Editing Matching pennies (section)

== Theory ==
{{Payoff matrix | Name = Matching pennies | Width=300
                | 2L = Heads      | 2R = Tails      |
1U = Heads      | UL = +1, −1     | UR = −1, +1     |
1D = Tails      | DL = −1, +1     | DR = +1, −1     }}
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 strategy|mixed strategies]] and a mixed strategy [[Nash equilibrium]].<ref>{{cite book |first=Robert |last=Gibbons |title=Game Theory for Applied Economists |publisher=Princeton University Press |year=1992 |isbn=978-0-691-00395-5 |pages=29–33 |url=https://books.google.com/books?id=8ygxf2WunAIC&pg=PA29 }}</ref>

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 strategy|mixed strategies]]: each player chooses heads or tails with equal probability.<ref>{{cite web|url=http://www.gametheory.net/dictionary/Games/Matchingpennies.html |title=Matching Pennies |publisher=GameTheory.net |url-status=dead |archive-url=https://web.archive.org/web/20061001133455/http://www.gametheory.net/Dictionary/Games/MatchingPennies.html |archive-date=2006-10-01 }}</ref> 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:

[[Image:Reaction-correspondence-matching-pennies.jpg|500px|thumbnail|center|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.]]

When either player plays the equilibrium, everyone's expected payoff is zero.

=== Variants ===
{{Payoff matrix | Name = Matching pennies | Width=300
                | 2L = Heads     | 2R = Tails      |
1U = Heads      | UL = +7, -1    | UR = -1, +1     |
1D = Tails      | DL = -1, +1    | DR = +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:
* For the Even player, the expected payoff when playing Heads is <math>+7\cdot x -1\cdot (1-x)</math> and when playing Tails <math>-1\cdot x +1\cdot (1-x)</math> (where <math>x</math> is ''Odd's'' probability of playing Heads), and these must be equal, so <math>x=0.2</math>.
* For the Odd player, the expected payoff when playing Heads is <math>+1\cdot y -1\cdot (1-y)</math> and when playing Tails <math>-1\cdot y +1\cdot (1-y)</math> (where <math>y</math> is ''Even's'' probability of playing Heads), and these must be equal, so <math>y=0.5</math>.
Note that since <math>x</math> is the Heads-probability of ''Odd'' and <math>y</math> is the Heads-probability of ''Even'', the change in Even's payoff affects Odd's equilibrium strategy and not Even's own equilibrium strategy. This may be unintuitive at first. The reasoning is that in equilibrium, the choices must be equally appealing.  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.