Week 6: The Secretary Problem

Optimal stopping, the 1/e rule, and why "you'll know when you know" is a bad strategy

27 March 2028

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You interview candidates one at a time, in random order, and must accept or reject each before seeing the next; rejected candidates can’t be recalled. This is the secretary problem, and its answer is exact: reject the first roughly 37% of your expected pool outright, gathering information only, then accept the next candidate who beats everyone seen so far. No other stopping rule does better against an unknown distribution.

This week is where the semester turns from description to prescription for the only time: not because the rule tells you when to commit, but because it tells you exactly how much looking is enough, which is the one thing search-cost intuition alone can’t answer. Tutorial 6 runs the rule against your own intuition, live.

Priya, from week 2, spent the intervening weeks running the experiment on herself for real: eighteen trivia-league candidates, taken in order, no do-overs. She rejected the first seven outright on principle — gathering information only, exactly as prescribed — and committed to the next one who beat the field, a mechanical engineer named Theo, whom she never would have met through any channel except the one she’d already spent two years testing to destruction. The math held up. Whether Theo enjoyed being “the next one who beat the field” is a separate question, and one week 8 is much better equipped to ask than week 6 is.

Outline#

• the problem, the 1/e derivation, and why the threshold is exact rather than approximate
• what the model assumes that real search doesn’t (no recall, sequential, one slot) and what breaks when those assumptions don’t hold
• what the rule is and isn’t good advice for
• why the rule only optimizes your half of the decision

One thing this framework won’t explain#

The rule tells you exactly when to stop looking. It says nothing about whether the person you stop on would also stop on you. The secretary problem is one-sided by construction — the interviewer chooses, the candidate has no vote — and every guarantee about the 37% threshold quietly assumes that whoever you pick, picks you back. Real dating doesn’t grant that assumption for free.

That’s a fifth kind of gap this course keeps finding, and it’s a different shape from the others: Weeks 1, 3, 4, and 5 each found something the model couldn’t price or distinguish. This one is missing a whole side of the interaction. It’s also the first gap with a partial answer already scheduled — Week 8’s stable matching is exactly what happens when you stop assuming the choice runs one way. Week 12 is still where all five come back at once.

Further reading#

• Ferguson, T. S. (1989). “Who Solved the Secretary Problem?” Statistical Science, 4(3), 282–289. Less a derivation than a history — the 1/e rule’s actual origin (Cayley, Kepler, and a 1960 Scientific American column) is more tangled than “the math says 37%” suggests, and this is the paper that sorts out who actually solved what.