  Binary decision making classifiers are not fair by default. Fairness
requirements are an additional element to the decision making rationale, which
is typically driven by maximizing some utility function. In that sense,
algorithmic fairness can be formulated as a constrained optimization problem.
This paper contributes to the discussion on how to implement fairness, focusing
on the fairness concepts of positive predictive value (PPV) parity, false
omission rate (FOR) parity, and sufficiency (which combines the former two). We
show that group-specific threshold rules are optimal for PPV parity and FOR
parity, similar to well-known results for other group fairness criteria.
However, depending on the underlying population distributions and the utility
function, we find that sometimes an upper-bound threshold rule for one group is
optimal: utility maximization under PPV parity (or FOR parity) might thus lead
to selecting the individuals with the smallest utility for one group, instead
of selecting the most promising individuals. This result is counter-intuitive
and in contrast to the analogous solutions for statistical parity and equality
of opportunity. We also provide a solution for the optimal decision rules
satisfying the fairness constraint sufficiency. We show that more complex
decision rules are required and that this leads to within-group unfairness for
all but one of the groups. We illustrate our findings based on simulated and
real data.
