Thought Toys · Strategy & computation · Exhibit 47
Applicants arrive one at a time in random order. You must hire or pass on each the moment you meet them — no callbacks. It sounds hopeless, yet one simple rule catches the single best applicant more than a third of the time, however long the line.
The line-up — bar height is quality, revealed left to right lookhiredthe best
Chance of hiring the very best vs. how long you look exactyour runs
Look at the first 37%, then leap
—
Every applicant has a hidden quality; the tallest bar is the best of the bunch. You interview them left to right and can only ever hire the person in front of you. The rule on trial: look at a fixed fraction and hire no one, just remember the best you've seen — then leap, hiring the first later applicant who beats that running best. Hit "Deal one" and watch it play out: the grey look-phase goes by, then the first applicant taller than everything before them is hired in amber. Green marks who was actually best. You win only if amber lands on green.
Any single deal is a coin toss of luck — but slide the look phase and hit "Run 500," and the pattern is iron. Look too little and you leap at someone merely good early on; look too long and the best has usually already walked past during the look phase. The sweet spot is 1/e ≈ 37%, and there the odds of catching the single best applicant are also about 37% — not just "pretty good," but provably optimal — no strategy of any kind does better. The same look-then-leap arithmetic is why "reject early options to calibrate, then commit" is such durable advice for hiring, apartments, and more.
improve/verify/47-secretary.js): at n=100 the
best cutoff is m=37 with P≈37.1%; a 200,000-deal Monte Carlo of the actual game matches
the formula; as n→∞ both the cutoff fraction and the win probability converge to 1/e;
and an independent backward-induction dynamic program — free to accept or reject at every
step reaches the identical optimum, so the 1/e rule is genuinely best, not merely good.
Negative controls: "take the first," "take the last," and "take a random one" each win
exactly 1/n of the time — at n=100 the 1/e rule beats them by more than 30×; and any
cutoff far from 1/e (too greedy or too patient) is measurably worse.
Also in Strategy & computation: Freeze too fast, stay stuck →
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