Weekly Problem: The Second-Chance Romance Strategy

The Problem

You’ve just been through a tough breakup and are ready to start dating again. Over the next year, you’ll meet 100 potential romantic partners one by one. After meeting each person, you must immediately decide whether to commit to them or move on forever. Once you pass on someone, you can never go back to them.

You want to maximize your chances of finding your best match. How should you approach this decision problem? Is there an optimal strategy that balances exploration (meeting enough people to understand your options) with commitment (not waiting too long and missing out on great partners)?

This is a classic mathematical challenge known as “The Secretary Problem” or “The Optimal Stopping Problem,” reimagined for the dating world.

Mathematical Solution

The optimal strategy, mathematically proven, is surprisingly simple:

1. Observe phase: Reject the first 37% of candidates (about 37 out of 100)
2. Selection phase: Choose the first candidate who is better than everyone you’ve seen so far

This 37% rule (sometimes called the 1/e rule, where e is Euler’s number) balances the risk of rejecting the best match early against the risk of waiting too long and ending up with someone suboptimal.

Why does this work? The observation phase helps you establish a “baseline” of what’s available. Then in the selection phase, you have enough information to make better decisions while still having plenty of candidates left.

The probability of success (finding the absolute best match) with this strategy approaches 1/e ≈ 37% as the number of candidates approaches infinity. While this might seem low, it’s mathematically proven to be the best possible strategy for this constraint.

Discussion Questions

1. How would your strategy change if you could go back to previously rejected candidates, but each time you reject someone, there’s a 10% chance they become unavailable?
2. What if the quality of candidates isn’t randomly distributed, but follows patterns (like better candidates appearing in certain social circles or seasons)?
3. In real life, we assess potential partners on multiple dimensions, not just a single score. How would you adapt the secretary problem to handle multiple attributes?

I’ll reveal the best reader solutions next week. Good luck with your dating strategy!

Solution

The Problem Revisited

You’ll meet 100 potential romantic partners one by one over the next year. After meeting each person, you must immediately decide whether to commit to them or move on forever. Your goal is to maximize the probability of selecting the very best match among all 100 candidates.

The Optimal Strategy Explained

As presented in the initial solution, the mathematically optimal strategy follows a simple 37% rule:

1. Observation Phase: Reject the first 37 candidates (37% of the total) regardless of how good they seem. Use this phase to establish a baseline and learn what’s available.
2. Selection Phase: After candidate #37, select the first person who is better than everyone you’ve seen so far. If no such person appears, select the last candidate (#100).

The Mathematical Proof

The derivation of this 37% rule comes from solving a recurrence relation that maximizes the probability of success as the number of candidates approaches infinity:

For large n (number of candidates), the optimal stopping point r satisfies:
r/n ≈ 1/e ≈ 0.368…

Where e is Euler’s number (≈ 2.718). This is why we get the approximately 37% rule.

The probability of success (finding the absolute best match) with this strategy approaches 1/e ≈ 37% as the number of candidates approaches infinity.

Why This Works

This strategy works because:

1. The observation phase builds knowledge about the distribution of candidate quality without committing prematurely.
2. The selection phase leverages this knowledge to make an informed decision while still having enough candidates remaining (63 people).
3. The 37% threshold optimally balances the risk of rejecting the best candidate early versus the risk of passing on good candidates and ending up with someone suboptimal.

Common Misconceptions

Several readers had insightful questions about the strategy:

1. “Isn’t 37% success rate low?” While 37% might seem low, it’s mathematically proven to be the best possible probability under these constraints. No other strategy can achieve a higher success rate for finding the absolute best candidate.
2. “What if I don’t care about finding the absolute best?” If you’re satisfied with finding someone in the top 10% of all candidates, different thresholds apply. The optimal strategy changes based on your satisfaction criteria.
3. “Does this work in real life?” Real dating involves many more variables than this idealized model, but the core principle—balancing exploration and commitment—remains valuable in many decision-making contexts.