each round, all players simultaneously say a number 1-8

if your number is strictly between the other two, you get a point

at the end, the person whose point total is strictly between the others wins.

That’s it. A few strategies immediately spring to mind, but I bet you could come up with some really complex ones.

Aside: what would you call little games like this? Nim, rock paper scissors, tic tac toe all kind of fit in the category. Almost game-theoretical exercises.

I don’t understand this MEDIAN game. Why wouldn’t everyone always pick 4 or 5? When would anyone score a point? What incentive is there to pick another number? Am I stupid?

Oh, I missed that part. I thought you were just playing for the highest score. I like that mechanic of winning by having the middle score. I wonder how that would apply to more “normal” games. I think it could work great for a lot of three player games. Imagine a three player T&E where the winner has the middle score.

Remember that the winner is the player whose score is strictly between both of their opponents’. With absolutely no attempt at mathematical rigor, it seems like you could eliminate one or two opponents using “always choose 4 or 5” by always selecting 1 or 8 until an opponent is winning, then changing strategy.

Two players who always choose 1 or 8 and one player who always chooses 4 or 5 will force a draw, unless there’s some kind of sudden death rule (in which case the 4/5 player would only be in contention during sudden death P=.5^8 - ie. if the 1/8 players managed to pick the same number in all 8 rounds, leaving the score 0-0-0, at which point they would continue to have no chance to win unless one of the players modifies their strategy)

short version: In each round, the player in second place scores a variable number of points. At the end of the fifth round, the player with the second-highest overall nonzero score wins.