An Axiomatic Theory of Fairness in Network Resource Allocation (0906.0557v4)

Abstract: 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 alpha-fairness, 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 generalized Jain's index that tunes the resolution of the fairness measure, a new understanding of alpha-fair utility functions, and an interpretation of "larger alpha is more fair". We also construct an alternative set of four axioms to capture efficiency objectives and feasibility constraints.

• The paper introduces five axioms that rigorously define fairness measures in network resource allocation.
• It unifies conventional metrics, showing that methods like α-fairness and Jain’s index naturally emerge as special cases of the framework.
• The study separates fairness from efficiency in utility functions, offering clear insights for balancing resource distribution and performance.

Axiomatic Theory of Fairness in Network Resource Allocation

The paper "An Axiomatic Theory of Fairness in Network Resource Allocation" by Tian Lan et al. constructs a formal framework to evaluate fairness in distribution protocols for network resources. The authors present a comprehensive set of axioms that provide a foundation for analyzing various fairness measures.

Axiomatic Foundation

The core contribution of the paper lies in its systematic development of a set of five axioms to define fairness measures. These are Continuity, Homogeneity, Asymptotic Saturation, Irrelevance of Partition, and Monotonicity. The authors propose that these axioms collectively form a robust means of assessing fairness, capturing the intuitive notion that a slight change in allocation should affect the fairness measure only slightly and that fairness should be independent of absolute magnitudes.

Unified Framework and Specific Measures

The paper establishes a framework whereby different fairness formulations such as $\alpha$-fairness, Jain’s index, and entropy, traditionally seen as diverse mechanisms for quantifying fairness, are now special cases of this axiomatic foundation. For instance, Jain's index is derived using a harmonic mean generator function, and the $\alpha$-fair utility functions emerge naturally from this formulation.

The paper demonstrates that the family of fairness measures is Schur-concave, implying that Pareto-improving allocations lead to higher fairness—a significant result enhancing the plausibility of using these measures to assess fairness in resource allocations robustly.

Implications and Generalizations

One of the striking aspects of the paper is its implication on $\alpha$-fair utility functions, which weigh distribution fairness alongside efficiency. This dual focus in traditional utility settings, the paper argues, masks the true influence of fairness against pure efficiency gains. Through the axioms, utility functions can be factored to explicitly separate fairness from efficiency, enhancing the understanding of how adjustments in $\alpha$ affect perceived fairness.

Further, by generalizing these axioms and relaxing them—such as removing the Axiom of Homogeneity—the authors explore a broader class of measures that integrate both fairness and efficiency. This approach offers a way to weigh the intrinsic trade-offs in resource allocation tasks without requiring fairness to be detached from efficiency considerations.

Conclusion

This paper advances the paper of fairness by anchoring the concept in a formal, axiomatic system. It clarifies and unifies various existing fairness measures and proposes new insights and results applicable in network resource allocation. Future work might delve into how this axiomatic foundation can be further developed or adapted when some of the assumptions, such as indivisibility or utility constraints, are relaxed. The paper illuminates new pathways for fairness evaluation and operationalizes fairness in a manner conducive to its integration with efficiency, shaping how fairness might be reconceptualized in future network systems.