**Assignment: Practice 1 (Spring, 2013) (Zero-Sum Games)**

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 | 3 | 2 | -1 |

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:

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 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.

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 game has a "saddle point," what cell of the matrix corresponds to the saddle point?

*Game I*

Row/Column |
I | II | III |

1 | (2, -2) | (6, -6) | (-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) | (-4, 4) |