Draft Lottery Designs

From Lottery Number to Causal Question

A draft lottery can create a source of plausibly random variation in eligibility for military service. But a lottery number is not itself military service, and eligibility does not guarantee that a person serves. A sound design therefore begins by defining who was subject to the lottery and tracing the assignment rule into actual exposure. Only then can the comparison be interpreted as an effect of eligibility or, under additional assumptions, an effect of service.

The study population matters. A defensible population might be all people in specified birth cohorts who were registered and eligible for a particular lottery before numbers were drawn. It should not be defined as only those who enlisted, were inducted, or completed service: those events occur after assignment and conditioning on them can select people based on factors related to outcomes. The target population may also be narrower than everyone assigned a number—for example, people in a specific cohort and jurisdiction—so conclusions should be limited accordingly.

Definition: Let $Z_i=1$ indicate that person $i$'s randomly assigned lottery number placed them in the category eligible for a draft call under the specified rule; let $Z_i=0$ indicate that the number did not. Let $D_i=1$ indicate actual military service during a prespecified period, and $D_i=0$ otherwise. Let $Y_i$ be a specified outcome measured after assignment, such as annual earnings in a defined follow-up year. The assignment effect compares outcomes by $Z_i$; the exposure effect concerns the effect of $D_i$.

The binary indicator $Z_i$ is a coding of the lottery rule, not a claim that all people with $Z_i=1$ were inducted. A lottery number may determine eligibility or priority, while medical screening, deferments, voluntary enlistment, administrative decisions, and other events determine actual service. Keeping $Z_i$ and $D_i$ distinct is essential: the randomized feature is the assignment, while service is generally selected in part by individual circumstances.

What the Lottery Can Identify

The direct randomized comparison is the intention-to-treat (ITT) effect of lottery eligibility. In a population with unconditional random assignment, it is the difference in mean outcomes between those assigned $Z_i=1$ and $Z_i=0$. It captures the effect of being placed in the eligibility category, including any consequences of eligibility that do not operate through completed service. It does not, by itself, equal the effect of military service.

To interpret assignment as an instrument for service, define potential service $D_i(z)$, the service status person $i$ would have under assignment $z$, and potential outcome $Y_i(d)$, the outcome under service status $d$. A standard binary-instrument interpretation requires the following conditions for the stated study population:

Random assignment: the lottery assignment is independent of the potential service and outcome values, either in the full population or conditional on the lottery's randomization strata.
Relevance: eligibility changes the probability of service, so the assignment contrast in service is not zero.
Exclusion: assignment affects the outcome only through actual service, represented by $Y_i(d)$ rather than a potential outcome that also depends directly on $z$. This can fail if eligibility changes outcomes through channels other than service.
Monotonicity: $D_i(1)\geq D_i(0)$ for every person: no one would serve when ineligible but avoid service when eligible, under the defined assignment and exposure.
Consistency and no interference: observed service and outcome correspond to the assigned potential values, and one person's assignment does not change another person's outcome in a way relevant to the analysis.

Under these assumptions, people fall into principal strata according to $(D_i(0),D_i(1))$: never-takers $(0,0)$, compliers $(0,1)$, and always-takers $(1,1)$. Monotonicity rules out defiers $(1,0)$. Lottery eligibility can change service only for compliers. Consequently, the instrumental-variable ratio identifies an average service effect for compliers, not necessarily the average effect for everyone in the study population.

\frac{E[Y_i\mid Z_i=1]-E[Y_i\mid Z_i=0]} {E[D_i\mid Z_i=1]-E[D_i\mid Z_i=0]} = E[Y_i(1)-Y_i(0)\mid D_i(1)>D_i(0)].

This formula in its displayed, unadjusted form requires unconditional random assignment. If assignment is randomized within strata with different assignment probabilities, the corresponding comparisons must be made within strata and combined using the same prespecified population weights for both outcomes and service. A pooled difference can mix different stratum compositions in the two assignment groups and need not have the interpretation shown.

Deriving the Stratum-Adjusted Wald Contrast

Let $S_i=s$ denote a lottery randomization stratum, such as a cohort or a group for which numbers were drawn separately. Define the target-population weight $w_s=P(S_i=s)$, with $\sum_s w_s=1$. Within each stratum, define the outcome and service contrasts:

\Delta_{Y,s}=E[Y_i\mid Z_i=1,S_i=s]-E[Y_i\mid Z_i=0,S_i=s], \qquad \Delta_{D,s}=E[D_i\mid Z_i=1,S_i=s]-E[D_i\mid Z_i=0,S_i=s].

Under random assignment within each stratum, consistency, exclusion, and monotonicity, $\Delta_{D,s}$ is the proportion of compliers in stratum $s$, and $\Delta_{Y,s}$ is that proportion multiplied by the average service effect among that stratum's compliers. To see why, random assignment allows each within-stratum assignment group to represent the same distribution of potential values. Under exclusion, the outcome contrast is the average of $Y_i(D_i(1))-Y_i(D_i(0))$. Never-takers and always-takers have no service change, so their contribution is zero. Compliers contribute $Y_i(1)-Y_i(0)$. The same argument applied to $D_i$ gives a contribution of one for each complier and zero for the other allowed types.

Thus the target-population standardized contrasts are $\Delta_Y=\sum_s w_s\Delta_{Y,s}$ and $\Delta_D=\sum_s w_s\Delta_{D,s}$. Their ratio is a complier-weighted average of the stratum-specific effects: strata with more compliers contribute more to the ratio. It is not generally the simple average of stratum-specific Wald ratios.

\frac{\Delta_Y}{\Delta_D} = \frac{\sum_s w_s P(\text{complier}\mid S_i=s)\, E[Y_i(1)-Y_i(0)\mid \text{complier},S_i=s]} {\sum_s w_s P(\text{complier}\mid S_i=s)}.

The derivation also explains the practical rule: use the same target weights for both contrasts. In a sample, the weights can be chosen to represent the study population, such as the observed share of each stratum in the full pre-assignment eligible cohort. Inference must reflect the lottery design and the estimation procedure; merely calculating a ratio does not guarantee that its uncertainty is small, especially when the first stage is weak.

Worked Example: Define the Population and Trace Assignment to Service

Suppose, hypothetically, a lottery assigns numbers to 1,000 registered people in a specified birth cohort. Before the draw, the study protocol defines the population as those 1,000 people. Numbers at or below 30 make a person eligible for a call; higher numbers do not. Of the 300 people assigned numbers at or below 30, 180 serve during the follow-up period. Of the 700 others, 70 serve, perhaps through voluntary enlistment.

Code eligibility as $Z_i=1$ for numbers at or below 30, and actual service as $D_i=1$ for service during follow-up. The observed service rates are $180/300=0.600$ in the eligible group and $70/700=0.100$ in the ineligible group. The first-stage difference is $0.600-0.100=0.500$, or 50 percentage points. This confirms that the assignment rule strongly predicts service in these hypothetical data.

Suppose the outcome is annual earnings in thousands of dollars in a specified post-lottery year, and the group means are $\$46$ thousand for $Z_i=1$ and $\$50$ thousand for $Z_i=0$. The ITT contrast is $46-50=-4$, or $-\$4,000$ per person per year. Under unconditional random assignment, this is the estimated effect of eligibility assignment on earnings for this population. If exclusion and monotonicity also hold, the Wald estimate is $(-4)/0.5=-8$, or $-\$8,000$ per year as a complier average effect of service.

The calculation does not establish the assumptions. The lottery must actually randomize eligibility in this cohort; eligibility must affect service; and it must not affect earnings except through service for the IV interpretation. The ITT remains an assignment effect even if exclusion is doubtful, but the service-effect interpretation would then be unsupported. These numbers are hypothetical and are not evidence about any actual cohort.

Why Stratum Composition Changes the Answer

Lottery numbers may be randomized separately across cohorts or other groups, and eligibility rates can differ across those groups. Conditional random assignment does not imply that the pooled $Z_i=1$ and $Z_i=0$ groups contain the same proportions of each stratum. The following example shows why the stratum definition and common weights are substantive, not cosmetic.

Worked Example: Correcting a Misleading Pooled Contrast

Consider two equally large hypothetical strata. In stratum 1, the probability of eligibility is 0.9 and everyone is a never-taker. In stratum 2, the probability is 0.1 and everyone is a complier. Assignment is randomized within each stratum. In stratum 1, the outcome is always 0. In stratum 2, the outcome is 100 without service and 110 with service. Thus the service effect for every complier is 10 outcome units, and the population share of compliers is 0.5.

Within stratum 1, the service contrast is $0-0=0$. Within stratum 2, it is $1-0=1$. With common population weights $w_1=w_2=0.5$, the standardized first stage is $0.5(0)+0.5(1)=0.5$. The outcome contrast is $0.5(0)+0.5(10)=5$, so the standardized Wald ratio is $5/0.5=10$, the stated complier effect.

Now pool without adjustment. Each stratum contains 0.5 of the population. Eligible people make up $0.5(0.9)+0.5(0.1)=0.5$ of the population. Among the eligible, the stratum 1 share is $0.45/0.50=0.9$ and the stratum 2 share is $0.05/0.50=0.1$. Their mean service is $0.1$, and their mean outcome is $0.9(0)+0.1(110)=11$.

Among ineligible people, stratum 1 contributes $0.5(0.1)=0.05$ and stratum 2 contributes $0.5(0.9)=0.45$, so their stratum shares are 0.1 and 0.9. Their mean service is 0 and their mean outcome is $0.1(0)+0.9(100)=90$. The pooled service contrast is therefore $0.1-0=0.1$, while the pooled outcome contrast is $11-90=-79$. The pooled ratio, $-79/0.1=-790$, plainly fails to recover the effect of 10. It is distorted by the assignment groups' different stratum composition. The common-weight stratum-adjusted contrasts recover the appropriate target-population comparison.

Design Checks, Alternatives, and Limitations

Before estimating an effect, reconstruct the lottery mechanism from records: who entered the draw, how numbers were generated, which rule defined eligibility, and whether assignment probabilities varied by stratum. Verify that the analysis includes people according to pre-assignment eligibility criteria rather than later service status. Examine sample sizes and assignment rates within each stratum, check for missing outcomes or exclusions associated with assignment, and inspect covariate balance as a diagnostic. Balance is not proof of randomization, but substantial unexplained imbalance can reveal coding errors, selective inclusion, or an inaccurate account of the draw.

Report the first-stage contrast by stratum as well as in the target-weighted summary. A near-zero first stage means assignment provides little information about service, making the ratio unstable and its uncertainty potentially large. Also consider whether the binary eligibility indicator faithfully represents the rule: changing the threshold, combining lotteries, or including people not subject to the same mechanism changes the design and potentially the target population.

A direct comparison of people who served with people who did not serve is generally not randomized; service selection may reflect health, preferences, family circumstances, or other determinants of outcomes. Covariate adjustment or matching may address measured differences but does not turn the service comparison into a lottery comparison. The ITT instead preserves the randomized assignment contrast and answers a policy-relevant question about eligibility. The IV ratio targets a service effect for compliers, but requires exclusion and monotonicity in addition to random assignment and relevance.

Exclusion deserves particular scrutiny. Draft eligibility could affect schooling, employment choices, migration, or other behavior even for someone who does not ultimately serve. If such paths affect the outcome independently of actual service, the lottery may still identify an effect of eligibility, but the ratio is not identified as the causal effect of service under the stated model. Similarly, interference can matter if one person's eligibility changes another person's probability of being called or the resources available to them. The study should state its exposure and outcome periods, address missing follow-up, and avoid generalizing a complier effect to populations with different service responses without further justification.

Common pitfall: “Assignment was randomized within each cohort, so the pooled eligible-versus-ineligible difference is randomized.” This is not necessarily true when cohorts have different eligibility probabilities. The groups can have different cohort composition, as the worked example demonstrates. Correct the reasoning by estimating within-stratum contrasts and standardizing both the outcome and service contrasts to the same target-population weights.

A useful reporting sequence is to state the population and assignment rule, show the assignment-to-service first stage, report the ITT, and only then present a Wald estimate with its assumptions and complier interpretation. This keeps the design's directly randomized conclusion separate from the stronger causal claim about service.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A hypothetical cohort has an assignment-to-service contrast of 0.25 and an outcome contrast of $-\$3,000$ per year. Under the IV assumptions, what is the Wald estimate, and whose effect does it represent?
Two equally weighted strata have $\Delta_{D,1}=0.2$, $\Delta_{D,2}=0.6$, $\Delta_{Y,1}=2$, and $\Delta_{Y,2}=3$. Calculate the standardized Wald ratio using common weights of 0.5.
Why does a balance check not prove that a lottery was correctly randomized?
If exclusion is implausible but random assignment is credible, which causal contrast remains directly interpretable?

Solutions. (1) The ratio is $-\$3,000/0.25=-\$12,000$ per year. With relevance, exclusion, monotonicity, and the other IV conditions, this is the average service effect for compliers, not necessarily for all assigned people. (2) The standardized outcome contrast is $0.5(2)+0.5(3)=2.5$; the standardized service contrast is $0.5(0.2)+0.5(0.6)=0.4$. The ratio is $2.5/0.4=6.25$ outcome units. (3) Balance is evidence about observed covariates only; unobserved differences, selective inclusion, incorrect assignment coding, or chance variation can remain. It is a diagnostic, not a proof of the lottery mechanism. (4) The ITT effect of assignment or eligibility remains directly interpretable under random assignment and appropriate outcome follow-up. The IV interpretation as a service effect requires the additional assumptions.

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