Guess two-thirds of the average

Question

A room full of rational players each secretly writes down a number from 0 to 100. The organizer averages all the numbers and multiplies by two-thirds. Whoever's guess is closest to that target wins.

If everyone reasons perfectly, what number should you pick, and why?

Accepted Answer

The unique answer for perfectly rational players is 0. Peel the logic one layer at a time. The target is 	frac{2}{3} of the average, and the average can be at most 100. So the target can be at most 	frac{2}{3}	imes 100 \approx 66.7. Any guess above 67 is pointless, it can never be closest. So no rational player picks above 67. Now iterate. If nobody picks above 67, the average is at most 67, so the target is at most 	frac{2}{3}	imes 67 \approx 44.7. So no one should pick above 45. Repeat: the ceiling on sensible guesses drops to \approx 30, then \approx 20, then \approx 13, and so on. Ea…

Guess two-thirds of the average

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