When to stop looking

You're hiring a secretary. N candidates show up one at a time in random order. After each interview you must decide on the spot — hire them or reject them forever. No going back. No comparing side-by-side. When do you stop looking and commit?

That's it. That's the whole setup. It's called the secretary problem, formulated in the 1950s and famous ever since — partly because the answer is so weirdly specific, and partly because the same shape of problem keeps showing up everywhere else.

Here's the rule:

Look at the first 37 % of candidates without hiring anyone. Then take the next one who beats everyone you've seen so far.

Why 37 %? Because 1/e ≈ 0.368, and that's the threshold that balances the two ways the rule can fail: rejecting the actual best because they fell into your look phase, versus settling too early before the best has even shown up. Tilt the threshold up and you waste good candidates on calibration. Tilt it down and you commit before you know what good looks like. The sweet spot — provably — is the reciprocal of e.

The look phase isn't picking — it's calibrating. You're not throwing those candidates away, you're learning what good looks like. Then the moment someone beats your calibration set, you commit.

Two clean phases:

• Look (first N/e candidates): reject everyone, but remember the best.
• Leap (the rest): take the first candidate who beats that best-seen-during-look.

Now here's the part that makes the rule famous. Your chance of actually landing the best candidate is also about 37 % — and that's true no matter how big N gets. With 10 candidates or 10 million, the rule wins you the top of the pile around 37 % of the time.

That's a wild bargain. Without the rule, picking randomly nets you the best one with probability 1/N. With 100 candidates that's 1 %. The 37 % rule, with the same information, scores 37 × better.

• Apartment hunting. Got 30 days to view places? Spend the first 11 just looking — turn down everything, no matter how good. Then commit on the next apartment that beats the best you saw in those 11 days.
• Dating (the "37 % rule of marriage"). If you'll seriously date between age 18 and 40, the look phase ends around 26. After that, commit to the next person who beats every ex.
• Parking. Drive past the first ~37 % of available spots before considering anything. Then take the next one better than the best you saw.

"You don't need to see all the options. You need to see enough to know what good looks like."

The math is the math. The hard part is trusting the calibration — actually rejecting that very nice 11th apartment, the one you would have been happy to live in.

The classic 37 % rule assumes the rules at the top of the chapter. Relax any one of them and the answer shifts:

• Recall allowed. If you can return to earlier candidates (maybe they'll still take the job, maybe they won't), success goes up. The look phase shrinks; the leap phase grows.
• Full information. If you actually know the score distribution in advance — say candidate quality is normally distributed with known mean and variance — you don't need a calibration phase at all. You set a fixed threshold from the start and accept the first candidate over it. Studied in depth by Robertson & Flannery and by Gilbert & Mosteller.
• Candidates can reject you. Now it's a two-sided matching problem — closer in spirit to stable-marriage than to optimal stopping.
• Cost per look. If interviewing costs money, the look phase shrinks again — every look has to be worth its weight.

Take the secretary problem and squint. Anywhere data arrives as a stream and you have to decide irrevocably on each item, you're inside an optimal-stopping problem:

• Online algorithms — the whole class. Inputs come one at a time; you commit per-item without seeing the future.
• A/B test stopping — when do you stop collecting data? Watch enough, but don't watch forever.
• Server selection — pick the first server from a rotating list whose latency beats a threshold you calibrated on the first batch.
• Online hiring — same as the original, just with résumés in a queue.
• Multi-armed bandits — the first phase of many bandit algorithms (UCB, ε-first) is essentially a look phase before the leap.

The algorithm is short enough that the type signature alone tells most of the story: stream in, maybe-pick out. Flip between the two languages — the shapes are nearly identical.

The secretary problem is one corner of a larger map. The next chapter is its opposite mood: when you can come back, when failure is cheap, when the question stops being "should I commit?" and becomes "should I try something new, or stick with what I already love?"