|Heads||+1, −1||−1, +1|
|Tails||−1, +1||+1, −1|
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 because each participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants. If the participants' total gains are added up and their total losses subtracted, the sum will be zero.
The game can be written in a payoff matrix (pictured right - from Even's point of view). 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 in Figure 1 below:
When either player plays the equilibrium, everyone's expected payoff is zero.
|Heads||+7, -1||-1, +1|
|Tails||-1, +1||+1, -1|
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.
Human players do not always play the equilibrium strategy. 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.
To overcome these difficulties, several authors have done statistical analysis of professional sports games. These are zero-sum games with very high payoffs, and the players have devoted their lives to become experts. Often such games are strategically similar to matching pennies:
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