Capacity Constraints And Administrative Rationing

When Scarcity Creates a Research Opportunity

A program with more eligible applicants than available places must ration access. Its capacity limit does not, by itself, create a natural experiment. The relevant question is how applicants are ordered and who receives an offer. A documented random tie-break among equally eligible applicants may create a credible comparison. A priority system based on urgency, application timing, or staff judgment may instead sort applicants in ways that also predict their outcomes.

The distinction between an offer and treatment receipt matters. Some people offered a place decline it, while some not initially offered may later enter from a waitlist. An offer can therefore serve as an instrument for actual participation only if its assignment has the properties needed for instrumental-variable analysis. The resulting estimate generally concerns applicants whose participation changes because of the offer, not every eligible person or every program applicant.

Audit the Allocation Rule Before Estimating an Effect

Start by reconstructing the actual allocation process, not simply the program’s written capacity. Identify the eligible applicant pool, the date at which eligibility and priority were determined, the ranking variables, the number of places, and the order in which offers were made. Then check how the process handled ties, refusals, missed deadlines, and later openings. Administrative records may reveal that a nominal lottery was preceded by discretionary screening, or that waitlist order changed after applicants’ circumstances were reviewed.

Different allocation mechanisms support different kinds of comparisons. A genuine lottery among eligible applicants can make offer assignment independent of potential outcomes within the lottery pool. A fixed priority score may leave a quasi-random comparison only among applicants tied at a boundary if the tie-break was genuinely random. By contrast, comparing people offered places with everyone not offered may be confounded when the first group had greater need, applied earlier, or received discretionary priority.

Even a randomized offer does not guarantee that treatment receipt is randomized. Take-up is selective: offered applicants may decline, and people without an initial offer may find another route into treatment. The offer can still be informative if it substantially changes treatment receipt and the other instrumental-variable assumptions are plausible. But the treatment effect inferred through that offer is not generally the average effect for all applicants.

These assumptions should be assessed against the institution’s process. Relevance can be checked by comparing treatment rates by offer status. Independence is supported by a credible random tie-break, not established merely because measured baseline variables look similar. Exclusion is questionable if an offer provides information, priority for other services, or financial support even when the applicant does not enroll. Monotonicity can fail if an offer causes some applicants to forgo treatment they otherwise would have obtained elsewhere. No interference is especially demanding when places are fixed: one applicant’s offer can displace another from a scarce slot.

Why the Wald Estimate Targets Compliers

With a binary offer and binary treatment, the Wald estimate is the difference in average outcomes by offer status divided by the difference in treatment rates by offer status. Under the assumptions above and randomized assignment within the target pool, it identifies the average treatment effect among compliers: applicants who receive the program if offered and do not receive it if not offered.

The logic can be shown by grouping applicants according to their pair of potential treatment statuses \((D(0),D(1))\). Under monotonicity, the possible groups are always-takers \((1,1)\), compliers \((0,1)\), and never-takers \((0,0)\). Defiers \((1,0)\) are ruled out. Independence makes the groups comparable across offer assignment. Exclusion means the offer does not directly change outcomes within a treatment status. Thus, the offer’s average outcome difference comes only from compliers, whose treatment changes from 0 to 1:

The offer’s effect on the treatment rate is the proportion of compliers, because only compliers change treatment status when the offer changes:

Dividing the first difference by the second cancels the complier proportion and gives the complier average causal effect. This derivation also shows why a weak difference in treatment rates is problematic: the estimate divides by a small first stage, so sampling uncertainty can be large. The offer’s effect on the outcome itself, without dividing by the treatment-rate difference, is the intention-to-treat effect of the offer.

Allocation Order and Tie-Breaking in Practice

A priority system can be reasonable for program administration and still be unsuitable for causal identification. For example, an agency may place applicants with the greatest measured need first. Comparing those offered places with lower-priority applicants then mixes the effect of treatment with differences in need. Application timing can pose a similar problem: people able to apply immediately may differ in employment, health, or access to information.

A tie-break is useful only if it is both consequential and credible. Establish what counted as a tie, whether all tied applicants had the same eligibility and priority, and how the winner was selected. A recorded random number generated before offers is different from a staff member choosing among tied applicants after reviewing their files. If the process was random only within a priority stratum, the comparison should respect that stratum; it should not be generalized to applicants with different priorities without additional assumptions.

Selective Take-Up and the Scope of the Effect

The waitlist often changes over time. Some people decline offers, others fail to respond, and later applicants may be contacted as places open. For analysis, define the offer \(Z\) and treatment \(D\) over a clear period and population. If the instrument is the initial offer, later treatment receipt must be recorded for both initially offered and unoffered applicants. Treating everyone not offered initially as untreated can misclassify waitlist entrants and distort the first stage.

It is also important to distinguish the effect of being offered a place from the effect of receiving the program. The intention-to-treat contrast answers the first question and remains meaningful even if exclusion is implausible. The Wald ratio aims at a treatment effect for compliers, but only when the additional assumptions hold. If an offer itself changes behavior—for example, applicants obtain other support after learning they are waitlisted—then the outcome effect of the offer may not operate solely through the program.

Diagnostics, Alternatives, and Limits

Useful diagnostics begin with the assignment mechanism. Check whether the stated number of offers matches capacity, whether timestamps and ranks precede outcomes, whether tie-break records are complete, and whether offers follow the recorded rule. Compare treatment rates by offer status to assess the first stage. Examine baseline covariates and pre-offer outcomes within the allocation strata; substantial differences can reveal implementation problems, although balance cannot prove independence. Investigate missing outcomes, refusals, crossovers, and whether staff overrode the queue.

A weak first stage, extensive unrecorded waitlist movement, or treatment availability outside the program can make the offer a poor instrument. Repeated applicants or households may create dependence that requires appropriate uncertainty estimation. With fixed capacity, one person’s gain may be another person’s lost place, violating a simple no-interference interpretation. The result may then describe the effect of an offer under that particular rationing system, rather than an effect that would persist if capacity expanded.

A lottery-based design is the closest alternative when the allocation process actually randomizes offers; it directly supports comparison by assignment. A regression discontinuity design may be relevant when offers follow a score cutoff, but it requires its own cutoff-based assumptions and a defensible local comparison. Observational adjustment, including propensity score methods, can address measured differences only under their assumptions and does not repair unmeasured selection by urgency or staff discretion. The choice among these approaches follows from the allocation rule, not from the fact that the program was oversubscribed.