In Short:

We apply our framework to re-evaluate algorithms used in hospital care management and show that our approach yields alternative algorithms that lie on the fairness-accuracy frontier, offering improvements along both dimensions.

Abstract

Decision-making processes increasingly rely on the use of algorithms. Yet, algorithms’ predictive ability frequently exhibits systematic variation across subgroups of the population. While both fairness and accuracy are desirable properties of an algorithm, they often come at the cost of one another. What should a fairness-minded policymaker do then, when confronted with finite data? In this paper, we provide a consistent estimator for a theoretical fairness-accuracy (FA) frontier put forward by Liang, Lu, Mu, and Okumura [2024, https://arxiv.org/abs/2112.09975]. To do so, we recognize that the FA-frontier is a part of the boundary of a convex set—the feasible set of group-specific expected losses associated with all possible algorithms—that can be fully represented by its support function. We provide an estimator of this support function and show that it converges to a tight Gaussian process as the sample size increases.

We express special points on the FA-frontier through the support function, and show that hypotheses that have received much attention in the fairness literature can be expressed as restrictions on the support function. Doing so allows us to provide test statistics for whether (i) fully excluding group identity from use in training the algorithm is optimal; and (ii) there are less discriminatory alternatives to an existing algorithm.

The answer to the first question informs a fairness-minded policymaker interested in assessing whether banning group identity has the potential to mitigate disparate impact. Liang et al. [2024, Proposition 6] show that banning group identity is uniformly welfare-reducing when two special points on the FA-frontier are located strictly on the opposite side of the 45° line. We use the support function of the feasible set to obtain closed form expressions for the coordinates of these special points. The test statistic corresponds to the null hypothesis that the two special points are weakly on the same side of the 45° line, and rejecting the null provides evidence against banning group identity. The answer to the question of whether there exists a less discriminatory alternative to a given algorithm—and hence whether the algorithm is on the FA-frontier—is a key ingredient to assess whether disparate impact occurred. Both the plaintiff (e.g., a job applicant) and the defendant (e.g., the hiring company) involved in a dispute on disparate impact can benefit from our testing procedure, as it provides a tool to determine whether business necessity may justify disparate impact. Our test statistic relies on the support function through judicious use of the separating hyperplane theorem.

We also show how to estimate the distance between the fairest point on the frontier and the group-specific expected losses associated with a given algorithm. This inference tool can be of interest to any fairness-minded agent (e.g., a college) willing to give up some level of accuracy to mitigate disparate impact (e.g., via affirmative action). The agent may assess the trade-off between promoting equity and achieving efficiency by comparing alternative algorithms in terms of their distance to the fairest point on the FA-frontier.

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