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%#

Using this strategy, optimal solution takes the form of the Look-Then-Leap-Rule, where you “set a predetermined amount of time for ‘looking’–that is, exploring your options, gathering data–in which you categorically don’t choose anyone, no matter how impressive. After that point, you enter the ‘leap’ phase, prepared to instantly commit to anyone who outshines the best applicant you saw in the look phase”¹. With the 37% rule, in 100 applicants, we reject the first 37 random applicants to build our ordinal ranking then hire the first best applicant so far.