A guide to formulating fairness in an optimization model

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.