Irrevocable decision
Named by 7 essays across one field — each of them below, with the objects they name alongside it.
When to stop looking
Candidates arrive one at a time in a random order. Each must be accepted or rejected on the spot, with no going back and no way to know what is still to come. The best possible rule is to look at about a third of them and then take the first one that beats everything seen — and it works about a third of the time, however many there are.
When the numbers are shown
The secretary rule wins a third of the time and cannot do better, because it is told only who is ahead. Show the actual values and say where they came from, and the same problem is won three times in five — by a standard that falls as the end approaches.
Giving up on the best
The secretary rule treats landing the second-best exactly as badly as landing the worst, which is a strange thing to want. Ask instead for the smallest average rank and the answer is about the fourth-best candidate — whatever the size of the field, and whether it is ten or ten million.
Half of what an oracle takes
Compare an online rule not against the best it could have done but against a rule that has seen every value in advance. One fixed threshold secures half of what the oracle collects, whatever the distributions are — and there is an example on which half is all there is.
The thresholds that nest
Allow a second acceptance in the secretary problem and the chance of holding the best rises from about 37 per cent to about 59. The best rule is still a threshold — but one threshold for each number of choices still in hand, the earlier ones starting sooner, and each additional choice buying less than the one before.
![What a rule collects when every value comes from the same distribution. uniform on [0, 1]: best rule 0.995, one threshold 0.635 of E[max] at n = 200; exponential: best rule 0.905, one threshold 0.678 of E[max] at n = 200; Pareto, tail exponent 2: best rule 0.802, one threshold 0.714 of E[max] at n = 200; Pareto, tail exponent 1.2: best rule 0.802, one threshold 0.682 of E[max] at n = 200.](/f/what-a-rule-collects-when-every-value-comes-from-the-same-distribution-cdb6768b.svg)
When every value comes from the same hat
A rule that sees values one at a time and must keep or discard each on the spot can guarantee half of what a prophet collects, and no more, when the values come from different distributions. When they all come from the same one, the guarantee rises to 0.745 — and a single fixed threshold, set so that each value crosses it with chance 1/n, already secures 1 − 1/e. For bounded values the best rule collects nearly everything; only a heavy tail, where one enormous value carries the prize, keeps the gap open.
The rank that remembers every value
Values arrive one at a time, each must be kept or discarded on the spot, and the aim is to keep one whose rank among all of them is low on average. Told only who is leading, the best rule gets 3.87. Shown the values, a rule gets below 2.33 — and how much lower the best possible rule goes is not known, because the rank of what is kept depends on every value seen, and the best rule may need to remember all of them.
Named alongside it
The objects these essays reach for when they reach for this one.
Optimal stoppingThreshold ruleExpectationBackward inductionDecision procedureConditional probabilitye, the numberCounting argumentPermutationBoundCompetitive ratioHarmonic series