Dating Theory Calculator (37% rule: secretary problem / optimal stopping)

Understanding the dating theory (37 % rule) calculator — concept and principle

This calculator applies the secretary problem, the classic puzzle of optimal stopping often called the 37 % rule. If you meet n candidates one at a time, must decide on the spot and cannot go back, how many should you pass on first to have the best chance of choosing the very best? It also shows the success chance of your own cutoff.

How it is calculated

Strategy: reject the first k candidates, then accept the first one better than everyone seen so far.

Chance of ending with the very best: P(k) = (k ÷ n) × Σ_(j = k+1)^n 1 ÷ (j − 1), and 1/n for k = 0

Chance you pass on everyone = k ÷ n (the best was among the first k)

• The calculator evaluates P for every k and reports the best one.
• As n grows, the best k approaches n ÷ e ≈ 36.8 % of n, and the success chance approaches 1/e ≈ 36.8 %.

The principle

The first k candidates are a sample that sets your standard. Too small a sample and you accept someone good but not the best; too large and the best has likely already gone by. The formula adds up, for each position j where the best might appear, the chance that the best of the earlier j − 1 candidates was inside your sample, so you did not stop too early. Balancing those risks gives the famous 1/e. The model assumes random order, honest ranking only, and that only the very best counts as success.

Worked example

Default inputs: 10 candidates, your cutoff 3.

1. P(3) = 3/10 × (1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9) = 0.3 × 1.3290 = 39.87 %.
2. Checking every k, the best number to pass on first is 3, with 39.87 %; your cutoff therefore gives 39.87 % too.
3. Chance you pass on everyone with cutoff 3 = 3 ÷ 10 = 30 %.
4. 37 % rule estimate = 10 ÷ e = 3.7; limit for very large n = 36.79 %.

How to read the result

Even the best strategy fails most of the time: about 60 % here. Its strength is that no other rule does better under these assumptions. If settling for someone very good is acceptable, or if you can go back to earlier candidates, the best cutoff becomes smaller.

Common mistakes

• Thinking the rule guarantees the best choice.
• Applying it when candidates arrive in a non-random order.
• Forgetting the risk of ending with no one.
• Treating the model as relationship advice.

Frequently asked questions

What is the 37 % rule?

Look at about 37 % of the options without committing, then pick the first one better than all of them.

How often does it find the best?

About 37 % of the time for large n; slightly more for small n, such as 39.9 % for n = 10.

Where else is it used?

In hiring, house hunting and any choice made in sequence without going back.

Sources: T. S. Ferguson (1989), "Who Solved the Secretary Problem?", Statistical Science 4(3): 282–289