Closed-form characterization of the fairness region for FPR, FNR, and PPV

Derive a closed-form expression for the size of the fairness region comprising all two-group binary classifiers whose False Positive Rate (FPR), False Negative Rate (FNR), and Positive Predictive Value (PPV) satisfy epsilon-relaxed parity constraints (|FPR1 − FPR2| ≤ ε_fpr, |FNR1 − FNR2| ≤ ε_fnr, |PPV1 − PPV2| ≤ ε_ppv) under a given prevalence difference ε_prev between the groups, analogous to the closed-form result established for FPR, FNR, and Accuracy.

Background

The paper introduces a "fairness region" as the space of models that satisfy small, practitioner-acceptable relaxations of parity across key error-based metrics. A closed-form characterization is provided when analyzing fairness in terms of False Positive Rate, False Negative Rate, and Accuracy, enabling practitioners to reason about feasibility directly.

When replacing Accuracy with Positive Predictive Value—more relevant in resource-constrained settings—the authors could not obtain a closed-form characterization due to non-linearities and potential discontinuities, resorting instead to computational approximations (dot planimetry and constraint programming). They explicitly leave the closed-form derivation as future work.

References

"...we next turn our attention to the problem in terms of \fpr, \fnr and \ppv, but find that it is difficult to analyze in closed-form. Instead, through principled approximations, we are able to provide much the same guidance to practitioners as a direct analytical solution would, and leave deriving a closed-form result to future work."

— The Possibility of Fairness: Revisiting the Impossibility Theorem in Practice  (2302.06347 - Bell et al., 2023) in Section 1, Introduction (Paper roadmap)