Choosing an Initial Play-the-Winner Urn

Why the Starting Urn Matters

A play-the-winner urn adapts allocations as responses accumulate, but it cannot adapt before the first assignment. That first assignment is determined entirely by the urn’s initial composition. Choosing the starting counts therefore sets the initial treatment probabilities and establishes the baseline from which later changes occur.

The choice is a design decision, not a conclusion drawn from trial outcomes. Equal starting counts give the two treatments equal chances on the first draw. Unequal counts deliberately favor one treatment at that point. The resulting probability does not guarantee that a particular treatment will be assigned first, nor does the initial urn by itself guarantee a desired allocation over the full trial.

These probabilities require only the starting counts and the uniform-draw rule. No assumptions about treatment response rates are needed to calculate them. To discuss what happens after the first assignment, however, we also need an outcome and update rule. In the examples below, responses are observed before the next allocation, the drawn ball is returned, and one ball is added according to the play-the-winner rule: a response adds a ball of the assigned treatment’s color, while a nonresponse adds a ball of the other color. The calculations involving that update use unit reinforcement.

Equal and Unequal Starting Counts

The initial probability depends on the ratio of the two counts. If the counts are equal and positive, the first assignment probabilities are both \(1/2\). If the counts differ, the treatment with more balls has the higher first-assignment probability. The probability ratio and odds should be read with the treatment order stated explicitly:

For example, a ratio of \(3/5\) means the odds are 3 to 5 in favor of A, or equivalently 5 to 3 in favor of B. It does not mean the probability of A is \(3/5\): the probability is \(3/(3+5)\).

Choosing Counts for a Desired Initial Probability

Suppose the design calls for a particular first-assignment probability \(q\) for treatment A, where \(0<q<1\). The starting counts should satisfy $$ \frac{u_{A,0}}{u_{A,0}+u_{B,0}}=q. $$ Rearranging gives \(u_{A,0}/u_{B,0}=q/(1-q)\). Since counts must be whole numbers, a desired probability may be exactly achievable or may need to be approximated. Multiplying both counts by the same positive integer leaves the initial probabilities unchanged.

What Initial Composition Does—and Does Not—Determine

The starting ratio determines the first-draw probabilities exactly. It also affects the scale of subsequent changes when the reinforcement amount is fixed: a single added ball is a larger fraction of a small urn than of a large one. Thus, choosing counts involves more than choosing the desired first probability. The total number of starting balls matters too.

This observation does not mean that larger starting counts always produce a preferable design. The counts determine how strongly early updates change the allocation probability, but whether that is desirable depends on the design goals and the response process. There is no universally correct initial composition. It should be specified deliberately, and the treatment labels and update convention must be implemented consistently.

A starting composition is not evidence that one treatment is more effective. For example, assigning more initial balls to A makes A more likely to be assigned first; it does not show that participants assigned to A are more likely to respond. Nor does an unequal initial composition, by itself, establish that an unequal allocation is scientifically or ethically appropriate. Those decisions require design justification beyond the urn arithmetic.

Diagnostics, Alternatives, and Limitations

A basic diagnostic is to calculate both probabilities from the same total and check that they add to one. If \(u_{A,0}+u_{B,0}=N_0\), then \(u_{A,0}/N_0+u_{B,0}/N_0=1\). Also check that counts are nonnegative integers, the total is positive, and the treatment labels have not been reversed. A discrepancy between the intended probability and the probability calculated from the actual counts is a design or implementation error, not a feature of adaptation.

In a simulation that repeatedly draws the first assignment from a freshly initialized urn, the proportion assigned to A will vary around \(u_{A,0}/N_0\). It need not equal that probability in a finite run. For \(M\) independent simulated first draws, the standard error of the simulated proportion is \(\sqrt{p(1-p)/M}\), where \(p=u_{A,0}/N_0\). A small discrepancy is therefore expected from simulation randomness; a large or systematic discrepancy can indicate a coding, initialization, or labeling error.

An equal-start urn and a fixed 1:1 allocation schedule are not interchangeable. Equal initial counts give equal probabilities for the first urn draw, but response-based updates can change later probabilities. Complete randomization with a fixed allocation probability does not adapt to responses, while permuted-block randomization is designed to control balance within blocks. A play-the-winner urn instead changes its composition according to the response-update rule and does not, merely by starting with equal counts, guarantee equal treatment totals throughout the trial.

The choice of initial counts also has practical limits. A target probability may not be exactly representable with a small total, and increasing all counts to preserve the ratio changes how much a fixed reinforcement affects later probabilities. Initial counts alone cannot predict the realized allocation sequence: that sequence also depends on random draws and observed responses. The starting composition is therefore a transparent specification of the beginning of the adaptive rule, not a complete summary of its behavior.

Common Mistakes

• Using a count as the probability. With 3 A balls and 5 B balls, \(P(T_1=A)\) is \(3/8\), not \(3/5\). The value \(3/5\) is the ratio of the probabilities.
• Reversing the direction of the odds. A probability ratio \(P(T_1=A)/P(T_1=B)=3/5\) is 3 to 5 in favor of A, not B. State the treatment after “in favor of.”
• Assuming the first probability stays fixed. The initial ratio determines the first draw. Subsequent probabilities can change when the urn is updated.
• Treating a larger count total as probability-neutral for the entire process. Scaling both starting counts preserves the first probability, but it changes the relative size of a fixed reinforcement.
• Interpreting initial preference as treatment evidence. More A balls make assignment to A more likely; they do not imply a higher response rate under A.