  To better understand discriminations and the effect of affirmative actions in
selection problems (e.g., college admission or hiring), a recent line of
research proposed a model based on differential variance. This model assumes
that the decision-maker has a noisy estimate of each candidate's quality and
puts forward the difference in the noise variances between different
demographic groups as a key factor to explain discrimination. The literature on
differential variance, however, does not consider the strategic behavior of
candidates who can react to the selection procedure to improve their outcome,
which is well-known to happen in many domains.
  In this paper, we study how the strategic aspect affects fairness in
selection problems. We propose to model selection problems with strategic
candidates as a contest game: A population of rational candidates compete by
choosing an effort level to increase their quality. They incur a cost-of-effort
but get a (random) quality whose expectation equals the chosen effort. A
Bayesian decision-maker observes a noisy estimate of the quality of each
candidate (with differential variance) and selects the fraction $\alpha$ of
best candidates based on their posterior expected quality; each selected
candidate receives a reward $S$. We characterize the (unique) equilibrium of
this game in the different parameters' regimes, both when the decision-maker is
unconstrained and when they are constrained to respect the fairness notion of
demographic parity. Our results reveal important impacts of the strategic
behavior on the discrimination observed at equilibrium and allow us to
understand the effect of imposing demographic parity in this context. In
particular, we find that, in many cases, the results contrast with the
non-strategic setting.
