Math 486. Febr. 5,  2004. Midterm 1.
5 problems, 15 pts each. Name________________________
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 2,6, or 7 stones in a move. Initial position: 1 pile, 10000 stones.
Solution.
 Pile size 13n 13n+1 13n+2 13n+3 (e.g..10K) 13n+4 13n+5 W/L L L W W L L take 2 or 6 2 or 7
 13n+6 13n+7 13n+8 13n+9 13n+10 13n+11 13n+12 W W W L W W W 2,6 2,6, or 7 7 6 2,6, or 7 7

2. Blackjack. Player  has 10 and 6. Dealer shows 10.  Cards left: 5, 5, 6, 7.,7,7.
Solution.
P has 16, P's position, initial position.   -\$11/30
P  draws                                     P stands at 16
P has 16,  D has 10, chance move.    -\$11/30                            P has 16, D has 10, chance move .-\$2/5= -\$0.40
/              \                                                                            /         |                       \
1/3                  2/3                                                              1/3           1/6                      1/2
/                                \                                                       /                     |                              \
P  stands at 21  (5,6,7,7,7) \$.9        P is over    -\$1                        D (15)  \$1/5      D (16)   \$1/5          D(17) -\$1 for P
D draws at 10, chance move                                                             D draws , chance move
1/5                   1/5                      3/5                                                    3/5    2/5       2/5     3/5
D (15) \$3/4    D (16) \$3/4         D (17)    \$1                        D (22) \$1          D (21) -\$1     D is over, \$1
3/4             1/4    1/4     3/4
D (22) \$1  D (21) \$0  D (23) \$1
Answer:  Player (P)  draws once, P's expected payoff is -\$11/30.

3. 2 player game in normal form.
 -7, 1 4,0 -1, 3 0,0 3, 3 5,-1 5,0 0, 5 1, 5 6,1 0, 5 4,-1 -2, 4 6, 0 0, 3 7,2 5,0 0, 3 6,0 4,6
Solution.
 -7, 1 4,0 -1, 3* 0,0 3, 3* 5,-1 5*,0 0*, 5* 1, 5* 6*,1 0, 5* 4,-1 -2, 4 6*, 0 0, 3 7*,2 5*,0 0*, 3 6*,0 4,6*

4. Extensive form, 3 players, A B, C.
initial position
chance move
/                                     \
0.2                                        0.8
/                                                 \
A                                                      B
/          \                                               /        \
B             C                                         B           C
/      \      /          \                                 /      \     /         \
1,2,3   0,-1,0    -1,-2,-3                -1,0,1    1,1,0      0,0,2
Solution.
initial position  1, 1.2, 0.6
chance move
/                                     \
0.2                                        0.8
/                                                 \
A    1,2,3                                         B 1.1.0
//          \                                              / /        \
B   1,2,3          C   0,-1,0             1,1,0    B           C  0,0,2
//      \      //          \                                 /     \ \     /         \\
1,2,3   0,-1,0    -1,-2,-3                -1,0,1    1,1,0      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    0
2   1  1               0 -1    1
2  1  2                1   1  -2
2  2  1                1   0    1
2  2  2             -1    0    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    0*
2   1  1               0* -1    1*
2  1  2                1*   1*  -2
2  2  1                1*   0*    1*
2  2  2             -1*    0    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 are two equilibria.