The Worst Hiring Process Ever Invented
In 1958, the mathematician Merrill Flood posed a problem to his colleagues that was so impractical, so absurdly constrained, so flagrantly divorced from any real-world hiring process that it could only have come from a mathematician. It goes like this.
You need to hire a secretary. You have a hundred applicants. Here are the rules — and I want to stress that these are terrible rules, the kind of rules that would get you hauled before an employment tribunal in most civilized countries, but bear with me because the math is gorgeous.
You interview candidates one at a time, in random order. After each interview, you can rank the current candidate against everyone you've already seen — but you have no idea where they fall in the overall pool. Maybe the first person is the best you'll ever meet. Maybe they're the worst. You don't know. You can't know. And here's the knife: you must decide, right there in the room, whether to hire this person or pass. If you pass, they're gone. No callbacks. No second chances. The door closes and stays closed.
Your goal? Hire the single best candidate out of all one hundred. Not "pretty good." Not "top five." The best. Second-best is failure.
The problem surfaced in Martin Gardner's Scientific American column in February 1960, where he called it "the game of Googol."1 But the mathematical question — how do you choose the best from a sequence you can only see once? — had been circulating among statisticians since at least the early 1950s. The name "secretary problem" stuck, an artifact of mid-century office culture that, like the math itself, has stubbornly refused to go away.
Now think about the extremes. If you hire the first person you see, you've got a 1-in-100 chance of getting the best. That's the lottery approach: no information, pure luck. If you interview everyone, waiting for some miracle of certainty — well, you can't go back, so you're stuck with candidate #100, who is almost certainly mediocre. That's the analysis-paralysis approach: all information, no action.
Somewhere between "commit immediately" and "wait forever" lies the sweet spot. The question is where. And the answer, when it arrives, is one of the most elegant results in all of mathematics.
The 37% Rule
The answer — and I promise I'm not making this up — is to reject the first 37% of candidates, no matter how brilliant they are. Let them go. Thank them for their time. Use them as your calibration set, your education, your tuition paid to the universe. Then, starting with the next candidate, hire the first person who's better than everyone you've seen so far.
That's it. That's the optimal strategy.
And it has a name so elegant it almost seems like a prank: the 1/e rule, where e is Euler's number, approximately 2.71828 — the same constant that governs compound interest, radioactive decay, and the shape of a hanging chain. Since 1/e ≈ 0.3679, you get the "37% rule."
Then pick the first one better than all r
And the probability of selecting the very best candidate with this strategy? Also about 1/e. Also about 37%.2
Now, 37% might not sound like great odds. You'll fail nearly two-thirds of the time. But context matters: if you picked randomly, you'd have a 1% chance. The 37% rule improves your odds by a factor of 37. And here's the clincher — it's optimal. No other strategy, no matter how clever, can beat it. This isn't a rule of thumb or a heuristic. It's a theorem.
As n grows — a thousand candidates, a million, a billion — the optimal fraction to reject stays at 1/e, and the success probability stays at 1/e. The problem scales perfectly. Whether you're choosing among 10 applicants or 10 million, the strategy is identical: explore 37%, then commit to the first thing that exceeds your benchmark.
There's something deeply satisfying about this. The universe, in its infinite complexity, has handed us a problem with a clean answer. Not an approximate answer, not a "depends on the situation" answer — a precise, universal, provably optimal answer. These are rare in life. Enjoy it.
The optimal strategy splits the search into two phases: learn for 37%, then leap at the first candidate who exceeds your benchmark.
Why e? (A Derivation You Can Follow)
Let's see where that 1/e comes from. I promise: no calculus beyond a single derivative, and even that you can skip if you trust me.
Suppose there are n candidates. Your strategy: reject the first r (the "look" phase), then hire the next one who's the best you've seen (the "leap" phase). What's the probability this strategy finds the overall best?
The best candidate is equally likely to be at any position. Call their position i. There are two cases:
Case 1: The best is in the look phase (i ≤ r). You rejected them. You're out of luck. This happens with probability r/n.
Case 2: The best is in the leap phase (i > r). You'll hire them only if no one between positions r+1 and i−1 was also "best so far." Put differently: the best of the first i−1 candidates must fall in the first r positions. The probability of this is r/(i−1).
So the total probability of success is:
Each term r / (n(i−1)) is the probability that the best candidate is at position i (which is 1/n) times the probability you'd hire them (which is r/(i−1)). The sum telescopes into something clean:
P(r) ≈ (r/n) · ln(n/r)
Now we want to maximize this. Let x = r/n (the fraction you reject). Then P ≈ −x · ln(x). Take the derivative, set it to zero: −ln(x) − 1 = 0, which gives x = 1/e. Plug back in: P = 1/e ≈ 0.368.
Euler's number e shows up not because we went looking for it, but because it's the natural constant of growth and decay — it appears whenever a system balances accumulation against expenditure. Here, you're balancing information gained (which grows with each candidate you see) against opportunity lost (which also grows, because each rejected candidate might have been the one). The equilibrium of those two competing forces sits at exactly 1/e. It's the same tension that makes e appear in compound interest, in the Poisson distribution, in the normal curve. It is, in a sense, nature's answer to the question: how much should you explore before you commit?
The success probability curve: reject too few and you're guessing blind; reject too many and the best is gone. The peak sits at exactly 1/e.
Try It Yourself
Enough theory. Let's play the actual secretary problem. Candidates appear one at a time. You see only how each ranks against those you've already met. Your job: find the best. The progress bar shows a green marker at the 37% threshold — the mathematically optimal point to switch from exploring to deciding.
Run the simulation a few times. The peak always hovers around 37% — confirming that the math isn't just theory. It's reality.
Dating, Apartments, and the Rest of Your Life
The secretary problem isn't really about secretaries. It never was. It's about every decision where your options arrive sequentially, where saying "no" is permanent, and where you're trying to find the best in a sea of unknowns.
The psychologist Peter Todd has argued that the 37% rule approximates what humans actually do when dating. If you figure you'll have about 12 serious relationships between ages 18 and 35 (let's be generous), the math says: date the first 4 or 5 with no intention of settling down. Use them to calibrate your expectations. Then commit to the next person who exceeds everyone you've dated so far.
Todd found that people's actual behavior tracks this pattern surprisingly well.3 We seem to have an intuitive sense of when the "exploration phase" should end — and it kicks in at roughly the right time. Evolution, it appears, approximated optimal stopping theory long before mathematicians formalized it.
Apartment hunting in a hot market is another classic. Listings appear and vanish within hours. You can't take a day to think about it — another couple is already signing the lease. The optimal strategy: spend the first 37% of your search time just looking. Don't take anything. Establish your benchmark. Then grab the first place that exceeds it.
Job offers work the same way, but in reverse — now you're the candidate, and you're deciding when to accept. The math doesn't care which side of the table you're on.
| Domain | Pool Size | Explore Phase | Then… |
|---|---|---|---|
| Hiring (100 applicants) | 100 | Interview first 37 | Hire next one who's best yet |
| Dating (age 18–35) | ~12 | Date first 4–5 freely | Commit to next best-yet partner |
| Apartment search (30 days) | ~30 days | Just browse for 11 days | Sign for next best-yet place |
| Any sequential choice | n | Observe first n/e | Take next record-setter |
The deeper lesson here is about the value of information. Those first 37% of candidates aren't wasted — they're your education. You're paying tuition. The tragedy of the over-eager is committing before you know what's possible; the tragedy of the over-cautious is learning the whole landscape and watching the best option walk out the door.
When the Rules Change
The classic secretary problem is elegant but fragile. Real life, as usual, is messier. What happens when we loosen the screws?
What if you can go back?
In the real world, rejected candidates sometimes accept a recall offer. Maybe the other job fell through. Maybe your ex answers the phone. If there's a probability p that a rejected candidate will come back, the optimal strategy shifts: you should explore less. The safety net makes you rationally bolder. When p = 1 — everyone comes back if you ask — the problem becomes trivial. Just interview everyone and pick the best.4
This has a beautiful real-world implication. In markets where recall is easy (online dating, where you can re-message someone), you should explore less aggressively than in markets where it's impossible (auction bidding, where the hammer falls and it's over).
What if looking costs something?
Each interview costs c — in money, time, or emotional energy. (If you've been on enough first dates, you know that c is not zero.) With search costs, you should stop earlier. The more expensive the search, the more willing you should be to settle. The 37% threshold drops, and the strategy smoothly transitions from "be picky" to "take what you can get" as costs rise.
This is the mathematical justification for a feeling everyone has: at some point, the process of searching becomes worse than accepting an imperfect result. The math just tells you precisely when.
What if "good enough" counts?
The original problem demands perfection — only the absolute best will do. But if you'd be happy with a top-10% candidate, the problem gets much easier and your success probability jumps dramatically. This is sometimes called the postdoc variant, because in real academic hiring, nobody insists on the literal best candidate; they want someone excellent.5
Herbert Simon won the Nobel Prize in Economics partly for the concept of satisficing — choosing the first option that meets a threshold of acceptability rather than searching for the absolute best. The secretary problem vindicates both approaches: if you want the best, the 37% rule is optimal; if you want someone great, you can explore less and succeed more often. The math tells you the price of perfectionism.
What if you don't know how many there are?
Perhaps the most realistic variant: you don't know n. You don't know how many candidates, apartments, or dates you'll encounter. Here, the math gets harder but the qualitative answer stays the same: use your early experiences to calibrate, then act when something exceeds your expectations. The 37% rule adapts to a time-based version — explore for the first 37% of your available time, then commit.6
Here's something counterintuitive: the more candidates there are, the better the 37% rule performs relative to random guessing. With 3 candidates, the optimal strategy barely outperforms chance — about 50% versus 33%. With 1,000 candidates, you're doing 370× better than random. Scale is the friend of strategy. In a world of abundant options, systematic thinking pays exponentially.
The Explore-Exploit Boundary
The secretary problem is, at its core, about a tension that defines much of human life: the tension between exploring and exploiting. Every time you try a new restaurant instead of returning to your favorite, every time you read an unfamiliar author instead of rereading a beloved book, you're navigating this tradeoff.
Computer scientists call it the explore-exploit tradeoff, and it shows up everywhere: in how Netflix recommends movies, how clinical trials allocate patients to treatments, how Google orders search results, how a baby decides whether to put something in its mouth. The secretary problem is the purest distillation of this tension — all exploration, then all exploitation, with a single sharp cutoff at 1/e.
And the answer it gives us is strangely comforting. You should explore. Not endlessly, not recklessly, but deliberately and with a plan. About 37% of the time. Then trust what you've learned and act decisively.
There's a melancholy hidden in the math. Even the optimal strategy fails 63% of the time. The best possible approach to life's biggest decisions still leads to the wrong answer more often than not. If that sounds depressing, consider the alternative: without any strategy, you'd succeed 1% of the time. And consider what "failure" means here — the 37% rule doesn't just maximize your chance of finding the best; conditional on failing, it still tends to find candidates near the top. The expected rank of your hire, even averaged over failures, is remarkably good.7
The secretary problem doesn't promise you'll find the best. It promises that your regret will be minimized — that you'll have given yourself the best shot possible, given the fundamental uncertainty of the world.
And maybe that's all any of us can ask of mathematics: not certainty, but the best possible odds in the face of irreversible choices and incomplete information.
Notes & References
- Gardner, M. "Mathematical Games: New Mathematical Diversions," Scientific American, February 1960. Gardner's column introduced the problem to a popular audience. The mathematical formulation traces to Flood and others in the early 1950s.
- The rigorous proof appears in Ferguson, T.S. (1989), "Who Solved the Secretary Problem?" Statistical Science, 4(3), 282–289. The 1/e result was independently discovered by Lindley (1961), Dynkin (1963), and others. Ferguson's paper is also a fascinating case study in how mathematical results get attributed — at least four people derived the answer independently.
- Todd, P.M. (1997), "Searching for the Next Best Mate," in Simulating Social Phenomena, Lecture Notes in Economics and Mathematical Systems, vol 456. His empirical findings on real dating behavior are both validating and slightly terrifying — we're running optimal stopping algorithms without knowing it.
- The recall variant was studied by Yang, M.C.K. (1974), "Recognizing the maximum of a sequence based on relative rank with backward solicitation," Journal of Applied Probability, 11(3), 504–512.
- For the threshold variant, see Bearden, J.N. (2006), "A new secretary problem with rank-based selection and cardinal payoffs," Journal of Mathematical Psychology, 50(5), 481–487. Also Freeman, P.R. (1983), "The secretary problem and its extensions: a review," International Statistical Review, 51(2), 189–206.
- The unknown-n variant is treated in Bruss, F.T. (2000), "Sum the odds to one and stop," Annals of Probability, 28(3), 1384–1391. Bruss's "odds algorithm" elegantly generalizes optimal stopping to cases where the number of options is itself uncertain.
- For a beautiful exposition of the secretary problem and the broader explore-exploit framework, see Chapter 1 of Christian, B. and Griffiths, T., Algorithms to Live By (2016). They show that the 37% rule's "failures" are not as bad as they sound — the expected rank of the selected candidate is far better than average.