Source Clarifications

Source locations refer to cited publication and cited publication for Capacity Constraints Make Admissions Processes Less Predictable.

Capacity, variability, and queue representation

• Theorem 1 no-zero and exact-one claims → positive, binding capacity $0<q<|U|$, where $q$ is capacity and $U$ the applicant universe (main lines 281–285; appendix no-zero and q-representative arguments). At $q=0$ everyone may be rejected; at $q\geq|U|$ everyone may be admitted. Both rules have instability and variability zero, so the exact nonzero conclusion excludes them.
• Lemma A.1: two disjoint size-$q$ pools → one size-$(q+1)$ pool in the monotonicity/q-acceptance contradiction (appendix lines 54–66). For $q>0$, q-acceptance admits every singleton; monotonicity would then admit all $q+1$ members of that pool, violating capacity. The proof therefore also covers universes smaller than $2q$.
• Theorem 2’s literal queue count → existence of a representation by one priority order in the variability-one characterization. Redundant copies of a queue cannot change the choice rule. The range $1\leq m\leq n$, where $m$ is realized variability and $n$ the number of queues, is checked on the positive, binding-capacity domain above.

Scores, strict orders, and assignment choices

• Unrestricted strict-order completeness → comparison of distinct applicants only (main lines 192–200). An applicant cannot strictly precede itself.
• Raw-score rank selection → a fixed ex-ante tie order, with $q>0$ and $q\leq|X|$ for the q-th threshold of pool $X$. This specifies the current selector, without asserting that raw predicted scores are distinct.
• Lemma A.8 and Theorems A.10–A.11: unspecified selection among optimal linear assignments → a fixed generic refinement of the primary objective selecting one admitted set (appendix lines 610–666). The source says weights have no ties; distinct weights can still give equal total assignment objectives. The refinement specifies a single choice in those cases.

Exact appendix corrections

• Lemma A.6, lines 422–433: self-equality → $C(X_1)=C(X_2)\Rightarrow V_C(X_1)=V_C(X_2)$ for a feasible, q-acceptant, substitutable rule. Here $C(X)$ is the admitted set and $V_C(X)$ its members displaceable by one insertion. The source’s one-instability condition supplies substitutability through Theorem 1.
• Corollary A.3, lines 333–338: $dk$ → $d=2k$ with $k>0$. Positive even tight instability is excluded; the $2k$ bound improves to $2k-1$.
• Theorem A.6 proof, lines 397–416: malformed union/deletion expression → $X=\bigcup_iX'_i=X_0\cup\{x'_1,\ldots,x'_m\}$, the pool formed by adjoining the selected applicants $x'_i$ to the base pool $X_0$.
• Lemma A.7 proof, lines 435–442: misplaced parenthesis → $X'\setminus\{x'\}=X$. The resulting removable-set equality also holds for pools smaller than capacity.
• Theorem A.5 proof, lines 314–331: applicant–set inequality → $x'\notin C(X')$. The new applicant’s membership indicator is zero.
• Theorem A.10 proof, lines 634–646: self-comparison → comparison of $C(X')$ with $C(X)$, as required for the capacity contradiction.
• Theorem A.11 proof, lines 656–666: repeated applicant index → the distinct selected applicants $x_{s_i}$ and $x_{s_j}$ under their shared slot order.