Anyone who has gone house hunting knows the dilemma. The place you just saw is pretty nice, but should you sign right away, or keep looking in case something better comes along? Rush, and you may miss a better chance; take too long, and the good places you passed up end up going to someone else.
The classic problem that works out mathematically when to stop and choose is the “Secretary Problem.” It asks how to maximize the probability of picking the very best person when you interview applicants one at a time and must decide on the spot whether to hire each one. Its answer is often called the “37% rule.”
In this article, we will look in turn at the rules of the secretary problem, the principle behind the 37% rule, the history of how the problem became known, and the points to keep in mind when applying it to real life.
When to Stop and Choose: The Secretary Problem
The Rules of the Secretary Problem
The secretary problem starts from a few simple rules. Taken together, these rules make choosing far harder than it seems.
(1) A fixed number of applicants
There is one position to fill, and you know in advance how many applicants there are in total. It is assumed that, if all the applicants were gathered together, they could be clearly ranked from best to worst.
(2) Interviews in random order
Applicants are interviewed one at a time in random order. The interviewer can only tell how the current applicant ranks among those seen so far, and has no way of knowing how good the applicants not yet interviewed are.
(3) Irreversible decisions
Right after each interview, you must decide whether to hire or reject the applicant, and anyone you reject cannot be called back. The goal is just one thing: to hire the single best person of all.
The Principle Behind the 37% Rule
The best strategy for this problem is surprisingly simple: just observe the first few, then make your move.
(1) Just look around at first
Reject the first roughly 37% of all applicants, no matter how much you like them. This stage is the time to set a benchmark for how good the applicants are. Look around too briefly and your benchmark will be low, so you may jump at someone ordinary; look around too long and the best person is more likely to have already been rejected.
(2) Pick the first one who beats the benchmark
From then on, hire the first applicant who is better than everyone you have seen before. Doing this gives you about a 37% chance of picking the best person. That is a very high probability, considering that picking at random would give you only a 1% chance of choosing the best out of 100.
(3) The 37% that comes from the number e
The figure 37% comes from 1/e, the reciprocal of the mathematical constant e, which is about 0.368. Whether there are 100 applicants or 100 million, this ratio stays almost the same. With 10 applicants, rejecting the first 3 and then choosing gives the highest probability, about 40%. However, since it is possible that no one beats the benchmark right to the end, this strategy also carries a risk of failure.
How the Secretary Problem Became Known
The secretary problem did not begin with a single paper; rather, it spread by being passed from person to person.
- The Fiancée Problem: The name introduced in a lecture in 1949 by the American mathematician Merrill M. Flood. It was not published as a paper at the time, but it spread widely among mathematicians by word of mouth.
- The Game of Googol: The version that Martin Gardner introduced in February 1960 in his mathematics column for the science magazine Scientific American. It is a game in which you turn over face-down slips of paper one by one and stop at the largest number, and it is known as the first printed record of the problem.
- Kepler’s Marriage Problem: After losing his first wife, the astronomer Johannes Kepler considered 11 prospective brides over two years, from 1611 to 1613. It is not the same problem, but it is often cited as an old example of a similar dilemma.
It has also been called by many other names, such as the marriage problem and the sultan’s dowry problem, and in 1989 the statistician Thomas S. Ferguson sorted out its tangled history in his article “Who Solved the Secretary Problem?”
Points to Keep in Mind in Real Life
The 37% rule is appealing, but there are a few things to consider before applying it directly to everyday life.
(1) When you don’t know the total
In real life, it is hard to know in advance how many applicants or candidates there are in total. Without knowing the total, the probability of picking the best one is generally lower, and many variants of the problem adapted to such situations have been studied.
(2) When the best isn’t necessary
The secretary problem counts only first place as success. In reality, though, hiring someone quite good is often satisfying enough, and when the goal changes this way, deciding earlier becomes the better choice.
(3) People who stop too early
When similar situations were tested in experiments, people generally tended to stop searching earlier than the optimal point. This is interpreted as the time and effort of examining each candidate influencing their choices. Choosing a gas station on the highway or searching for plane tickets without comparing enough options and deciding in a hurry are cited as similar examples.
The secretary problem shows in numbers the balance between looking further and choosing now. Even if the answer is not always 37%, the attitude of looking around enough, setting a benchmark, and then committing helps in many moments of choice.
Knowing when to stop is the first step toward a choice without regrets.
