Waiting Buys Exactly Two Moments: Endogenous Identification in Prediction-Driven Allocation

Rama AlHamidi; Rasul Khanbayov; Erchin Serpedin; Hasan Kurban

cs.LGOct 16, 2026

Waiting Buys Exactly Two Moments: Endogenous Identification in Prediction-Driven Allocation

Authors: Rama AlHamidi, Rasul Khanbayov, Erchin Serpedin, Hasan Kurban

Organizations: Department of Computer and Electrical Engineering, Texas A&M University at Qatar, Doha, Qatar · College of Science and Engineering, Hamad Bin Khalifa University, Doha, Qatar · Department of Computer and Electrical Engineering, Texas A&M University, College Station, TX, USA

Abstract

A planner with a fixed budget of preventive help must decide whom to help and when. Waiting reveals risk but removes intervention opportunities as higher-risk individuals leave first. We characterize this tradeoff under repeated noisy signals and risk-dependent attrition: in the canonical model, a length- $τ$ observation prefix identifies exactly the latent risk moments $m_j=\mathbb E[p^j]$ for $j=0,\ldots,2τ-1$ , buying two new moments per additional period. These identify average next-period attrition among survivors but need not identify allocation welfare or attrition among those prioritized for treatment. Our observationally equivalent constructions establish unavoidable timing regret at any sample size for a fixed observation length, while exact certificates determine whether all compatible populations share a unique optimal date. Matched experiments separate the effects of estimation, bootstrap averaging, and regret weighting. Longer observation often resolves timing ambiguity, but the optimal date may already have passed. In our experiments, bootstrap averaging and early stopping improve decisions, while regret weighting adds no improvement in pooled mean squared regret: better statistical fit need not yield better allocation welfare. The code and artifacts are available at https://github.com/KurbanIntelligenceLab/waiting-buys-two-moments.

Figures & tables

Figure 1: Waiting buys exactly two moments, but the riskiest people leave first. After t periods the planner observes exactly the first

2t-1

moments of the risk distribution (orange tiles), so its reconstruction (orange) of the true distribution (black) sharpens with every period of waiting (blue arrow). The people who generate those clues are the highest-risk ones, and they drop out first (dashed outlines): waiting improves the ranking of a population that has already lost the people most worth helping (red arrow).

Figure 2: Identical observations, different optimal dates. The two populations have matching observation probabilities through $\tau=2$ (a), but different welfare curves (b): $Q_{0}$ is optimized at $s=5$ , whereas $Q_{1}$ is optimized at $s=1$ . The illustrated recommendation $s=5$ is optimal for $Q_{0}$ but incurs relative welfare loss under $Q_{1}$ . Dates are evaluated as policies for a future population with the same risk distribution.

Welfare uncertainty
	Observation length
Measure	$\tau=2$	$\tau=3$	$\tau=4$
Median exhibited welfare spread	$42.1\%$	$27.3\%$	$15.2\%$
Interquartile range	$27.8$ – $63.8\%$	$14.0$ – $38.8\%$	$2.7$ – $27.4\%$
Largest exhibited welfare spread	$85.8\%$	$74.3\%$	$50.5\%$
Timing disagreement

Table 1: Identification ambiguity across observation lengths. Selected population pairs match moments through degree

2\tau-1

and therefore generate identical prefix observation laws. Compatibility, welfare trajectories, and regret separation are verified in exact rational arithmetic for all 90 pairs. Welfare spread is the maximum relative difference between the two welfare curves across candidate dates. Regret bounds are exhibited lower bounds, not necessarily sharp over the compatible set. Median regret bounds are calculated among pairs with disjoint optimal-date sets.

All dates: $1,\ldots,T-1$
Outcome	$\tau=2$	$\tau=3$	$\tau=4$
Best date identified	$\mathbf{1/30}$	$\mathbf{16/30}$	$\mathbf{27/30}$
Best date ambiguous	$29/30$	$14/30$	$3/30$
Available dates: $\tau,\ldots,T-1$
Best available date identified	$\mathbf{3/30}$	$\mathbf{18/30}$	$\mathbf{28/30}$
Act now ( $s=\tau$ )	$3/30$	$14/30$	$27/30$

Table 2: Do the observations determine the best intervention date? Counts across 30 configurations per observation length. Identified means one date is uniquely optimal for every compatible population; ambiguous means no date is optimal for all of them. All cases were resolved after the extended pass. Indented rows partition the identified available-date cases.

Reference policies
	$N=10^{3}$			$N=10^{5}$
Procedure	MSR $\downarrow$	MSR gain $\uparrow$	$p_{10}$ $\uparrow$	MSR $\downarrow$	MSR gain $\uparrow$	$p_{10}$ $\uparrow$
Oracle	0.00	+4.61	1.0000	0.00	+1.60	1.0000
Act immediately	8.64	$-4.02$	0.8535	8.85	$-7.26$	0.8201
Single fit and bootstrap aggregation
Certified NPMLE	4.61	+0.00	0.8672	1.60	+0.00	0.9658

Table 3: Overall decision performance. Results at

\tau=2

on 1,169 matched runs across the two sample sizes. MSR is mean squared relative regret multiplied by

10^{3}

; MSR gain is its reduction relative to certified NPMLE.

p_{10}

is the 10th percentile of welfare efficiency. Early-stopped EM uses 55 iterations.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

	$N=10^{3}$		$N=10^{5}$
Estimator	MSR	$p_{10}$	MSR	$p_{10}$
Certified NPMLE	3.37	0.8933	1.55	0.9876
Parametric Beta	4.03	0.8536	3.81	0.8536
EM, 30 iterations	0.19	0.9964	0.05	0.9964
EM, 55 iterations	0.23	0.9964	0.00	0.9969
EM, 80 iterations	0.27	0.9964	0.00	0.9969

Appendix

Table A1: Estimator choice at

\tau=2

. MSR is multiplied by

10^{3}

;

p_{10}

is the tenth percentile of welfare efficiency. Rounded zeros do not imply zero regret.

Rule	$p_{10}$	Median	Mean	MSR	Exact (%)
Act immediately	0.8535	0.9475	0.9251	8.693	6.66
Worst-case envelope	0.8535	0.9475	0.9251	8.693	6.66
Achieved-welfare LP	0.8535	0.9539	0.9299	8.032	8.43
Maximin welfare	0.8536	0.9933	0.9604	5.006	32.35

Appendix

Table A2: Approximate ambiguity-aware procedures.

p_{10}

, median, and mean summarize welfare efficiency; MSR is multiplied by

10^{3}

. Exact is agreement with the reference optimal date. All rows use the same 7,189 runs with nonempty estimated sets, out of 7,200 planned runs.

Procedure	$N=10^{3}$	$N=10^{5}$
Certified NPMLE	0.0648	0.0387
EM, 30 iterations	0.0375	0.0383
EM, 55 iterations	0.0357	0.0339
EM, 80 iterations	0.0373	0.0341
Bagged NPMLE	0.0558	0.0302
Regret calibration	0.0576	0.0302

Appendix

Table A3: Worst-member mean regret over 30 compatible pairs. Procedures share pre-failure information and matched trials. Values are relative-regret fractions; bold marks the lowest point estimate per column.

Procedure	$N=10^{3}$	$N=10^{5}$
EM, 55 iterations	$-0.0292\;[-0.0362,-0.0242]$	$-0.0048\;[-0.0109,-0.0022]$
Bagged NPMLE	$-0.0090\;[-0.0153,-0.0032]$	$-0.0085\;[-0.0140,-0.0047]$
Regret calibration	$-0.0072\;[-0.0141,-0.0009]$	$-0.0085\;[-0.0140,-0.0047]$

Appendix

Table A4: Paired differences from certified NPMLE in pooled worst-member mean regret. Negative values favor the listed procedure. Brackets show exploratory, unadjusted 95% percentile-bootstrap intervals conditional on the fixed pairs and retained trials.

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