Math 486.  Febr. 4, 2010.  Midterm 1.
5 problems, 15 pts each.                      Name____________________
m1 .....   /75.  total    ................./.....

Find an equilibrium in pure strategies and the corresponding payoff. In Problems 1 and 2, the bet is \$1.

1. Restricted Nim. Last move wins. Players alternate and can take 1,2,4, or 5 stones in a move from a pile. Initial position: 2 piles, 100 and 200 stones.

2. Blackjack. Player (P) has hard 17. Dealer (D)  shows 10.  Cards remaining including D's face-down are 4, 4,4, 7, 7,7.

3. 2 player game in normal form.

 7, 1 4,0 -1, 3 0,0 3, 3 -3,4 5,-1 5,0 -2, 5 1, 4 3,1 0,0 0, 5 4,-1 -2, 4 6, 0 0, 3 1,1 1,1 5,0 0, 3 6,0 3,3 3,3

4. Extensive form, 3 players, A B, C.

initial position  B

/                                         \
/                                                 \
A                                                      B
/          \                                               /        \
B             C                                         B           C
/      \      /          \                                 /      \     /         \
1,2,3   0,-1,0    -1,-2,1                -1,0,1    2,-1,0      0,0,2

5.  Game with 3 players, A, B, C. in normal form.
strategy                     payoff
A  B  C               A   B   C
1   1  1               0  -1    1
1  1  2                1   1  -2
1  2  1                1   0   0
1  2  2                 0    0   1
2  1  1                 0  -1    1
2  1  2                1   1  -2
2  2  1                1   0  -1
2  2  2                 -1   1   0
3  1  1                 0 -1    1
3  1  2                 -1   1  -2
3  2  1                1  -1   0
3  2  2                 -1   0   0