Source clarifications: Wisdom and Foolishness of Noisy Matching Markets

Source: Wisdom and Foolishness of Noisy Matching Markets. Proposition and lemma numbers below are the numbers rendered in the paper’s attenuation and amplification appendices.

Appendix Proposition 1: lower-tail orientation and polynomial rate

What changes: the displayed integral from vS to infinity → the matched mass at values at or below vS, with the displayed O(C^{-K(β,γ)}) rate. The sentence immediately after the display, the preceding reduction, and Theorem 1 all concern students below vS; by contrast, the displayed upper-tail integral converges to the positive supply when match probability converges to one above vS.
Additional condition: beta-max concentration alone → beta-max concentration plus MemLp (fun x : ℝ => x) 2 D for one iid noise draw. The Case-2 lower-tail Chebyshev calculation is a statement about a single draw, whereas beta-max concentration controls maxima. Under this explicit finite-second-moment condition, the formalization proves the stated polynomial lower-tail rate uniformly over the literal stable-market instances.

What changes: the Proposition 7(ii) near-one rate under beta-max concentration alone → the same rate with the finite iid one-draw second moment above. The proof applies Chebyshev to one draw. The formalization uses the lower-tail complement valid in the presence of atoms and the squared Chebyshev denominator; it retains the displayed exponent.
What changes: the Proposition 8 lower-tail high-cutoff integral rate under beta-max concentration alone → the same O(C^{-K(β,γ)}) rate with that inherited one-draw finite-second-moment condition. Proposition 8 invokes Proposition 7(ii), so it needs the same condition. The corrected result retains the paper’s below-vS integral and exponent.

What changes: the initially defined open block C((P*, infinity)) → the closed block C([P*, infinity)) used by the dense-cluster proof. At a cutoff tie, the two sets differ. The proof’s dense cluster begins at P*, and its subsequent integral argument also uses the closed block. The formalization retains Proposition 3’s two displayed rates, Proposition 4’s lower-tail integral rate, and the strict low-side separation; this is a boundary convention, not a change to the market primitives.

What changes: an informal use of p_mu(v,F_i) with immediately preceding calculations left implicit → the actual iid cutoff-affordance probability together with precisely the endpoint, capacity, and tail-ratio estimates used in the local derivation. Proposition 10 retains its positive sigma witness, positive epsilon, and strict open interval. Proposition 14 retains its strict large-firm probability-difference bound. Proposition 15 retains the eta((v*,v_+)) = sqrt(epsilon) selection and its small-firm bound below v*. These are explicit proof-context inputs rather than additional economic assumptions.

What changes: the printed capacity algebra in Appendix Propositions 2 and 5 → the correctly signed capacity calculation. The displayed rates are unchanged.
What changes: the intermediate event comparisons in Appendix Propositions 3 and 4 → the valid Chebyshev inclusions at the stated endpoints. The displayed rates are unchanged.
What changes: the maximum-growth equality in Appendix Lemma 6 → a triangle-inequality and independent-sample argument, with atom-safe tail events. Its o(log n) conclusion is unchanged.
What changes: the central-window and all-real bridge in the amplification appendix → the displayed central interval together with a finite long-tail shift argument from an interior anchor. Theorem 2’s all-real conclusion is retained.

What changes: a conditional-noise description stated for every value vector → a conditional law on the student-law support, almost everywhere. This is the measure-theoretic reading needed for the affordability probabilities and integrals in Theorems 3 and 4.