Recall the game “matching pennies“. Player I has to chose between ‘0’ or ‘1’, player II has to chose between ‘0’ and ‘1’.No player knows what is the choice of the other player before making his choice. Player II pays to player I one dollar if the two number match and gets one dollar from player I if they do not match.
The optimal way to play the game for each of the player is via a mixed strategy. Player I has to choose ‘0’ with probability 1/2 and ‘1’ with probability 1/2. And so is player II.
Now, suppose that the game is the same except that if the outcomes match and both player chose ‘1’ player II pays 5 dollars to player I and not one dollar. (All other payoffs remain at one dollar.)
Test your intuition: What is the probability player I should play ‘1’. (More than 1/2? less than 1/2? more than 1/3? less than 1/3? more than 2/3? less than 2/3?)
According to Abraham Neyman, failing to have the right intuition in this case caused Barcelona to fail in the second game against Inter Milano, in the semi finals of the European champion leagues last year. Do you see a connection? I will come back to that in a later post.