**Matrix Games**

Prepared by:

Joseph Malkevitch

Mathematics Department

York College (CUNY)

Jamaica, NY 11451

email:

__malkevitch@york.cuny.edu__

web page:

__http://york.cuny.edu/~malk/__

1. The matrix below shows an example of a game (payoffs in dollars) involving two players, row and column.:

Column I | Column II | |

Row 1 | (3, -3) | (-11, 11) |

Row 2 | (-2, 2) | (10, -10) |

2. The matrix below shows an example of a game (payoffs in dollars) involving two players, row and column.:

Column I | Column II | |

Row 1 | (3, -3) | (-2, 2) |

Row 2 | (-15, 15) | (10, -10) |

3. The matrix below shows an example of a game (payoffs in dollars) involving two players, row and column.:

Column I | Column II | |

Row 1 | (-3, 3) | (11, -11) |

Row 2 | (2, -2) | (-10, 10) |

In the games above, an entry such as (-10,10) means a payoff of -10 to the row player and a payoff of 10 to the column player. Because these two entries sum to 0, the game is called a zero-sum game.

Which if any of these games is "fair?" It is not even so clear what one might mean by this. However, one idea is that if the game is played many times the average winnings of each player is 0. Thus, a game is fair if no player achieves a non-zero flow of money after many, many plays of the game. Of course, after a particular number of plays of the game it is unlikely that one player might not be ahead in earnings for a fair game. However, on average, the long term earning of each player in a fair game should be zero.

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