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

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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