When the target is above the average

Question

Every player in a large group picks a number from $0$ to $100$ . This time the winner is closest to $1.2$ times the average of all numbers chosen (a multiplier above one, not below).

What number do you pick, and where is the equilibrium now?

Accepted Answer

The equilibrium is everyone picks 100. With a multiplier above 1, the target always sits above the current average (since 1.2 	imes 	ext{avg} \ge 	ext{avg} for any nonnegative average). So whatever you believe the average is, you want to guess higher than it, which pushes the whole group upward. Iterated elimination now runs from the bottom up: - The average is at least whatever the smallest rational pick is; since the target exceeds the average, any low pick is beaten by a higher one. Low numbers are dominated. - The pressure keeps ratcheting up until it hits the ceiling. At everyone pick…

When the target is above the average

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