When to Stop
Twenty candidates will pass before you, one at a time — flats, hires, suitors, exits; the mathematics does not care which. Each holds up a score. Say yes, and the door opens for them. Say no, and no is forever: passed candidates do not return. Reach the last, and you take the last. Your goal is not a good one. It is the best one.
Now watch the mathematics
If all you can judge is how each candidate ranks against the ones before, there is a provably optimal strategy, and it is beautifully blunt: look, then leap. Spend the first 37 per cent of the field looking — choose nobody, remember the best you saw — then take the very next candidate better than everyone so far. Below, the exact odds for every possible looking phase. As the field grows large the best looking phase converges on exactly 1/e of it — 36.8 per cent, with Euler’s number hiding in your love life — and succeeds just as often. With a field of twenty the discrete optimum lands on seven candidates, and the curve finds it: watch the peak. Blind luck manages 5 per cent.
The rule carries over to many irreversible choices made in sequence, and its deeper lesson is a kindness: some failure is the fee, not the fault. Even played by the rule, you lose most of the time — so a bad outcome does not prove you chose badly, and the looking you did before leaping was not waste. It was the strategy.
One confession. This game shows you scores out of 100, which is more than the classic problem allows, and knowing the scale changes the answer. A 97 among the first few is worth taking on sight, because there is so little room above it. Played with that knowledge, perfectly, this game can be won about 62 per cent of the time, against look-then-leap’s 38. The 37 per cent rule is for when all you can do is compare, which is how flats, hires and suitors usually arrive: no scale, only the ones before.
More play on the Playground; and for the other kind of stopping, where the question is whether a machine will let you, try The Stop Button.