Suppose two people are playing a simple game with nickels and quarters. At the same time, they each put out either a nickel or a quarter. If at least one player plays a nickel, player 1 gets both coins. Otherwise, player 2 gets both. Naturally, both players wish to gain as much money as possible. How should they play in order to do this?

We can assign payoff matrices to such games that define the payoffs that players will get based on the strategies they use. In this example, each player has only two strategies--put out a nickel or put out a quarter. Here is a payoff matrix for player 1:

Player 2 | |||

Nickel | Quarter | ||

Player 1 | Nickel | 5 | 25 |

Quarter | 5 | -25 |

The rows represent player 1's possible strategies, and the columns represent player 2's possible strategies. If player 1 and player 2 both play nickels (the top left entry), player 1 wins player 2's nickel so gains 5 cents. On the other hand, if both play quarters (the bottom right entry), player 2 wins player 1's quarter, so player 1 loses 25 cents.

Notice that every entry in the first row is greater than all of the entries in that column. In other words, playing a nickel is always at least as good as playing a quarter for player 1. So, playing a nickel is called a dominant strategy, and it dominates the strategy of playing a quarter. It is never advantageous to play a dominated strategy, so we can reduce our payoff matrix to reflect this:

Player 2 | |||

Nickel | Quarter | ||

Player 1 | Nickel | 5 | 25 |

Now, the nickel strategy for player 2 also dominates. So, playing nickels is the best strategy for both players. Notice that, if either plays quarters, he will not gain more money than if he had just played nickels.