Three-card stopping rule: choose a later record

A three-card stopping rule can be checked exactly when the available information is kept separate from the hidden answer. This fictional exercise offers three items in an equally likely random order. You must choose one as it arrives, without recalling a rejected item. You learn only how the current item compares with items already seen. Rejecting the first, accepting the second if it is better than the first, and otherwise taking the last succeeds in three of the six possible orders. The probability is one half under these stated rules.

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Define what can be observed before choosing

There are three distinct hidden ranks: one is worst, two is middle and three is best. These numbers are used only in the later audit of a completed choice. They are never shown to the participant while decisions are being made. The participant receives a relative comparison: at the second item, it is possible to learn whether it is better or worse than the first. Seeing the first item supplies no information about whether it is best, middle or worst among all three.

Assume each of the six possible rank orders has probability one sixth. The participant knows this probability model and the total of three items, but receives no other identifying feature correlated with rank. At each arrival, choosing the current item ends the run. Rejecting it makes that item unavailable permanently. If the first two were rejected, the third must be chosen. These information and action rules define the optimization question. A procedure that can look back or recognize the absolute best would answer a different question.

Success means choosing hidden rank three. Every other final choice is a failure for this declared objective, even if the selected item is middle rather than worst. We are maximizing the probability of that binary success, not the average numerical rank, a monetary payoff or a worst-case guarantee. No actual randomizer, shuffled product batch or observed person's judgment is being authenticated. The equal weights are assumptions of the paper model, and the calculations below follow from them.

Write the rule without revealing the hidden ranks

Always reject the first item after observing it. When the second arrives, compare it with the first. If the second is better, accept it and stop. If it is worse, reject it and take the third when it arrives. Calling the second a record means only that it is better than everything seen so far; at that point the seen comparison is just with the first. It does not mean the participant has learned that the second is best among all three.

The third is a fallback if the second was worse. It is selected even if it fails to exceed an earlier item. No extra test is available after rejecting both previous items, because one choice is required and only the third remains. Keep that forced-last clause with the record instruction. A rule that refuses the third whenever it is not a record could finish without choosing anything, which is outside the present exercise. The fallback is part of the policy rather than an exception added after a disappointing result.

Audit every equally weighted order

For the audit only, write each hidden order as its three ranks from first to last. In order 1, 2, 3, the second exceeds the first, so the rule chooses rank two at position two. It fails because rank three arrives later. In order 1, 3, 2, the second again exceeds the first, so the rule chooses position two. This time it selects rank three and succeeds. The identical visible comparison at the second position can therefore accompany different hidden outcomes.

In order 2, 1, 3, the second is worse than the first. Reject it and take the third, selecting rank three successfully. In order 2, 3, 1, the second is better than the first and is accepted. Its hidden rank is three, giving another success. Neither trace requires the participant to identify the first as middle. That label belongs to the later audit; the actual decision uses only the permitted comparison with the item already observed.

In order 3, 1, 2, the first is secretly best but has already been rejected. The second is worse, so the rule reaches the third and chooses rank two, failing. In order 3, 2, 1, the second is also worse than the first, and the forced third has rank one. That run fails too. These failures do not make the rule incorrectly executed. They show the cost of an irrevocable decision made without absolute-rank information.

The successful orders are 1, 3, 2; 2, 1, 3; and 2, 3, 1. There are three successes among six equally weighted orders, yielding probability three sixths, or one half. Taking the first position without comparison would succeed only when rank three appears first, which happens in two orders. Always waiting for the last would likewise succeed in two orders. Each fixed-position rule has probability one third under the same uniform distribution.

Check the second-position decision when it is better

To prove the rule is optimal under the permitted information, examine the decision after the first item has been rejected. If the second is better, the hidden order must be one of 1, 2, 3; 1, 3, 2; or 2, 3, 1. These three orders remain equally likely when conditioned on that comparison. The participant cannot distinguish them using any additional absolute value, name or feature, because the observation contract supplies none.

Accepting the better second item succeeds in two of those three cases: it is rank three in 1, 3, 2 and 2, 3, 1. Rejecting it forces selection of the third, which succeeds only in 1, 2, 3. Thus acceptance has conditional success probability two thirds, while waiting has one third. Accepting is the better action at this information state. This conclusion uses the equal-weight model; merely listing three possible cases without their conditional weights would not establish the comparison.

Check the second-position decision when it is worse

If the second is worse than the first, the possible hidden orders are 2, 1, 3; 3, 1, 2; and 3, 2, 1. Accepting the second cannot select the best in any of them. There is already an observed item better than it, even though that earlier item is no longer available. The conditional success probability of immediate acceptance is therefore zero. This conclusion does not require identifying which of the three hidden orders occurred.

Waiting for the third succeeds in the first of those three cases, 2, 1, 3. Its conditional success probability is one third. Waiting is therefore better than acceptance at this information state. Notice the two uses of past observation: the rejected first item cannot be recovered, but its comparison remains available as information. No-recall prevents selecting an old item; it does not require forgetting how the current one ranked against it.

The better-second and worse-second comparison states each occur in three of the six original orders, so each has probability one half. The best continuation after rejecting the first therefore has success probability one half times two thirds, plus one half times one third. That sum is one half. The rule attains it by accepting in the better state and waiting in the worse state. Each branch has been evaluated using exactly the information available there.

Compare continuing with accepting the first

At the first position, the only possible immediate action is to accept or reject the current item. Its hidden rank is equally likely to be one, two or three, so immediate acceptance succeeds with probability one third. Rejection followed by the best second-position actions succeeds with probability one half. Rejection is the better first action. Together with the two branch comparisons, this establishes the stated policy's optimality among decisions based on the allowed observations.

There is also a small exhaustive control. A deterministic decision table specifies whether to accept the first, whether to accept a better second, and whether to accept a worse second. Each entry has two possibilities, giving eight written tables. Applying each to all six hidden orders gives forty-eight traces. The largest success count is three out of six. Tables that accept the first never use their second-position instructions, but including those redundant written tables does not omit any permitted deterministic policy.

Independent randomized actions cannot improve this maximum. At either second-position state, randomizing between acceptance and waiting makes the success probability an average of the two action values already calculated. Such an average cannot exceed their larger value. At the first position, accepting with probability alpha and otherwise following the best continuation gives (alpha × 1/3) + ((1 − alpha) × 1/2). For alpha between zero and one, this is no greater than one half.

The keychain scene disclosure guide keeps a fictional event fixed while choosing when its narration states the object's location. That is an editorial information-order choice. It supplies no equal-weight hidden ranks, irreversible acceptance policy or success-probability optimum for this exercise.

Change the information and see why the conclusion changes

Suppose the actual rank numbers were visible on arrival. The participant could reject ranks one and two and accept rank three whenever it appeared, succeeding in every order. That does not refute the one-half optimum, because it violates the original observation restriction. A picture of three cards labeled with their absolute ranks during play would therefore communicate the wrong model. The audit numbers should remain in the completed-run record, separate from what the participant was allowed to see at a decision.

The uniform-order assumption is just as material. For a different model that always presents hidden order 3, 1, 2, the stated rule rejects the best immediately and eventually takes rank two. It never succeeds. Taking the first would always succeed in that changed distribution. The original result is not an adversarial guarantee and does not say the rule wins in every sequence. Its optimality belongs to the six equally weighted orders and the exact relative-information rules specified above.

Ties, recall, a different item count or extra identifying information would also require a new problem. This article proves no general formula for longer sequences and makes no hiring, shopping or assessment recommendation about real people or products. Keep the total of three, strict hidden ranking, equal weights and forced final choice beside the rule. Those conditions allow another reader to reproduce both its successes and its failures without supplying information that the participant never had.

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The five-mark stop game guide calculates endpoint probabilities and mean steps for a fixed absorbing rule. Here the endpoint is not a position on a random walk. The task chooses when to accept one item, and compares policy success probabilities under a finite observation contract.

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Retain the hidden audit beside the visible decisions

The finished worksheet records the six equally weighted hidden orders, each permitted better-or-worse observation, the selected position and whether hidden rank three was chosen. Keep the three successful traces beside the branch values two thirds versus one third and zero versus one third. They justify rejecting the first and choosing a later record under this three-card stopping rule. The full-information and fixed-order countermodels explain why neither visible absolute ranks nor another distribution can be silently added to that conclusion.