The 37% rule for apartment hunting

Question

You will view 12 apartments one at a time in random order. After each viewing you must sign now or walk away for good; landlords do not hold units. You want the single best apartment of the twelve. How many should you view before you start signing, and what is your chance of landing the best?

Accepted Answer

The optimal policy views and rejects the first r - 1 apartments purely to calibrate, then signs the first "record" that beats all of them. The win probability for cutoff r over n apartments is P(r) = \frac{r-1}{n}\sum{i=r}^{n}\frac{1}{i-1}. The asymptotic optimum rejects about n/e options: here 12/e \approx 4.4, so try a learning phase of 4. Checking the neighbors confirms it: - r = 5 (reject the first 4): P = 	frac{4}{12}\left(	frac14 + 	frac15 + \cdots + 	frac{1}{11}ight) \approx 	frac13 (1.187) \approx 0.396. - r = 4 (reject the first 3): P \approx 	frac{3}{12}(1.520) \approx 0.380…

The 37% rule for apartment hunting

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