Imagine you’re hiring a new secretary. One hundred candidates will line up for interviews, arriving one by one in random order. After each interview, you must make a decision immediately: hire this candidate on the spot or reject them forever and move on. You cannot recall someone you rejected. The challenge? You want to maximize the probability of hiring the very best candidate.
This fascinating puzzle is known as the Secretary Problem, also called the Marriage Problem or the Sequential Hiring Decision Problem. It blends probability, decision theory, and even philosophy about decision-making in life.
Origins of the Secretary Problem
The Secretary Problem has its roots in probability theory and optimal stopping theory. Early references appeared in the 1950s and 1960s, with mathematicians and statisticians intrigued by its real-world parallels.
Although the scenario is framed in terms of hiring a secretary, the same structure appears in many contexts:
- Choosing a life partner (hence the nickname marriage problem).
- Deciding when to sell an asset in a fluctuating market.
- Picking the right apartment in a competitive rental market.
- Accepting or rejecting offers in online dating apps.
At its core, the problem asks: When should you stop searching and commit to the best option you’ve seen so far?
The Rules of the Game
- Candidates arrive in random order.
- You can rank candidates relative to those you’ve already seen, but you don’t know their absolute quality in advance.
- You must decide immediately after each interview to accept or reject the candidate.
- Rejected candidates cannot be recalled.
- Your goal: maximize the probability of selecting the very best candidate overall.
The Surprising Optimal Strategy
Mathematicians proved that the optimal strategy follows a neat formula:
- Reject the first ~37% of candidates (observe only).
- Then, hire the first candidate who is better than all the ones you’ve seen so far.
This is sometimes called the “37% rule” because as the number of candidates approaches infinity, the cutoff fraction tends toward 1/e ≈ 0.3679 (≈ 37%).
Example: 100 Candidates
- Interview the first 37 people and reject them all (no matter how good they seem).
- Starting with candidate 38, hire the first person who is better than everyone in the first 37.
- Following this strategy gives you about a 37% chance of hiring the single best candidate.
While 37% may not sound impressive, it’s mathematically the highest possible success rate in this problem structure. Any other strategy performs worse on average.
Why 37%? The Math Behind It
The reasoning involves probability and calculus:
- If you stop too early, you don’t have enough data to know what a “good” candidate looks like.
- If you wait too long, you risk missing the best candidate who already passed by.
- The sweet spot balances exploration (gathering data) and exploitation (acting on what you’ve learned).
This trade-off is why the strategy is often studied in optimal stopping theory — a mathematical field concerned with the timing of decisions.
Real-Life Applications
The Secretary Problem isn’t just an academic curiosity. Variants of it appear in surprising areas:
- Dating and Marriage: If you’re choosing a life partner from many possibilities, should you settle early or keep looking? The 37% rule suggests you should use the first third of relationships to “learn” and then commit when you find someone better than all before.
- Apartment Hunting: Touring rentals in a city? The same rule applies: don’t grab the first decent place, but don’t wait until the last minute either.
- Online Auctions: Sellers deciding whether to accept bids now or wait for potentially better offers face the same challenge.
- Hiring Decisions in Business: Recruiters sometimes unknowingly use similar cutoffs when screening a pool of candidates.
Criticisms and Limitations
Of course, the Secretary Problem is a simplification of real life. In reality:
- We often have imperfect information (we can’t perfectly rank candidates).
- The number of candidates may be unknown in advance.
- Sometimes we can reconsider rejected options (e.g., reaching out to a past candidate again).
- Humans have emotions, intuition, and biases that don’t fit neatly into equations.
Still, the model is powerful because it reveals how probability can guide rational decision-making under uncertainty.
Fun Variants of the Problem
Over time, mathematicians have explored creative twists:
- Unknown Number of Candidates: What if you don’t know how many total applicants will appear?
- Costs of Waiting: Each interview has a cost — how does that affect the strategy?
- Multiple Selections: What if you want not just the best candidate but the top 5%?
- The Online Dating Twist: Some versions allow “recall” of past candidates, but at an increasing cost.
These variants bring the problem closer to real-world decision-making and highlight how mathematical insights can adapt to practical settings.
The Philosophy of Optimal Stopping
The Secretary Problem teaches more than math — it gives us a metaphor for life decisions:
- Don’t commit too early. You need time to learn the landscape.
- Don’t wait forever. Opportunities pass, and hesitation can cost you.
- Balance patience and action. The art of decision-making lies in knowing when to switch from observing to doing.
As the saying goes: “He who hesitates is lost, but fools rush in.” The Secretary Problem shows how math can help find the middle ground.
Conclusion
The Secretary Problem — whether framed as hiring a secretary, choosing a spouse, or finding the perfect apartment — is one of the most elegant examples of applied probability.
It illustrates how mathematical reasoning helps us make decisions under uncertainty, and why balancing exploration with commitment is crucial in both mathematics and life.
So the next time you’re faced with a big choice, remember the 37% rule: observe, learn, and then be ready to act when the right opportunity arrives.