**Homework 3: Game Theory (2018)**

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. Given the decision matrix below:

Decision maker ROW/Nature (Column) | I | II | III | IV |

1 | 12 | -1 | 1 | 0 |

2 | 5 | 1 | 7 | -18 |

3 | 3 | 2 | 4 | 3 |

4 | -20 | 0 | 0 | 16 |

a. What action would the decision maker choose if he/she selects the best of the worst outcomes?

b. Which action would the decision maker choose if he/she selects on the basis of assuming that nature's states are equally likely?

c. Do any of the rows in this matrix above dominate other rows?

d. Construct the Savage regret matrix associated with this decision problem. What action would the decision maker choose to minimize the maximum regret?

2. Given the non-zero sum game below

Column I |
Column II | |

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

Row 2 | (11, -8) | (-9, -7) |

a. Draw a motion diagram for this game.

b. Are there any row or column dominations in this game?

c. Determine all of the Nash equilibria for this game, assuming any exist.

d. Suppose the (4,3) entry is changed to (x,y) where x and y are positive integers distinct from the other entries in the matrix. What are the smallest values of x and y such that playing Row 1 and Column I as pure strategies yields a Nash equilibrium?

e. Using the values you find for x and y from d., is there a mixed strategy Nash equilbrium for the matrix?