Guess half of the average

Question

Every player in a large group picks a number from $0$ to $100$ . The winner is whoever is closest to half the average (one-half, not two-thirds) of all the numbers chosen.

What number do you pick? Give the equilibrium and what actually tends to win.

Accepted Answer

The equilibrium is still 0. The multiplier 	frac12 is below 1, so the same iterated elimination of dominated strategies applies. Half of an average that is at most 100 is at most 50, so picks above 50 are dominated. But then the target is at most 25, so picks above 25 are dominated, and the ceiling collapses 50 	o 25 	o 12.5 	o \dots 	o 0. The unique Nash equilibrium is everyone choosing \mathbf{0}. What actually wins is lower than in the two-thirds game. Real crowds reason a finite number of levels, and each level now multiplies by 	frac12 instead of 	frac23, so the numbers fall faste…

Guess half of the average

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