**Practice 3 (Spring, 2016)**

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. (i). For the following game against nature, determine which row to play if one picks the row which leads to

a. best of the best outcomes

b. best of the worst outcomes

c. worst of the worst outcomes

d. worst of the best outcomes

Action/State of nature | State 1 | State 2 | State 3 |

Action 1 | 4 | -5 | -2 |

Action 2 | -1 | 13 | 4 |

Action 3 | 20 | -50 | 2 |

Action 4 | -20 | 9 | 12 |

ii. (a) What action would you take assuming the probability of each state of nature is the same?

(b) What action would you take assuming the probability of the first two states of nature are 1/10 and 1/8?

(c) What action should you take assuming the probability of the last two states of nature are 1/12 and 3/10?

2. (a) Compute the Savage regret matrix associated with the game above.

(b) Compute the action which corresponds to minimizing the maximum regret.

3. Assume the non-zero sum game below is based on cardinal utilities for payoffs.

a. Write down the associated ordinal payoffs matrix associated with this matrix game assuming that 4 taken as high (best) and 1 as low (worst). (We use ranks from 1 to 4 because there are 4 outcomes.)

Column I |
Column II | |

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

Row 2 | (3, -2) | (-13, -3) |

b. Determine if the game above has any pure strategy Nash equilibria by constructing a motion diagram for the game.

c. Does the game above have any mixed strategy Nash equilibria?

4. (a) Find all possible Nash equilibria for the matrix below:

Column I | Column II | |

Row 1 | (-2, 10) | (-10, -15) |

Row 2 | (-3, -2) | (13, -4) |

(b) If you were Row how would you play this game?

(c) If you were Column how would you play this game?