**Game Theory Practice 1 (Zero-Sum Games) (Spring, 2016)**

Prepared by:

Joseph Malkevitch

Department of Mathematics

York College (CUNY)

Jamaica, New York 11451

email:

__malkevitch@york.cuny.edu__

web page:

__http://york.cuny.edu/~malk__

1. The matrix below shows the payoffs of a zero-sum game from Row's point of view.

.

Row/Column |
I | II | III |

1 | 4 | 3 | -2 |

2 | 9 | -3 | -4 |

3 | 0 | 5 | 7 |

Write down a version of the game matrix where the payoff entries in the cells have the form (a, b) where a is Row's payoff and b is Column's payoff.

2. For each of the games below (and the one above):

a. Indicate if any of the rows dominate other rows

b. Indicate if any of the columns dominate other columns

c. Does the game have a value in "pure strategies?" If the zero-sum game has a value, what is this value?

d. If there are dominating rows and/or columns write down the matrix that results after carrying out a dominating row and column analysis. (This means crossing out rows and/or columns that are dominated by other rows or columns.)

e. If after completing dominating row and column analysis you obtain a 2x2 game, find the optimal mixed strategy for each player. (This means designing a spinner for each of the players and determining the "value" of the game that occurs when these spinners are used to play the game optimally.)

f. If the zero-sum game has a "saddle point," what cell of the matrix corresponds to the saddle point? (This means using the short notation which shows that payoffs are from the point of view of Row that the cell entry is minimum in its row and maximum in its column.)

*Game I*

Row/Column |
I | II | III |

1 | (2, -2) | (7, -7) | (-5, 5) |

2 | (-3, 3) | (4, -4) | (-6, 6) |

3 | (3, -3) | (-3, 3) | (2, -2) |

*Game II*

Row/Column |
I | II | III |

1 | (-2, 2) | (-1. 1) | (-2, 2) |

2 | (-1, 1) | (5, -5) | (3, -3) |

3 | (5, -5) | (1, -1) | (3, -3) |

Game III

Row/Column |
I | II | III |

1 | (1, -1) | (-2, +2) | (8, -8) |

2 | (4, -4) | (3, -3) | (7, -7) |

3 | (-2, 2) | (0, 0) | (-5, 5) |