The Secretary Problem & the 37% Rule

You are hiring an assistant, and the rules are cruel. Applicants arrive one at a time, in random order. You interview each, and the moment the interview ends you must either hire this person on the spot or reject them forever — no “let me sleep on it,” no calling someone back, no shortlist. You can rank any two people you have already met, but you have no idea how good the ones still to come will be. Your goal is not “hire someone decent.” It is to hire the single best applicant in the entire pool. Anything less counts as total failure. How on earth do you play a game this hostile?

This is the secretary problem, and its answer is one of the most quietly shocking results in decision theory: you can win — land the exact best person — about 37% of the time, no matter whether the pool has 10 applicants or 10 million. This lesson builds that answer from the ground up. We define the game precisely, watch the two “obvious” strategies collapse, derive the look-then-leap rule, grind through the small cases by hand, and then meet the constant that runs the whole show: 1/e.

The setup: a game rigged against you

Picture a sealed deck of cards, each printed with a different number, all face down. I flip them one at a time. After each flip you say “stop” or “keep going.” Stop, and you win whatever is on the current card and the game ends. Keep going, and that card is burned forever — you can never return to it. You want to stop on the single highest card in the deck. That is the secretary problem wearing a costume, and every real version shares the same four rules.

Let us pin them down, because the whole difficulty lives in these constraints:

• Sequential. You see the N candidates one at a time, in a uniformly random order. Every arrangement is equally likely.
• Comparative only. You can rank the candidates you have seen against each other, but you get no absolute score — you never learn “this one is a 9.4/10.” All you know is relative standing so far.
• Irreversible. Each candidate is accepted or rejected the instant you see them. Accept, and the game ends. Reject, and they are gone.
• No recall. A rejected candidate cannot be summoned back, ever.
• Best-or-nothing. You win only if you stop on the single overall best. Second-best is a loss, identical to worst.

Now watch your intuition offer two strategies, and watch both die.

The greedy grab: “take the first one that looks good.” The trouble is good compared to what? The first candidate is, by definition, the best you have seen — because they are the only one you have seen. If “looks good” means “best so far,” you will hire candidate #1 every single time. Your odds of that person being the true best are exactly 1/N — for 100 applicants, a dismal 1%.

The picky hold-out: “wait for someone amazing.” Here you set a sky-high bar and refuse to settle. But you have no absolute scale, so “amazing” can only mean “beats everyone so far” — and if you demand that too late, the genuine best may have already walked past you early and been burned. Hold out too long and you are forced to take the very last candidate sight-unseen, whose chance of being best is, again, 1/N.

The look-then-leap rule

Since the game gives you information over time, the optimal strategy has a beautifully simple shape: spend the early candidates learning, then pounce. This is the look-then-leap rule, and it has exactly one knob.

Here it is in full. Pick a number r. Then:

1. Look phase (calibrate). Interview the first r candidates and reject them all, no matter how good. These are pure reconnaissance. Their only job is to show you what the field looks like. Remember the best score among these r — call it the bar.
2. Leap phase (commit). Starting with candidate r + 1, hire the first candidate who is better than the bar — that is, the first one who beats everyone in your sample. If nobody ever clears the bar, you are stuck hiring the last candidate.

Think of it like tasting wine before ordering a case. The first few glasses are throwaways — you are not buying, you are building a sense of “good.” Once you have a feel for the range, you commit to the first bottle that beats everything you have tasted. The r sample sets the standard; the leap phase acts on it.

A candidate who is the best you have seen so far is called a record (or a candidate, confusingly, in the classic literature). The look-then-leap rule is simply: ignore every record during the look phase, then grab the very first record of the leap phase. The entire art of the problem reduces to one question — how big should r be?

Working it by hand: N = 3

Abstract rules are slippery, so let us actually compute. Take N = 3 candidates with quality scores 1, 2, 3, where 3 is the best and “beats the bar” simply means a higher number. There are 3! = 6 equally likely arrival orders, each with probability 1/6. We will test three stopping rules and count, for each, how many of the 6 orders end with us hiring the top candidate, 3.

The three rules for N = 3:

• r = 0 — look at nobody, just hire the first candidate. (The greedy grab.)
• r = 1 — reject the first candidate, remember their score as the bar, then take the first later candidate who beats it (a higher score).
• r = 2 — reject the first two, then take the last candidate only if it beats the best of those two. With only one candidate left, r = 2 simply forces you onto whoever is in position 3.

The table below writes out all six orders (the sequence of scores seen), and for each rule marks the score you end up hiring. A hire of 3 is a win (✓).

Let us trace the r = 1 column so the logic is transparent, since it is the winner. Reject the first candidate, set the bar at its score, then hire the first later score above the bar:

• Order 1, 3, 2: reject the 1 (bar = 1). Next is 3, which beats 1 → hire 3. Win.
• Order 2, 1, 3: reject the 2 (bar = 2). Next is 1 (skip, not above 2); then 3 beats 2 → hire 3. Win.
• Order 2, 3, 1: reject the 2 (bar = 2). Next is 3, which beats 2 → hire 3. Win.
• Order 1, 2, 3: reject the 1 (bar = 1). Next is 2, which beats 1 → hire 2. Loss (the best, 3, was still coming but we already leapt).
• Order 3, 1, 2: reject the 3 (bar = 3). Next is 1 (skip); then 2 (skip) — nobody beats 3 → forced onto the last, hire 2. Loss (the best was sacrificed to the sample).
• Order 3, 2, 1: reject the 3 (bar = 3) → nobody beats it → forced onto the last, hire 1. Loss.

So r = 1 wins on exactly three of the six orders — those where the best candidate is not first (so it survives the sample) and no lesser candidate sneaks above the bar before it arrives. That is 3 out of 6 = 50%.

The verdict for N = 3 is clean: looking at exactly one candidate first (r = 1) wins 50% of the time, beating both the pure greedy grab (r = 0, 33%) and the forced hold-out (r = 2, 33%). Even at this toy size, “look a little, then leap” already dominates both extremes.

For completeness, N = 4 works the same way but with 24 orderings. Grinding through them, the optimal rule is to look at one candidate (r = 1), and it wins about 45.8% of the time. Notice the win rate has dropped from 50% — as N grows the game gets harder — but it is falling toward a floor, not toward zero. That floor is the punchline of the next section.

The 37% rule and why e appears

Now let N grow. For each N there is a best r, and the natural question is: what fraction of the pool should the look phase cover? Run the numbers for larger and larger N and a stunning pattern emerges — the optimal ratio r/N settles down to a constant, and so does the win probability. Both converge on the same number:

1/e ≈ 0.3679 ≈ 37%.

That is the 37% rule: as N grows, look at (reject) the first ~37% of candidates, then leap at the first record after that. And the probability you land the exact best also tends to ~37%. The constant e ≈ 2.71828 — the base of natural logarithms, the same one that governs compound interest and radioactive decay — falls out of a hiring puzzle. Why?

The intuition. Everything hinges on a tug-of-war over the sample size r:

• The sample must be big enough to set a demanding bar. A tiny sample gives a weak standard that early mediocrities clear, so you leap too soon.
• The sample must be small enough that it probably has not already eaten the best candidate. Every candidate you put in the look phase is a candidate you have promised to reject — including, sometimes, the champion.

The win happens when two things line up: (a) the true best is not in the first r (so it is still available to hire), and (b) when the best does arrive, no earlier leap has already fired on a false record. Condition (b) works out to needing the best-of-everyone-so- far, at the moment just before the true best, to sit inside the sample — so the bar is high enough that nothing between the sample and the true best falsely clears it.

The light derivation. Write the look fraction as x = r/N. A standard approximation for the win probability of the look-then-leap rule, for large N, is:

P(win) ≈ x · ln(1/x) = (r/N) · ln(N/r).

Read it as a product of two competing terms. The factor x = r/N is the chance the best candidate falls after the sample (it must, or it is unhireable) — this grows as you look more. The factor ln(1/x) captures the chance no false record leaps early — this shrinks as you look more. Their product is a hump: zero at x = 0 (you leap instantly on junk) and zero at x = 1 (you reject everyone and lose), with a single peak in between. To find the peak, ask where the product stops rising. That maximum sits exactly at:

x = 1/e ≈ 0.3679,

and plugging back in gives P(win) ≈ (1/e) · ln(e) = 1/e · 1 ≈ 0.3679 as well. The same constant governs both the ideal look fraction and the resulting win rate — a rare and elegant coincidence. (The full proof replaces the approximation with an exact sum and takes a careful limit, but the hump-with-a-peak-at-1/e picture is the honest heart of it.)

Here is the fact worth tattooing on your decision-making: the ~37% win rate barely moves as N explodes. With N = 3 it is 50%, N = 4 about 46%, and it descends smoothly toward 1/e, sitting at roughly 37% by the time N reaches a few dozen — and it stays there. Whether you are choosing among 100 apartments or 10 million, look-then-leap catches the very best roughly 37 times out of 100. A blind, one-shot, no-recall game that feels hopeless is in fact winnable more than a third of the time, at any scale.

What the 37% actually buys you

Step back and appreciate the size of the win. In a game where you are blindfolded, allowed no take-backs, and required to find the exact best out of a huge crowd, catching that best person 37% of the time is astonishing. Compare it to the honest alternatives:

The naive strategies decay to nothing as the pool grows — with a million candidates, grabbing the first (or a random pick) wins one time in a million. Look-then-leap refuses to decay. It converts the flood of candidates from a curse into fuel: more candidates means a richer sample, which means a sharper bar. That is why the win rate flatlines at ~37% instead of collapsing. You are not beating the odds by luck; you are extracting the maximum information the game will ever let you have and spending it at exactly the right moment.