**Mathematics 483 ** (Game Theory)

Homework Assignment ii

prepared by:

Joseph Malkevitch

Mathematics Department

York College (CUNY)

Jamaica, NY 11451

email: joeyc@cunyvm.cuny.edu

web page: http://www.york.cuny.edu/~malk

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

Column I | Column II | |

Row 1 | (6, -6) | (-3, 3) |

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

a. Rewrite this game with a matrix where each cell has a single entry given from Row's point of view.

b. What is row's best/worst strategy?

c. What is column's best/worst strategy?

2. The matrix below shows an example of a zero-sum game (payoffs in dollars) involving two players, row and column., with the payoffs shown from row's point of view:

Column I | Column II | |

Row 1 | 6 | -4 |

Row 2 | -5 | 7 |

a. If Row plays Row 2 and Column plays Column II what is Row's payoff?

b. If Row plays Row 2 and Column plays Column II what is Column's payoff?

c. If Row plays Row 1 and Column plays Column I what is Column's payoff?

d. What is Row's best/worst strategy?

e. What is Column's best/worst strategy?

3. a. Determine if the zero-sum game below, with payoffs from Row's point of view has a saddle point.

b. If there is a saddle point what is the significance of this?

Column I | Column II | |

Row 1 | -19 | 30 |

Row 2 | 2 | 4 |

c. What advice would you give row about how to play this game?

d. What advice would you give column about how to play this game?

e. Is this a fair game?

4. Apply dominant row/dominant column analysis to the zero sum games below to determine, if possible, simpler versions of the original games. Do any of these games have a value, in the sense that they either reduce to a single cell or have a saddle point. If the game has a value what is this value. Furthermore, compute the best/worst strategy for each game for row and for column.

a.

2 | -2 | 15 |

3 | 6 | 3 |

b.

-1 | -2 | 1 |

0 | -1 | -3 |

1 | 1 | -1 |

c.

2 | -4 | 3 | 2 |

-7 | 3 | 3 | 0 |

0 | -4 | 2 | 1 |

1 | -5 | 3 | 0 |

d.

1 | 0 | 3 |

-2 | -3 | -1 |

-4 | -2 | 4 |

e.

1 | -1 | 2 |

-1 | 1 | 3 |

-1 | -1 | -1 |

f.

6 | 4 | -2 | 2 | 0 |

3 | 1 | 3 | -1 | -1 |

-1 | 2 | -4 | 0 | 2 |

0 | -1 | 0 | -4 | -2 |

2 | 5 | 1 | -1 | 4 |

5. Given the zero-sum game below:

4 | -3 |

-3 | 2 |

a. Do you think this is a fair game?

b. Design the optimal spinner for Row and for Column to play this game many times in an optimal manner.

c. Find the value of this game?

d. What is each players best/worst strategy?

6.

100 | -2 |

-50 | 1 |

a. Do you think this is a fair game?

b. Design the optimal spinner for Row and for Column to play this game many times in an optimal manner.

c. Find the value of this game?

d. What is each players best/worst strategy?