Abstract-We present five axioms for fairness measures in resource allocation. A family of fairness measures satisfying the axioms is constructed. Special cases of this family include α-fairness, Jain's index, and entropy. Properties of fairness measures satisfying the axioms are proven, including Schurconcavity. Among the engineering implications is a generalized Jain's index that tunes the resolution of fairness measure, a new understanding of α-fair utility functions, and an interpretation of "larger α is more fair". We also construct an alternative set of axioms to capture system efficiency and feasibility constraints. Given a vector x ∈ R n + , where x i is the resource allocated to user i, how fair is it? One approach to quantify the 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]- [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 the entropy function. Some of these fairness measures map x to normalized ranges between 0 and 1, where 0 denotes the minimum fairness, 1 denotes the maximum fairness (often corresponding to an x where all x i are the same) and a larger value indicates 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. How are these fairness measures are related? Is one measure "better" than any other? What other measures of fairness may be useful? An alternative approach that has gained attention in the networking research community since [7] , [8] is the optimizationtheoretic approach of α-fairness and the associated utility maximization. Given a set of feasible allocations, a maximizer of the 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 proportionally fair, and a maximizer of the α-fair utility function as α → ∞ is maxmin fair. More recently, α-fair utility functions have been connected to divergence measures [9] , and in [10], [11], the parameter α was viewed as a fairness measure in the sense that a fairer allocation is one that is the maximizer of an α-fair utility function with larger α -although the exact role This work has been supported in part by NSF CNS-0905086 and CNS-0519880. of α in trading-off fairness and throughput can sometimes be surprising [12] . While it is often held that α → ∞ is more fair than α = 1, which is in turn more fair than α = 0, it remains unclear what it means to say, for example, that α = 3 is more fair than α = 2. Clearly, these two approaches for quantifying fairness are different. On the one hand, α-fair utility functions are continuous and strictly increasing in each entry of x, thus its maximization results in Pareto optimal resource allocations. On the other hand, scale-invariant fairness measures (ones that map x to the same value as a normalized x) are unaffected by the magnitude of x, and an allocation that does not use all the resources can be as fair as one that does. Can the two approaches be unified? To address the above questions, we develop an axiomatic approach to fairness measures. 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 axioms, we can generate a family of fairness measures from generator functions 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 f β 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. In particular, for β ≤ 0, well-known fairness measures (e.g., Jain's index and entropy) are special cases of our construction, and we generalize Jain's index to provide a flexible tradeoff between "resolution" and "strictness" of the fairness measure. For β ≥ 0, α-fair utility functions can be factorized as the product of two components: our fairness measure with β = α and a function of the total throughput that captures the scale, or efficiency, of x. Such a factorization also quantifies a tradeoff between fairness and efficiency in achieving Pareto dominance with the maximum possible α, and facilitates a clearer understanding of what it means to say that a larger α is "more fair" for general α ∈ [0, ∞). The axiomatic construction of fairness measures also illuminates their engineering implications. Any fairness measure satisfying the five axioms can be proven to have many prop-978-1-4244-5837-0/10/$26.00
An axiomatic approach to fairness is given in REF , with an insightful discussion unifying (and extending) Jain's fairness and α-fairness.
An Axiomatic Theory of Fairness in Network Resource Allocation
2010 Proceedings IEEE INFOCOM
2010 Proceedings IEEE INFOCOM
Computer Science