Math 486. Febr. 7,  2008. Midterm 1.                                                                                                   m1                          total
5 problems, 15 pts each.                      Name____Leon__Vaserstein_________________                            /75.                              /125.
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Find an equilibrium and the corresponding payoff. In Problems 1 and 2, the bet is $1.

1. Restricted Nim. Last move wins. Players can take 2,5, or 8 stones in a move. Initial position: 1 pile, 24 stones.
Solution.   For any number of stones < 25, we write a winning move (the number of stones we take)  in the case when
the player who starts at this position has a winning strategy; otherwise, we write  L.

0
 1
 2
 3
 4
 5
 6
 7
 8
 9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24


2
 2
5
 5
8
 2 or 8

2 or 8
 2
5
 2
8
 2

2
 2
L
 L
 W
 W
 L
 W
 W
 L
 W
 W
 L
 L
 W
 W
 L
 W
 W
 L
 W
 W
 L
 L
 W
 W
 L

So the player  who starts at 24 loses $1  if the second player uses an optimal strategy.


2. Blackjack. Player (P)  has hard 16. Dealer (D)  shows 10.  Cards remaining including D's face-down are 4, 6, 6, 7, 7,7,7.
Solution.
                                                   P has 16, P's position, initial position.   -$11/21
                                           P  draws                                     P stands at 16
         P has 17,  D has 10, chance move.    -$5/7                          D has 10 and  opens the second card, chance move .  -$11/21
                                 /              \                                                                            /         |                       \
                           1/7                  6/7                                                              1/7           2/7                      4/7
                         /                                \                                                       /                     |                              \
P  stands at 20  (6,6,7,7,7,7) $1        P is over    -$1                        D (14)  -$1      D (16)   $2/3         D(17) -$1  for P
D draws at 10, chance move                                                            D draws from shoe, chance move
1/3                    2/3                                                                     2/3    1/3       1/6      5/6
D (16) $1          D (17)    $1                                    D (21) -$1          D (20) -$1     D is over, $1
4/5           1/5   
D (23) $1  D (22) $1  D goes over
Answer:  Player (P)  stands, P's expected payoff is -$11/21.

3. 2 player game in normal form.
-7, 1 4,0 -1, 3 0,0 3, 3
5,-1 5,0 -2, 5 1, 4
 3,1
0, 5 4,-1 -2, 4 6, 0 0, 3
7,6 5,0 0, 3 6,0 3,6
Solution.
-7, 1 4,0 -1, 3* 0,0 3*, 3*
5,-1 5*,0 -2, 5*
 1, 4
 3*,1
0, 5* 4,-1 -2, 4 6*, 0 0, 3
7*,,6* 5*,0 0*, 3 6*,0 3*,6*
Three equilibria (in bold).

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    0,1,0      0,0,2
Solution.
                            initial position   B  (1,2,3)
                              
                   /   /                                     \
                   / /                                           \
            A    1,2,3                                         B 0.1.0
       //          \                                              / /        \
  B   1,2,3          C   -1,-2,1             0,1,0    B           C  0,0,2
    //      \      /        \ \                                 /     \ \     /         \\
 1,2,3   0,-1,0    -1,-2,1                   -1,0,1      0,1,0      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             -1    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   0    0
3  2  2             -1    0    1
Solution.
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             -1*    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*   0*    0
3  2  2             -1*    0    1*
There is one equilibriaum