Best-of-five, the secretary cutoff

Question

Five distinct-valued offers are revealed one at a time in random order. After each, you must take it now (ending the game) or reject it forever. If you reach the fifth you must take it. You win only if you take the single highest offer.

What cutoff maximizes your chance of catching the best, and what is that probability?

Accepted Answer

The optimal policy skips the first r - 1 offers, then accepts the first "record" (an offer better than all before it). For n offers the win probability of cutoff r is P(r) = \frac{r-1}{n}\sum{i=r}^{n}\frac{1}{i-1}. With n = 5: - r = 1 (take the first): P = 1/5 = 0.200. - r = 2: P = 	frac15\left(	frac11 + 	frac12 + 	frac13 + 	frac14ight) = 	frac15 \cdot 	frac{25}{12} = 	frac{25}{60} \approx 0.417. - r = 3: P = 	frac25\left(	frac12 + 	frac13 + 	frac14ight) = 	frac25 \cdot 	frac{13}{12} = 	frac{26}{60} \approx 0.433. - r = 4: P = 	frac35\left(	frac13 + 	frac14ight) = 	fr…

Best-of-five, the secretary cutoff

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