Question

Dominance: Ten players simultaneously and noncooperatively each pick a real number in [0,1]. Let their choices be X1, X2, ....X10. Let X be the average of all the x's. The game proceeds as follows. Everyone picks their value x. Then the average X is calculated. Whichever of the players guesses closest to 2/3 the value of X wins $1 and all other players get nothing. If there is a tie, then they flip coins for a single winner. Can this game be solved by eliminating dominated strategies? If so, explain how. If it cannot, explain why.

Answer 1

No strictly dominant strategy.

Guessing any number that lies above 66.67 is dominated for every player since it cannot possibly be 2/3rds of the average of any guess. These can be eliminated. Once these strategies are eliminated for every player, any guess above 44.45 is strictly dominated for every player since no player will guess above 66.67 and 2/3 of 66.67 is approximately 44.45. This process will continue until all numbers above 0 have been eliminated.