A Guide to Formulating Equity and Fairness in an Optimization Model

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Keywords: equity modeling, survey on fairness definitions and formulations

We provide a survey of various schemes that have been proposed for formulating the ethics-related criteria of fairness and equity, including those that integrate efficiency and equity concerns. The survey covers inequality measures, Rawlsian maximin and leximax criteria, alpha fairness and proportional fairness (also known as the Nash bargaining solution), Kalai-Smorodinsky bargaining, and recently proposed utility-threshold and equity-threshold schemes for combining utilitarian with maximin or leximax criteria. We introduce an n-person model for the equity-threshold criterion. The paper also examines statistical fairness metrics that are popular in machine learning, including demographic parity, equalized odds, accuracy parity, and predictive rate parity. We present what appears to be the best practical approach to formulating each criterion in a linear, nonlinear, or mixed integer programming model. We analyze the mathematical properties of the various formulations, presenting new results in several cases, and indicate some of the strengths and weaknesses of each. We also cite relevant philosophical and ethical literature where appropriate.