Source Clarifications: Fair Allocation through Selective Information Acquisition

Source anchors refer to audit/cited publication.

Full-vector posterior reading for allocation policies

The source defines the allocation rule as $A(\hat{\mathbf U})$ of the full post-screening estimate vector (lines 285–291). Read together with its realized-to-posterior utility comparison (lines 693–697), the displayed estimate for applicant $i$ is the lender’s posterior utility after observing that full vector:

$$\hat U_i = \mathbb E[U_i \mid \sigma(\hat{\mathbf U})]$$

almost surely. This is the source-model reading used for Definition 1, Theorem 2, Appendix Lemmas 4–5 and 7, and Appendix Theorem 6. It preserves the paper’s full-vector allocation class and makes the corresponding realized-to-posterior comparison applicable to every permitted allocation.

Appendix Lemma 5 interpolation display

The proof display at lines 725–730 defines an unused quantity with an undefined $t_i$ and then gives an interpolation probability that does not follow from the displayed target. Let $A$ be expected utility strictly above the threshold and $B$ be expected utility at the threshold. For a target $\upsilon$ with $A \leq \upsilon \leq A+B$ and $B>0$, admit the boundary with probability $\alpha=(\upsilon-A)/B$. The resulting expected utility is $A+\alpha B=\upsilon$, and $\alpha\in[0,1]$. If $B=0$, the target is $A$ and any boundary probability works. The same argument with expected costs proves the cost branch. Thus the Lemma’s stated conclusion is unchanged.

Appendix Theorem 6 threshold-domain proof route

Appendix Definition 3 allows a cost-aware threshold to use any real cutoff or either infinite endpoint (lines 626–636), and Appendix Lemma 5 consequently may return any such cost-attaining witness (lines 704–712). The printed proof of Appendix Theorem 6 immediately applies Lemma 4’s equal-cost case to that witness (lines 735–754), but Lemma 4 states that case only for a finite threshold $t>0$ (lines 643–650). Thus the source does not establish the threshold-domain premise needed for that displayed proof step. This is a proof-route issue only: it does not change the statement of Appendix Theorem 6 or of the main-text Theorem 2 that follows from it.

The Lean proof uses the needed stronger route. It proves the equal-cost utility comparison for every finite cutoff, handles the two endpoint threshold cases separately, and only then combines that result with exact cost attainment. The resulting theorem has the same paper-facing conclusion without adding an economic assumption.