An Axiomatic Theory of Fairness in Resource Allocation

We develop a set of five axioms for fairness measures in resource allocation: the axiom of continuity, of homogeneity, of saturation, of partition, and of starvation. We prove that there is a unique family of fairness measures satisfying the axioms, which is constructed and then shown to include Atkinson’s index, αfairness, Jain’s index, entropy, and other “decomposable” global measures of fairness as special cases. We prove properties of fairness measures satisfying the axioms, including symmetry and Schur-concavity. Among the engineering implications is a generalized Jain’s index that tunes the resolution of fairness measure, a decomposition of α-fair utility functions into fairness and efficiency components, and an interpretation of “larger α is more fair” and of Rawl’s difference principle. The axiomatic theory is further extended in three directions. First, we quantify fairness of continuous-dimension inputs, where resource allocations vary over time or domain. Second, by starting with both a vector of resource allocation and a vector of user-specific weights, and modifying the axiom of partition, we derive a new family of fairness measures that are asymmetric among users. Finally, we develop a set of four axioms by removing the axiom of homogeneity to capture a fairness-efficiency tradeoff. We present illustrative examples in congestion control, routing, power control, and spectrum management problems in communication networks, with the potential of a fairness evaluation tool explored. We also compare with other work of axiomatization in information, computer science, economics, sociology, and political philosophy.