We present 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 set of fairness measures satisfying the axioms, which is constructed and shown to include α-fairness, Jain's index, and entropy as special cases. Properties of fairness measures satisfying the axioms are proven, including Schur-concavity and symmetry. 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”.

We further extend the axiomatic theory in three directions. First, the results are extended to quantify fairness of continuous-dimension inputs, where resource allocations vary over time or domain. Second, by modifying the axiom of partition, we derive a new family of fairness measures, which are asymmetric among users and depend on user-specific weights. Finally, a set of four axioms is developed by removing axiom of homogeneity to capture a fairness-efficiency tradeoff. We present illustrative examples in congestion control, routing, power control, and spectrum management problems. We also compare with other work of axiomatization in information and economics, and explore connections with Rawls' theory of justice.