MAT199 · Erich Prisner · Franklin College · Fall 2008

Test 1

First version

Consider the following two-player game with players Ann and Beth, where Ann has three choices, A1, A2, or A3, and Beth has even four choices, B1, B2, B3, and B4.
 B1 B2 B3 B4 A1 2,2 3,2 3,3 2,3 A2 4,1 3,4 5,2 4,1 A3 1,1 2,3 4,2 3,2

1. (1 point) What is the payoff for Beth if Ann plays A2 and Beth plays B3?
2. (1 point) Find the Maximin moves for both Ann and for Beth.
3. (1 point) Which moves are strictly dominated by which moves?
4. (1 point) Which moves are weakly dominated by which moves?
5. (1 point) Draw the best response digraph.
6. (1 point) Are there any Nash equilibria?
7. (1 point) Which move do you think Ann will play? Which move do you think Beth will play? Give a reason.
8. (1 point) The last question refers to the following zero-sum game:
 L R U 2 1 D 3 4
What is Beth's best response to Ann's move "U"? What is Beth's best response to Ann's move "D"?
1. 2
2. for Ann it's A2, for Beth it's B2 or B3.
3. A3 is strictly dominated by A2, B1 is strictly dominated by B3.
4. A3 is weakly dominated by A2, A1 is weakly dominated by A2, B4 is weakly dominated by B3, and B1 is weakly dominated by B2, B3, and B4.
5. A1--->B3, A1--->B4, A2--->B2, A3--->B2, B1--->A2, B2--->A1, B2--->A2, B3--->A2, B4--->A2.
6. the pair (A2, B2) is a Nash equilibrium.
7. Probably Ann will play A2 and Beth will play B2. This is the Nash equilibrium, but the moves are also maximin moves.
8. "R" is Beth's best response to Ann's "U", and "L" is Beth's best response to Ann's "D"

Second Version

Consider the following two-player game with players Ann and Beth, where Ann has three choices, A1, A2, or A3, and Beth has even four choices, B1, B2, B3, and B4.
 B1 B2 B3 B4 A1 2,2 3,2 2,3 3,3 A2 1,1 2,3 3,2 4,2 A3 4,1 3,4 4,1 5,2

1. (1 point) What is the payoff for Beth if Ann plays A2 and Beth plays B3?
2. (1 point) Find the Maximin moves for both Ann and for Beth.
3. (1 point) Which moves are strictly dominated by which moves?
4. (1 point) Which moves are weakly dominated by which moves?
5. (1 point) Draw the best response digraph.
6. (1 point) Are there any Nash equilibria?
7. (1 point) Which move do you think Ann will play? Which move do you think Beth will play? Give a reason.
8. (1 point) The last question refers to the following zero-sum game:
 L R U 1 3 D 2 4
What is Beth's best response to Ann's move "U"? What is Beth's best response to Ann's move "D"?
1. 2
2. for Ann it's A3, for Beth it's B2 or B4.
3. A2 is strictly dominated by A3, B1 is strictly dominated by B4.
4. A2 is weakly dominated by A3, A1 is weakly dominated by A3, B3 is weakly dominated by B4, and B1 is weakly dominated by B2, B3, and B4.
5. A1--->B3, A1--->B4, A2--->B2, A3--->B2, B1--->A3, B2--->A1, B2--->A3, B3--->A3, B4--->A3.
6. the pair (A3, B2) is a Nash equilibrium.
7. Here it is not so clear. The Nash equilibrium pair is (A3, B2), but A3 is not the maximin move for Ann.
8. "L" is Beth's best response to Ann's "U", and "L" is also Beth's best response to Ann's "D"