Editing Matching pennies (section)

=== 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.