Matching pennies

Instead, the unique Nash equilibrium of this game is in mixed strategies: each player chooses heads or tails with equal probability.

The best-response functions for mixed strategies are depicted in Figure 1 below: When either player plays the equilibrium, everyone's expected payoff is zero.

The +7 possibility for Even is very appealing relative to +1, so to maintain equilibrium, Odd's play must lower the probability of that outcome to compensate and equalize the expected values of the two choices, meaning in equilibrium Odd will play Heads less often and Tails more often.

Laboratory experiments reveal several factors that make players deviate from the equilibrium strategy, especially if matching pennies is played repeatedly: Moreover, when the payoff matrix is asymmetric, other factors influence human behavior even when the game is not repeated: The conclusions of laboratory experiments have been criticized on several grounds.

[9][10] To overcome these difficulties, several authors have done statistical analyses of professional sports games.

These are zero-sum games with very high payoffs, and the players have devoted their lives to become experts.

Figure 1. Best response correspondences for players in the matching pennies game. The leftmost mapping is for the Even player, the middle shows the mapping for the Odd player. The sole Nash equilibrium is shown in the right hand graph. x is a probability of playing heads by Odd player, y is a probability of playing heads by Even. The unique intersection is the only point where the strategy of Even is the best response to the strategy of Odd and vice versa.