**Assignment: Practice 3 (Spring, 2013)**

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. a. Draw the motion diagram associated with the two non-zero-sum games shown below.

b. Which if any of the cells (results of playing pure strategies) in these two games might you describe as "stable" or "equilibrium" points and explain why you think they are "stable."

Game X

Column I | Column II | |

Row 1 | (6, 6) | (-10. 20) |

Row 2 | (20, -10) | (-3, -3) |

Game Y

Column I | Column II | |

Row 1 | (3, 3) | (-6, 4) |

Row 2 | (4, -6) | (-7, -7) |

(c) List the Nash equilibria for each of these games. (Look for Nash equilibria in pure strategies and mixed strategies.)

2. a. Carry out a dominating row analysis for the game against Nature shown below:

b. Which action by Row would represent a best/worst action?

c. Write down the Regret Matrix associated with the game against Nature below and determine which action would minimize the maximum regret.

You-Row\Nature | I | II | III | IV |

1 | 7 | -4 | 5 | -1 |

2 | -2 | 2 | 3 | 2 |

3 | 5 | -6 | 1 | -3 |

4 | 2 | 13 | 0 | -5 |

d. If you personally were required to play this game against Nature, which row action would you take and explain your reason why.

3. Given the 2x2 zero-sum game matrix below where a, b, c, and d represent *positive* real numbers and the matrix is such that there are no dominating rows or columns and no saddle point. (Payoffs from Row's perspective.)

Column I | Column II | |

Row 1 | a | -b |

Row 2 | -c | d |

a. A student in your class claims that if a + b = c + d then the game is "fair." Do you agree? If not, explain why.

b. What is the condition (conditions) on the values a, b, c, and d (subject to no dominating rows, columns or saddle point) that will make this game fair? Can you state the condition(s) you find in "easy" terms?