Optimal Stopping (37% Rule)
In the classic sequential hiring model, the 37% Rule is a large-pool strategy: observe the first 37% without choosing, then accept the next candidate better than everyone seen so far. It depends on a known pool, random order and no recall of rejected candidates.
THE IDEA
Optimal stopping problems involve deciding when to stop evaluating a series of options to maximize the probability of selecting the best one. The classic 'secretary problem' assumes you interview applicants in random order, know the number of applicants, can judge their relative ranks, and cannot recall rejected applicants. The optimal solution is the Look-Then-Leap Rule: look without committing at the first 37% of applicants, then leap at the first applicant who is better than all previously seen. As the pool grows, the probability of selecting the best approaches 37%; the exact optimal cutoff and success probability depend on a finite pool’s size.
The Secretary Problem
First appearing in Martin Gardner's 1960 Scientific American column, the secretary problem asks how to maximize the chance of hiring the best secretary from a sequential pool. The key is balancing looking (gathering data) and leaping (committing).
Whence 37%?
The optimal strategy converges to 37% as the pool grows. This is derived from a simple rule: set a 'look' phase for the first 37% of applicants, then accept the next best-yet. The math shows this balances the risks of stopping early (missing the best) and stopping late (waiting for a better candidate that doesn't exist).
Variants
- Rejection: The excerpt gives a model in which proposing after 25% of the search yields a 25% success rate. It does not specify the acceptance assumptions needed for this result.
- Recall: The excerpt quotes a 61% observation period and a 61% success rate for a recall variant, without specifying how likely earlier candidates are to remain available. If every candidate could always be recalled, one could inspect the whole pool and choose the best.
- Full Information: With objective scores (e.g., percentiles), use a Threshold Rule: accept if above a certain percentile depending on remaining options. The cited model gives a 58% success rate; this is not a guarantee for arbitrary score distributions.
- Selling a House: Offers with known dollar values and waiting costs; set a threshold once based on cost of waiting, then accept the first offer above it.
- Parking: On an infinite street with occupancy rate, use Look-Then-Leap: pass spots until a certain distance from destination, then take the first empty spot. The switch point depends on occupancy.
- Quitting (Burglar Problem): For repeated risky actions (e.g., robberies), stop after a number of successes roughly equal to success probability divided by failure probability. The quoted 90% success chance produces a cutoff of 9 robberies in the model, whose payoff and loss assumptions are not supplied here.
The Role of Time
Real-life decisions often include time costs, causing people to stop early. Waiting can bring useful information, but it can also cost money, time or an opportunity. Those costs help determine when a search should end.
When to use it
Hiring the best candidate from a pool of applicants.
Choosing a life partner (with caveats about model assumptions).
Selling a house or other asset with sequential offers.
Finding a parking spot in a crowded urban area.
Deciding when to quit a risky venture or investment.
FROM THEORY TO EVERYDAY LIFE
In practice
College Freshman's Dilemma
Brian, a college freshman, felt uncertain about his long-distance high-school relationship. His counselor advised 'gather data,' recognizing the dilemma as an optimal stopping problem: how many people to date before knowing the best match?
Mathematician's Proposal
Michael Trick, a graduate student, applied the 37% Rule to his love life, deciding to propose at age 26.1. Though rejected, he later found his spouse in a bar—a reminder that models aren't perfect.
Selling a House with Math
Laura Albert McLay used optimal stopping to sell her house, holding out for the right offer despite temptations to accept early ones. Her knowledge of the math gave her confidence to wait.
Parking in Los Angeles
The parking problem uses a threshold based on occupancy and distance from the destination. With fewer empty spaces, searching farther away can be sensible; it is not a fixed 37% rule for every street. The excerpt notes that Donald Shoup bikes to avoid the problem.
A MOMENT TO REFLECT
Put it to work.
Think of a recent decision where you had to choose from a sequence of options (e.g., apartment hunting, job offers). Did you stop too early or too late? How might the 37% Rule have helped?
Consider a situation where you had to decide when to quit an activity (e.g., a project, a game). How does the burglar problem's logic apply to your risk assessment?
How does the cost of time affect your decision-making in searches? Can you identify a time when boredom or impatience led you to stop prematurely?
Often confused with
References
- Scientific American column (1960) — Martin Gardner (1960)
- The Oligarchs — David Hoffman (2002)
- The High Cost of Free Parking — Donald Shoup (2005)
CONTINUE THE CONVERSATION
Source texts
- Algorithms to Live By The Computer Science of Human Decisions (Brian Christian, Tom Griffiths)