37% Rule

Using the #secretary problem, we get closer and closer to 37% as the number applicants increase.

Number of Applicants	Take the Best Applicant After	Chance of Getting the Best
3	1 (33.33%)	50%
4	1 (25%)	45.83%
5	2 (40%)	43.33%
6	2 (33.33%)	42.78%
7	2 (28.57%)	41.43%
8	3 (37.55%)	40.98%
9	3 (33.33%)	40.59%
10	3 (30%)	39.87%
20	7 (35%)	38.42%
30	11 (36.67%)	37.86%
40	15 (37.5%)	37.57%
50	18 (36%)	37.43%
100	37 (37%)	37.10%
1000	369 (36.9%)	36.81%

How to optimally choose a secretary¹

Algorithms to Live By by Brian Christian and Tom Griffiths - Optimal Stopping
↩︎

2021-11-30

Secretary Problem#
- Thus, the optimal cutoff tends to \(n/e\) as \(n\) increases, and the best applicant is selected with probability \(1/e\) or roughly 37%. Hence, the 37% Rule#.
- With this algorithm, there is a 37% chance of optimally finding the best one in any given amount of applicant pool virtually consistent. However, there is, in fact, a 67% chance of failure and thus very tempting to steer away from the solution. But, put things into perspective, brute force solution isn’t remotely any better than the 37% Rule#. For instance, reviewing a hundred applicants by random with no optimization algorithm would only result to only 1% success rate, that is when the best applicant is the last one in the pool, and with a million applicants, only a whopping 0.0001% chance of finding the best one, considering the interviewer have infinite amount of time to review all applicants. Using the optimal solution, the chances of finding the best applicant approaches 37% as the applicant pool grows, or put simply, the more applicants to review the more consistent the result is going to be.
Optimal Stopping#
- The concern about optimal stopping is not choosing the right thing, but when to stop looking. Stopping too early you risk of leaving opportunities undiscovered, while stopping too late, you hold out for opportunities that might never come. Optimal stopping requires the right balance between the two. Luckily for us, mathematicians have figured out a general solution for this: The 37% Rule#
- As an example we use the #secretary problem with its simplest solution. Finding the best applicants while settling nothing for less gets complicated as the number of applicants reviewed increase. By definition, in an applicant of three, the first applicant is the best applicant reviewed so far as it is incomparable. By settling on the first, you have 33.33% of choosing the best applicant but with the goal of choosing the best one in mind, it would be optimal to skip for the second one because then there is a 50/50 chance of getting the best/worst than the first applicant–best probability we could ever hope for. Using the same logic, our odds of picking the best one decreases as the number of applicants increase, and as the number of applicants approach \(\infty\), the chances of getting the best one converge to approximately 37%#
🧠 Index 🧠#