A Parametric, Second-Order Cone Representable Model of Fairness for Decision-Making Problems
Kaarthik Sundar, Deepjyoti Deka, Russell Bent
TL;DR
This paper introduces $\varepsilon$-fairness, a parametric, convex fairness model that can be embedded into optimization problems via a single SOC constraint, without altering problem complexity. It grounds the model in finite-dimensional norm equivalence and establishes a closed-form link to the Jain et al. fairness index, enabling direct interpretation of fairness levels. The authors prove monotonicity of the objective with respect to fairness and define a feasibility domain, providing practical means to navigate efficiency-fairness trade-offs. A detailed case study on a damaged DC power network demonstrates how $\varepsilon$-fairness shapes load-shed distributions and quantifies the associated efficiency loss, offering a scalable tool for fair decision-making in engineering systems. The work lays a foundation for applying SOC-based fairness across domains and motivates future extensions to weighted fairness and combinatorial problems.
Abstract
The article develops a parametric model of fairness called "$\varepsilon$-fairness" that can be represented using a single second-order cone constraint and incorporated into existing decision-making problem formulations without impacting the complexity of solution techniques. We develop the model from the fundamental result of finite-dimensional norm equivalence in linear algebra and show that this model has a closed-form relationship to an existing metric for measuring fairness widely used in the literature. Finally, a simple case study on the optimal operation of a damaged power transmission network illustrates its effectiveness.