10th May 2026
Choosing is easy when every option can be compared at once. It becomes harder when options arrive one at a time and rejection is permanent. Waiting helps establish what a good option looks like, but the best opportunity might pass during that learning period. Optimal stopping studies this tension.
The classical secretary problem sets precise rules. A known number of candidates appear in uniformly random order, with no ties. Only relative ranks among candidates already seen are available, and rejection is permanent. The objective is to maximise the probability of selecting the single best candidate. [1]
For a large candidate pool, observe and reject roughly the first 37 per cent, then accept the first later candidate who is best so far. If none qualifies, select the final candidate; that fallback cannot win because the best was rejected earlier. The initial group establishes the benchmark.
The threshold balances two ways of failing. Reject too few candidates initially, and an ordinary candidate may look impressive enough to select. Reject too many, and the best candidate is likely to have appeared before selection was allowed. For a large candidate pool, the success probability is approximately:
where x is the fraction initially rejected, P(x) is the probability of selecting the best candidate, and ln is the natural logarithm. The expression reaches its maximum at x=1/e. The constant e is approximately 2.718, so both the optimal rejection fraction and the limiting success probability are about 36.8 per cent. [1]
That probability can seem surprisingly low. Even the optimal strategy misses the best candidate most of the time. For one hundred candidates, though, choosing a fixed position without using observations succeeds only one per cent of the time. Optimality describes what is achievable under the rules, including their severe limits on information.
The 37 per cent rule depends on those limits. If earlier candidates remain available, the order carries information, or a very good candidate is almost as valuable as the best, the calculation changes. An unknown number of future options creates another problem. Different assumptions lead to different stopping policies. [2]
This is why the result should be applied carefully to everyday searches. Buying a home or recruiting a colleague rarely matches every assumption of the classical model. Its useful contribution is a precise account of the trade-off between learning and acting. Before choosing a stopping rule, the decision needs a definition of success, a description of the available information and a clear statement of which opportunities can be revisited.
[1] Thomas Ferguson — Who solved the secretary problem
[2] Thomas Ferguson — Optimal stopping and applications chapter two
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