A guide to formulating fairness in an optimization model

Abstract

Optimization models typically seek to maximize overall benefit or minimize total cost. Yet fairness is an important element of many practical decisions, and it is much less obvious how to express it mathematically. We provide a critical survey of various schemes that have been proposed for formulating ethics-related criteria, including those that integrate efficiency and fairness concerns. The survey covers inequality measures, Rawlsian maximin and leximax criteria, convex combinations of fairness and efficiency, alpha fairness and proportional fairness (also known as the Nash bargaining solution), Kalai–Smorodinsky bargaining, and recently proposed utility-threshold and fairness-threshold schemes for combining utilitarian with maximin or leximax criteria. The paper also examines group parity metrics that are popular in machine learning. We present what appears to be the best practical approach to formulating each criterion in a linear, nonlinear, or mixed integer programming model. We also survey axiomatic and bargaining derivations of fairness criteria from the social choice literature while taking into account interpersonal comparability of utilities. Finally, we cite relevant philosophical and ethical literature where appropriate.

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“…This is also known as the Robin Hood index. It is the fractional total utility to be redistributed to achieve perfect equality Chen and Hooker (2023). The Hoover index or Robin Hood index can be seen in figure 6 below as the maximum vertical distance between the Lorenz curve and the 45-degree line Kennedy et al (1996).…”

“…Other distance metrics such as the Manhattan distance, Euclidean distance, and the Chebyshev distance are compared in great detail in Mora et al (2016). The use of many fairness measures as constraints in optimization were explored in Chen and Hooker (2023) specifically we focused on this paper's research on Gini Coefficient and Hoover Index. The goal of this paper is to minimize cost and distance while maximizing the coverage of demand by placing rehabs in optimal regions in Oklahoma to best help addicts and alcoholics across the state while also comparing different distance metrics in conjunction with two of the fairness measures.…”

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“…where the log(•) function is applied to the objective function to guarantee a proportionally fairness rate selection solution [41], [42]. Note that the value of 1 is added to the sum of rates inside the logarithm to achieve a feasible solution in cases where s∈S R s,c (k) = 0.…”