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
Chiang, Peng, Lichtenstein, McDaniel, O'Neil, Thomas, Wright, Kleinberg, Tardos & Garg's proposed fix to 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.
Source
“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.”
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