On Fair Selection in the Presence of Implicit and Differential Variance

Discrimination in selection problems such as hiring or college admission is often explained by implicit bias from the decision maker against disadvantaged demographic groups. In this paper, we consider a model where the decision maker receives a noisy estimate of each candidate's quality, whose variance depends on the candidate's group -- we argue that such differential variance is a key feature of many selection problems. We analyze two notable settings: in the first, the noise variances are unknown to the decision maker who simply picks the candidates with the highest estimated quality independently of their group; in the second, the variances are known and the decision maker picks candidates having the highest expected quality given the noisy estimate. We show that both baseline decision makers yield discrimination, although in opposite directions: the first leads to underrepresentation of the low-variance group while the second leads to underrepresentation of the high-variance group. We study the effect on the selection utility of imposing a fairness mechanism that we term the $\gamma$-rule (it is an extension of the classical four-fifths rule and it also includes demographic parity). In the first setting (with unknown variances), we prove that under mild conditions, imposing the $\gamma$-rule increases the selection utility -- here there is no trade-off between fairness and utility. In the second setting (with known variances), imposing the $\gamma$-rule decreases the utility but we prove a bound on the utility loss due to the fairness mechanism.