1. What is 'two-person zero-sum game' ?

2. Define the following:

- Saddle point
- Pure strategy.
- Pay-off matrix.
- Optimal strategies

3. Explain briefly the importance of the principle of dominance.

4. What are the advantages & limitations of game theory?

1. Consider the game whose pay-off matrix is given below. Find its
solution.

Player B | ||||
---|---|---|---|---|

Player A | I | II | III | |

I | -4 | -6 | 3 | |

II | -3 | -3 | 6 | |

III | 2 | -3 | 4 |

2. Two companies A and B are competing for the same product. Their different strategies are given in the following pay-off matrix:

Company B | ||||
---|---|---|---|---|

Company A | I | II | III | |

I | 2 | -2 | 3 | |

II | -3 | 5 | -1 |

Determine the best strategies and find the value of the game.

3. Solve the following games:

(a)Player B | ||||||
---|---|---|---|---|---|---|

Player A | I | II | III | IV | V | |

I | 4 | 0 | 1 | 7 | -1 | |

II | 0 | -3 | -5 | -7 | 5 | |

III | 3 | 2 | 3 | 4 | 3 | |

IV | -6 | 1 | -1 | 0 | 5 | |

V | 0 | 0 | 6 | 0 | 0 |

(b

Player B | ||||
---|---|---|---|---|

Player A | I | II | III | |

I | -2 | 15 | -2 | |

II | -5 | -6 | -4 | |

III | -5 | 20 | -8 |

(c)

Player B | ||||
---|---|---|---|---|

Player A | I | II | III | |

I | 2 | -1 | 3 | |

II | 2 | -1 | 2 | |

III | -1 | 0 | 0 | |

IV | 2 | 0 | 4 |

(d)

Player B | |||||
---|---|---|---|---|---|

Player A | I | II | III | IV | |

I | -5 | 3 | 1 | 20 | |

II | 5 | 5 | 4 | 6 | |

III | -4 | -2 | 0 | -5 |

(e)

Player B | |||||
---|---|---|---|---|---|

Player A | I | II | III | IV | |

I | 3 | -5 | 0 | 6 | |

II | -4 | -2 | 1 | 2 | |

III | 5 | 4 | 2 | 3 |

4. Use the dominance principle to solve the following game:

0 | 0 | 0 | 0 | 0 | 0 |

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

4 | 3 | 1 | 3 | 2 | 2 |

4 | 3 | 7 | -5 | 1 | 2 |

4 | 3 | 4 | -1 | 2 | 2 |

4 | 3 | 3 | -2 | 2 | 2 |

5. Solve the following games:

(a)

Player B | ||||
---|---|---|---|---|

Player A | I | II | III | |

I | -5 | -1 | -1 | |

II | 4 | 0 | 2 | |

III | -5 | 2 | 0 |

(b)

Player B | ||||
---|---|---|---|---|

Player A | 6 | 8 | 3 | 13 |

4 | 1 | 5 | 3 | |

8 | 10 | 4 | 12 | |

3 | 6 | 7 | 12 |

6. Solve the following game algebraically

Player B | ||||
---|---|---|---|---|

Player A | I | II | III | |

I | 4 | 2 | 4 | |

II | 2 | 4 | 0 | |

III | 4 | 0 | 8 |

7. Reduce each of the following games by using the rule of dominance and then solve the reduced game by any of the method you have studied:

(a)B1 | B2 | B3 | |
---|---|---|---|

A1 | 3 | 8 | 5 |

A2 | 6 | 2 | 7 |

A3 | 4 | 5 | 6 |

(b)

B1 | B2 | B3 | B4 | B5 | |
---|---|---|---|---|---|

A1 | 8 | 7 | 6 | -1 | 2 |

A2 | 12 | 10 | 12 | 0 | 4 |

A3 | 14 | 6 | 8 | 14 | 16 |

8. Two players A & B, without showing each other put a coin on a table with head or tail up. If the coins show the same side (both head or tail), the player A takes both the coins, otherwise B gets them. Construct the matrix of the game and solve it.

9. In a game of matching coins with two players, suppose A wins one unit of the value when there are two heads; wins nothing when there are two tails and loses 1/2 units of value when there is one head and one tail. Determine the pay off matrix, the optimal strategies for both the players.