Two-Finger Dakota

The best strategy is to play one finger and guess two 7 times and to play two fingers and guess one 5 times.

If your opponent plays one and guesses two or plays two and guesses one, neither wins nor loses.

If your opponent plays one and guesses one, he wins 2 chips. This will happen 7 times in 12. The remaining 5 times he will lose 3 chips each time. Overall, you will win 1 chip every twelve hands, on average.

If your opponent plays two and guesses two, he wins 4 chips. This will happen 5 times in 12. The remaining 7 times he will lose 3 chips each time. Overall, you will again win one chip every twelve hands, on average.

Game rules as published in The Ascent of Man, J. Bronowski. This solution is credited to John von Neumann.