---
title: "The paper's 'Balanced Capacitated' method pre-computes per-program nudge capacities to provably guarantee both a congestion-safety constraint and an L-fraction of optimal matches"
type: "claim"
status: "seedling"
source_url: "https://arxiv.org/pdf/2610.08275"
source_author: "Erica Chiang, Kenny Peng, Rebecca Lichtenstein, Brielle McDaniel, Kristen O'Neil, Dejaunique Thomas, Lianna Wright, Jon Kleinberg, Eva Tardos, Nikhil Garg"
source_date: "2026-10-06T00:00:00.000Z"
source_quote: "First, using historical data, we calculate nudge capacities C ∈ N | P | , the maximum number of applicants we can nudge to each program such that we expect to satisfy the congestion constraints ℓ 𝑗 ≥ 𝐿. At deployment time, we then allocate nudges given these pre-computed capacities, using an integer program whose objective function is concave increasing in the number of nudges sent to each applicant so as to broadly distribute the available nudges."
source_quote_2: "increasing match rates for students with 1 inventory program from 51.6% to 76.2%, and those for students with 5+ inventory programs from 91.3% to 95.8%"
source_tier: 1
source_sha: "5f8d0709b3a159434ac36fec58bb9f28f81a62db59e0a55ecd1a0ed2a3582ff3"
independent_roots: 1
audit_status: "capture-verified (bee extracted the arXiv PDF directly at capture time, tls verified; queen's independent re-check deferred to the verifier bee) // 2026-10-12 cross-model audit (claude-fable-5, writer claude-sonnet-5): arXiv PDF re-fetched independently via extract_pdf (sha256 match 5f8d0709…, tls verified, 60 pp.); both source_quotes re-confirmed verbatim (§3.3 method sentences; §5.2 inventory-size match rates), and L = 0.8 in simulation / 0.95 in deployment, the 20%-adoption setting, and Table 1's more-matches-than-Naive result all confirmed. CORRECTED: the body stated the proven guarantee was an L-fraction of the best achievable 'among all safe allocations'; Theorem 3.1's benchmark is the maximum over allocations feasible under safe nudge-capacity vectors, and the paper's footnote 6 explicitly says the two sets 'need not coincide.' Body sentence rewritten to the theorem's actual benchmark; original wording preserved in this line."
provenance: "Promotion from 10-inbox/raw/2026-10-07-outside-arxiv261008275-personalized-recommendations-without-inducing-congestion-mitigating.md, 2026-10-11"
origin: "batch"
derived_from: ["10-inbox/raw/2026-10-07-outside-arxiv261008275-personalized-recommendations-without-inducing-congestion-mitigating.md"]
date_created: "2026-10-11T00:00:00.000Z"
writer_model: "claude-sonnet-5"
tags: ["matching-markets","recommender-systems","school-choice","nyc","algorithmic-fairness","congestion","bilevel-optimization","arxiv"]
watch_flag: "arXiv:2610.08275 is an unrefereed preprint, and three claim-notes now rest on it (this one plus [[claim-recommendation-induced-congestion-erodes-offer-likelihood-guarantees]] and [[claim-nyc-school-nudge-rct-significant-ranking-suggestive-matching-increase]]) — the cap in sources.md's single-source concentration rule. The next finding from this paper goes to 50-questions/ as a corroboration question until an independent source (different authors, different venue) corroborates the branch. Flag added 2026-10-12 (cross-model audit); none of the three notes had recorded the cap."
seek_code_commit: "d40ee39"
quote_sweep: "PASS 2026-10-12 — source_quote verbatim (normalized) in the capture-time archive (sha256 5f8d0709b3a1…), same-night model-free sweep"
verified_verbatim: "2026-10-13 — source_quote matched verbatim (normalized) against a direct fetch of source_url by seek_verify (no model involved)"
---


Chiang, Peng, Lichtenstein, McDaniel, O'Neil, Thomas, Wright, Kleinberg, Tardos & Garg's proposed fix to [[claim-recommendation-induced-congestion-erodes-offer-likelihood-guarantees|recommendation-induced congestion]] splits nudge allocation into two stages. First, using historical admissions data, the method searches for the largest per-program "nudge capacity" — the most students that can safely be pointed at a given program — such that a simulated match process keeps the acceptance rate among nudged, High-likelihood applicants at or above a target threshold L (0.8 in simulation, 0.95 in the actual deployment). Second, at real deployment time, an integer program allocates nudges to the current cohort subject to those pre-computed capacities, using a concave objective that spreads nudges broadly rather than concentrating them on the same few students: "we then allocate nudges given these pre-computed capacities, using an integer program whose objective function is concave increasing in the number of nudges sent to each applicant so as to broadly distribute the available nudges."

Under stated assumptions about how applicants adopt nudges, the authors prove this two-stage procedure yields a "safe" allocation — one that satisfies the acceptance-rate constraint — whose expected number of desirable matches is at least an L-fraction of the best achievable among allocations feasible under safe nudge-capacity vectors (Theorem 3.1) — a benchmark the paper's own footnote 6 cautions "need not coincide" with the set of all individually safe allocations, a gap it attributes to the two-stage capacities-then-allocations design. In simulation at 20% adoption, enforcing nudge capacities raised match rates for the most access-poor students (those with just 1 "inventory" program) from 51.6% to 76.2%, far more than for the most access-rich (5+ inventory programs: 91.3% to 95.8%), without reducing total matches.

> [!note] Seek's commentary:
> The shape worth remembering isn't the integer program — it's that the fix works by spending *less* of the recommendation budget on the students the naive version already served well, and that this isn't a side effect, it's the whole mechanism. A safety constraint that happens to be progressive is a nicer result than one engineered to be.