Optimal Stopping & the Secretary Problem

You are apartment hunting in a hot city. Places get listed and taken within a day, so each viewing is a one-shot deal: say yes on the spot and you sign, say no and it’s rented to someone else by tomorrow. You cannot line all the flats up, compare them at leisure, and pick the best — you meet them one at a time, in a random order, and you must decide before you’ve seen what’s coming. Commit too early and you might be signing away a much better place two viewings from now. Wait too long and the good ones are all gone, and you’re stuck with whatever is left in week six.

This is not a real-estate problem. It is the shape of a whole class of decisions: hiring when candidates take other offers, selling a house as bids trickle in, choosing when to stop dating and settle down, deciding when you’ve researched enough and should just decide. In every one, options appear in sequence, each must be accepted or rejected on the spot, and a passed option cannot be recalled. The instinct is to treat “when do I stop looking?” as a matter of gut feeling or nerve. It isn’t. It has a mathematically optimal answer — and the answer is beautiful.

The headline result is the secretary problem, and its solution is the 37% rule: to have the best shot at picking the single best of N options seen one at a time, spend the first 37% of them just looking — reject them all, no matter how good, but remember the best one you see. Then leap at the first option that beats everyone in that sample. Do this and you land the very best choice about 37% of the time — a staggeringly high hit rate for a decision made blind, and vastly better than grabbing the first decent option or dithering until the end. That recurring 37% ≈ 1/e is not a coincidence; it falls straight out of the mathematics of when to stop.

This course builds the whole model, rung by rung. It opens with the exact secretary problem — the 37% rule, why the number e appears, worked by hand on small cases so you can see the mechanism, not just trust the formula. It then extracts the general pattern — explore-then-commit: a look phase that calibrates a bar from the options you sample, and a leap phase that takes the first option to clear it. It shows what changes when you want the best expected value rather than the single best — a gentler goal where near-misses count, the optimal look shrinks below 37%, and the bar you demand should fall as your options run out (be picky early, forgiving late). It walks the variations that matter in real life — being able to go back to a passed option (recall), knowing the distribution of quality up front (use a reservation value instead), a cost to keep searching, an uncertain or unknown N, and offers that can be rejected. And it closes on transfer and honest limits — where the look-then-leap lens genuinely helps (hiring, house-hunting, when to stop researching), the pop-culture “37% then commit” version of dating and its real caveats, and where the pure rule optimises the wrong thing and quietly lies.

Because it does lie, if you take it literally. The clean 37% rule maximises the probability of the single best option and gives zero credit for a near-perfect second place — which is almost never what a real person wants. It assumes strict irreversibility and no recall, which is often false (you frequently can go back). It assumes you can rank options but don’t know the distribution — and if you do know the distribution, a reservation value beats it. Hold those edges, and what you keep is a genuine decision tool: recognise when a choice truly has the sequential, no-going-back shape; deliberately split it into a calibration phase and a commit phase; set your bar from what you see early; and lower it as the runway shortens. Stop looking at the right moment — and leap.