Math 486.  Febr. 3, 2011.  Midterm 1 5 problems, 15 pts each.

Name____________________                        m1 ….  /75.  total    ......./.....

Show your work. Write your name on the class list and on every page you

1. For every given number t, solve for   x, y the system

x+ty = 2, tx+y = -2
2-5. Find an equilibrium and the corresponding payoff.
2. Restricted Nim. Last move wins. The bet is \$1.  Two player, A and B. Players

alternate.  A  can take 1 or 2 stones in a move from a pile.  B can take  1, 2,  or 3

stones in a move from a pile.   A starts with one pile, 100 stones.

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

4. Extensive form, 3 players, A B, C.
initial position  B
/                                         \
/                                                 \
A                                                       \
/          \                                       0.5     /      \ 0.5
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