Well-known notions such as αfairness, Jain’s index and entropy are shown to be special cases. Properties of fairness measures satisfying the axioms are proven, including Schur-concavity. Among the engineering implications is a new understanding of α-fair utility functions and an interpretation of “larger α is more fair”. 1. QUANTIFYING FAIRNESS Given a vector x ∈ R+, where xi is the resource allocated to user i, how fair is it? Given a vector x ∈ R+, where xi is the resource allocated to user i, how fair is it? One approach to quantify a degree of fairness associated with x is through a fairness measure, which is a function f that maps x into a real number. Various fairness measures have been proposed throughout the years, e.g., in [1, 2, 3, 4, 5, 6]. These range from simple ones, e.g., the ratio between the smallest and the largest entries of x, to more sophisticated functions, e.g., Jain’s index and entropy function. Some of these fairness measures map x to normalized numbers between 0 and 1, where 0 denotes the minimum fairness, 1 denotes the maximum fairness, often corresponding to an x where all xi are the same, and a larger value indicate more fairness. For example, min-max ratio [1] is given by the maximum ratio of any two user’s resource allocation, while Jain’s index [3] computes a normalized square mean. Are these fairness measures related to each other? Is one of them measures “better” than the other? What other measures of fairness may be useful? Another approach that has gained attention in the networking research community since [7, 8] is α-fairness and the associated utility maximization. A maximizer of α-fair utility function satisfies the definition of α-fairness. Two well-known examples are as follows: a maximizer of the log utility function (α = 1) is proportional fair (i.e., any change in x has a negative total normalized change), and a maxiPermission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Copyright 200X ACM X-XXXXX-XX-X/XX/XX ...$5.00. mizer of the α-fair utility function with α →∞ is max-min fair (i.e., it is impossible to increase an xi without decreasing some xj ≤ xi). More recently, α-fair utility functions have also been connected to divergence measures [9]. As in [10, 11], parameter α can be viewed as a fairness measure in the sense that a fairer allocation is one with a larger α, although the exact role of α in trading-off fairness and throughput can sometimes be surprising [12]. While it is often held as selfevident that α → ∞ is more fair than α = 1, which is in turn more fair than α = 0, it remains unclear what does it mean to say that α = 3 is more fair than α = 2? These two approaches for quantifying fairness are obviously different. Since α-fair utility functions are continuous and strictly increasing in each entry of x, maximizing them results in Pareto optimal resource allocations. On the other hand, fairness measures that maps x to a normalized range are independent to the absolute values of x. What notions of fairness or efficiency is captured by each approach? Can the two approaches be unified? To address the above questions, we develop an axiomatic approach to measure fairness. We discover that a set of five axioms, each of which simple and intuitive, can lead to a useful family of fairness measures. The axioms are: the axiom of continuity, of homogeneity, of asymptotic saturation, of irrelevance of partition, and of monotonicity. Starting with these five self-evident truths, we can generate a family of fairness measures from a generator function g: any increasing and continuous functions that lead to a well-defined “mean” function (i.e., from any Kolmogorov-Nagumo function [16]). For example, using power functions with exponent β as the generator function, we derive a unique family of fairness measures fβ that include all of the following as special cases, depending on the choice of β: Jain’s index, maximum or minimum ratio, entropy, and α-fair utility, and reveals new fairness measures corresponding to other ranges of β. We note that our approach, unlike other well known axiomatic constuctions such as the Nash bargaining solution [18] and Shapley value [19], specifies a broad class of fairness measures rather than an optimality concept. This is a direct consequence of one axiom which by construction removes the concept of efficiency from fairness. However, we demonstrate that in many optimization theoretic constructions, a fairness measure is incorporated into the objective function. For example, we show that an α-fair utility function (which includes proportional fairness [7] and thus the Nash Bargaining Solution as a special case) can be factorized as the product of two components: our fairness measure with α = β and a function of the total throughput. We also