Derive the population-level exponential saturation law

Derive from first principles the exponential saturating form of the true-outcome fairness gap as a function of prediction-error scale, including the fitted saturation parameters, by modeling how the identity of the marginal individual pair changes under repeated independent prediction-noise realizations.

Background

The paper establishes a local sensitivity formula for the true-outcome fairness gap with respect to prediction errors affecting the two marginal individuals in the allocation linear program, together with a universal error-independent ceiling. Empirically, however, the population-level gap is well fit by an exponential saturation curve of the form c(1-e{-σ/τ}), with specific fitted values for c and τ.

The authors explicitly state that their theoretical results do not derive this functional form or its parameter values. Doing so would require a distributional model for how repeated prediction-noise draws alter the identities of the marginal individuals; the paper leaves this population-level derivation unresolved rather than imposing additional assumptions.

References

Neither theoretical result derives Eq.~eq:fitted-curve's exponential saturating form, or the specific values of $c$ and $\tau$, from first principles. Doing so would require a distributional assumption on how the identity of the marginal pair changes under repeated independent draws of the noise $\varepsilon$ -- the population-level analog of the per-draw mechanism Theorem~\ref{thm:local-sensitivity} characterizes -- which we leave as an open direction (Section~\ref{sec:discussion}) rather than introduce unjustified assumptions to close it here.

— Fairness under Prediction Error in Predict-then-Optimize Pipelines  (2610.18593 - Osei, 13 Oct 2026) in Section 5, Bound Validation, subsection “What this does not achieve”; Section 6, Discussion

Formalizing this threshold, and using it to derive the population-averaged small-$\sigma$ slope directly (rather than validating the per-draw mechanism only, as done here), is an open technical question.

— Fairness under Prediction Error in Predict-then-Optimize Pipelines  (2610.18593 - Osei, 13 Oct 2026) in Section 6, Discussion, subsection “The active-set stability threshold”