Update: Apparently this game was invented already by Douglas Hofstadter who called it “Mediocrity” and it is published in Hofstadter’s book Metamagical Themas: Questing for the Essence of Mind and Pattern. It is also called “Hruska.” (See here and here.)
Ehud Friedgut reminded me of the game MEDIAN which I proposed many years ago.
There are three players and they play the game for eight rounds. In every round all players simultaneously say a number between 1 and 8. A player whose number is (strictly) between the other two get a point. At the end of the game the winner is the player whose number of points is strictly between those of the others.