  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.
