Matching pennies

This article is about the two-person game. For the confidence trick, see coin-matching game.

	Heads	Tails
Heads	+1, −1	−1, +1
Tails	−1, +1	+1, −1
Matching pennies

Matching pennies is the name for a simple example game used in game theory. It is the two strategy equivalent of Rock, Paper, Scissors. Matching pennies is used primarily to illustrate the concept of mixed strategies and a mixed strategy Nash equilibrium.

The game is played between two players, Player A and Player B. 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) Player A keeps both pennies, so wins one from Player B (+1 for A, -1 for B). If the pennies do not match (one heads and one tails) Player B keeps both pennies, so receives one from Player A (-1 for A, +1 for B). This is an example of a zero-sum game, where 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 Player A's payoffs listed first.

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:

File:Reaction-correspondence-matching-pennies.jpg

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

The matching pennies game is mathematically equivalent to the games "Morra" or "odds and evens", where two players simultaneously display one or two fingers, with the winner determined by whether or not the number of fingers match. Again, the only strategy for these games to avoid being exploited is to play the equilibrium.

Of course, human players might not faithfully apply the equilibrium strategy, especially if matching pennies is played repeatedly. In a repeated game, if one is sufficiently adept at psychology, it may be possible to predict the opponent's move and choose accordingly, in the same manner as expert Rock, Paper, Scissors players. In this way, a positive expected payoff might be attainable, whereas when either player plays the equilibrium, everyone's expected payoff is zero.

Nonetheless, statistical analysis of penalty kicks in soccer—a high-stakes real-world situation that closely resembles the matching pennies game—has shown that the decisions of kickers and goalies resemble a mixed strategy equilibrium.