The Secretary Problem: The 37% Rule for Optimal Stopping

Every day, we face decisions where we cannot go back. You're apartment hunting and you've just toured a genuinely nice place — but you have 20 more scheduled. Do you take it? Or keep looking and risk losing it forever?

This isn't just a frustrating life experience. It's a precise mathematical problem with an exact solution.

The Secretary Problem is one of the most elegant puzzles in all of mathematics. It sounds like a riddle from a job interview. It looks like a question about dating or apartment hunting. But beneath its surface lies a deep truth about decision-making under uncertainty — and the answer involves one of the most famous constants in mathematics.

The magic number is 37%.

The Concept

Imagine you're a manager who needs to hire a secretary. You have 100 applicants who will interview one by one in a random order. After each interview, you must immediately decide: hire this person or move on. You cannot revisit a rejected candidate. You can rank the current person against everyone you've already seen, but you have no idea how they compare to candidates you haven't met yet.

What's your strategy?

Most people's instincts are wrong. "Wait for the obviously great one" — but how will you know they're great if you have no baseline? "Hire the first decent candidate" — but what if someone far better is coming? The classic errors are moving too fast (committing before you know what's out there) and moving too slowly (holding out so long the best options are gone).

The optimal strategy is this:

1. Observe the first 37 candidates without hiring anyone. Use this as your calibration phase — you're building a benchmark for what "good" looks like in this particular pool. 2. Then, from candidate 38 onward, hire the very first person who is better than everyone you saw in step one.

That's it. That's the algorithm.

The number 37 isn't arbitrary. It comes from 1/e, where e is Euler's number (approximately 2.718). If you have n candidates, observe the first n/e of them — which works out to roughly 36.8%, or about 37%. Skip too few and you're flying blind when you start selecting. Skip too many and the best candidates are already gone.

The History: A Problem Born in Folklore

The Secretary Problem has a murky but fascinating origin story — fitting for a puzzle about searching and not knowing what's out there.

Mathematician Merrill Flood first described something like it in a 1949 lecture, calling it the "fiancée problem." The problem circulated in mathematical folklore for over a decade, passed among researchers at cocktail parties and in hallways, without anyone publishing it formally.

Then in February 1960, Martin Gardner featured it in his legendary Mathematical Games column in Scientific American. He'd learned of it from John Fox and Gerald Marnie, who called their version the "game of googol" — a nod to the absurdly large numbers involved if you scale the pool up. Gardner asked readers for the solution, then published mathematician Leo Moser's elegant proof, and the world took notice.

(A related but different stopping problem had appeared as far back as 1875 in the work of Arthur Cayley, the Victorian mathematician famous for matrix theory. But Cayley's version had different rules, and Flood's formulation is the one that became canonical.)

Since Gardner's column, the problem has been renamed and rebranded constantly: the marriage problem, the sultan's dowry problem, the best choice problem, the fussy suitor problem. The core mathematics is the same under all the names.

Why It Matters

This strategy seems counterintuitive — you're deliberately passing on potentially great candidates in the first 37% of your pool, with no guarantee you'll ever see anyone as good. But the math shows it gives you the highest possible probability of landing the best candidate overall: roughly 37% (that 1/e appearing again).

Think about why this makes intuitive sense:

If you hire too early, you risk picking someone mediocre before you've seen what's out there.
If you wait too long, the best candidate may already be gone.
The 37% mark is the exact sweet spot where these two errors perfectly balance.

And here's something that surprises most people: the strategy works just as well whether you have 10 candidates or 10,000. The optimal fraction to skip is always 1/e — about 37%. The rule is universal, scale-free, and timeless.

That universality is the deepest thing about optimal stopping. The problem doesn't care how many options you have. The geometry of the situation is always the same.

The Details: Why 1/e?

Let's look at why the answer is 1/e — and why it's so beautiful that this number appears twice.

Suppose you have n candidates and decide to observe the first r without hiring, then take the next person who beats all of them. What's the probability this lands you the best candidate?

The best candidate can be anywhere in the lineup. For your strategy to succeed: 1. The best candidate must appear after your observation window (positions r+1 through n). 2. The best candidate among positions 1 through the winner's arrival must be inside your observation window — otherwise, someone mediocre in the gap would fool you into stopping too early.

Careful probability accounting gives:

P(success) ≈ (r/n) × ln(n/r)

To maximize this, take the derivative with respect to the ratio r/n and set it to zero. The maximum occurs when r/n = 1/e, yielding a maximum success probability of exactly 1/e ≈ 36.8%.

That 1/e showing up as both the optimal fraction to skip and the resulting success probability is one of those mathematical moments that feels like the universe winking at you. The same constant governs how long to look and how likely you are to succeed.

Three Surprising Places Optimal Stopping Appears

Apartment Hunting

You have one week to find an apartment in a new city and 20 places scheduled. Tour the first 7 without committing — that's 37% of 20, rounded down. Then take the first apartment that beats all seven. This gives you the best mathematical chance of landing the top option, even though you had no idea how good the pool was when you started.

The same logic applies to buying a house, accepting a job offer, or even choosing which restaurant to try in an unfamiliar city.

Dating and Long-Term Relationships

The "marriage problem" framing asks: if you'll have a certain number of serious relationships over your lifetime, how many should you experience before committing? The 37% rule says to treat roughly the first third as calibration — real relationships, real feelings, real learning — and then commit to the next person who is genuinely better than everyone before them.

Of course, real relationships are far messier than math. Rejected partners don't always vanish forever. Compatibility isn't a single ranking. Feelings and circumstances change. But the underlying insight is real: some period of observation before commitment leads to better outcomes than either jumping at the first option or endlessly searching for something perfect.

Financial Options and Timing Markets

Optimal stopping theory has deep roots in finance. Pricing an American stock option — one that you can exercise at any time before expiration — is mathematically equivalent to a stopping problem. When do you exercise? Too early and you leave money on the table. Too late and conditions may turn against you. The same mathematical structure that governs the secretary problem governs the optimal exercise of financial instruments.

The Limitations (and Why They're Interesting)

The Secretary Problem's model is deliberately stark:

One-way rejections: once you say no, it's final.
Perfect ranking: you always know if the current candidate is better than previous ones.
Single winner: the classic version gives no credit for landing second-best.
Known pool size: you know n in advance.

Real-world decisions relax all of these. Companies call back candidates. Rankings are subjective and multidimensional. Pools are often unknown in size. And "good enough" frequently beats "holding out for perfect."

Researchers have extended the Secretary Problem to handle every variant. Unknown pool size? There's a modified strategy. Can you recall rejected candidates with some probability? The math shifts. Do you want to maximize your expected rank rather than just your chance at the top? Different answer. Want to "satisfice" — find someone above a threshold rather than the absolute best? Different again.

Each extension is beautiful mathematics in its own right. But the bare-bones original problem stays famous precisely because its stark constraints expose the pure logic of optimal stopping — the universal tension between exploring (gathering information) and exploiting (committing based on what you know).

This explore-exploit tension isn't just a math puzzle. It's the fundamental challenge of learning anything in a world that doesn't wait for you.

Takeaways

The 37% Rule: in any sequential selection with no recall, observe the first 37% of candidates to calibrate, then take the first candidate better than your best so far.
The magic of 1/e: the optimal stopping fraction and your resulting success probability are both approximately 1/e ≈ 36.8% — the same constant that appears in compound interest, probability theory, and calculus.
Scale-free and universal: whether you have 10 candidates or 10,000, the optimal strategy stays the same fraction. The rule doesn't care how big your pool is.
Calibration is not wasted time: the observation phase builds the benchmark you need. You cannot know what "good" looks like without seeing enough options first.
The explore-exploit tradeoff: every decision under uncertainty faces this tension — search more or commit now? The Secretary Problem gives it a precise mathematical form, and the answer (37%) is the same wherever the problem appears.

Resources: - Secretary Problem — Wikipedia - Who Solved the Secretary Problem? — Thomas S. Ferguson - Knowing When to Stop — American Scientist - Optimal Stopping and the Secretary Problem — To Summarise