This is equivalent to the game Westley played with Vizzini. You know, if Westley didn't cheat. I like to call it "Sicilian Chess" for that reason, though that's just me.
There is a game where one player wants to predict the action of the other, and the other player wants them to fail (as a fixed sum game)
It has payoffs
1,-1| -1,1
-1,1|1,-1
Or equivalent.
I believe that it has a nash equilibrium of choosing randomly.