Matching pennies
Matching pennies is the name for a simple game used in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match (both heads or both tails), then Even keeps both pennies, so wins one from Odd (+1 for Even, −1 for Odd). If the pennies do not match (one heads and one tails) Odd keeps both pennies, so receives one from Even (−1 for Even, +1 for Odd).
Matching Pennies is a zero-sum game, since one player's gain is exactly equal to the other player's loss.
The game can be written in a payoff matrix (pictured right). Each cell of the matrix shows the two players' payoffs, with Even's payoffs listed first.
Matching pennies is used primarily to illustrate the concept of mixed strategies and a mixed strategy Nash equilibrium.
This game has no pure strategy Nash equilibrium since there is no pure strategy (heads or tails) that is a best response to a best response. In other words, there is no pair of pure strategies such that neither player would want to switch if told what the other would do. Instead, the unique Nash equilibrium of this game is in mixed strategies: each player chooses heads or tails with equal probability. In this way, each player makes the other indifferent between choosing heads or tails, so neither player has an incentive to try another strategy. The best response functions for mixed strategies are depicted on the figure 1 below:
When either player plays the equilibrium, everyone's expected payoff is zero.
Varying the payoffs in the matrix can change the equilibrium point. For example, in the table shown on the right, Even has a chance to win 7 if both he and Odd play Heads. To calculate the equilibrium point in this game, note that a player playing a mixed strategy must be indifferent between his two actions (otherwise he would switch to a pure strategy). This gives us two equations:
Note that is the Heads-probability of Odd and is the Heads-probability of Even. So the change in Even's payoff affects Odd's strategy and not his own strategy.
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