Math 486.  Febr. 4, 2010.  Midterm 1.
5 problems, 15 pts each.                      Name_______Dr.V_______
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.

Solution. First we do one pile

 stones 0 1 2 3 4 5 6 7 8 9 10 11 take L 1 2 L 1 or 4 2 or 5 L 1 or 4 2 or 5 L 1 or 4 2 or 5

Period is 3. A winning strategy:  keep the number of stones divisible by 3. You loose (L)  against a perfect player if you start with the number divisible by 3.

Now we consider two piles. Still we have period 3 for W and L in both directions.

A winning strategy: keep the numbers of stones congruent each other modulo 3. It suffices to take 1 or 2 stones in a move to win.

In particular, in the position (100,200) take either 1 or 4 from the pile of 200 or 2 or 5 from pile of 100.

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.

see pdf by H.K.

D's face down card is 7  D stands at 17  payoff is \$0.

/

0.5

/

P stands at 17  - --0.5-- ->  D's face down card is 4  D drws at 14  D wins. P's payoff is -\$1.

stand    / expected payoff is -\$0.5.

P   -\$0.15

draws \

-\$0.15       P draws at 17 --- 0.5--->  P gets 7. The payoff is -\$1.                       D gets 7 and stands at 17. \$1..              D gets 7 and ties. \$0.

\ 0.5                                                                                                        / 0.6                                           | 0.75

\$0.7        P gets 4. P has 21. P cannot do better. P stands at 21.D draws.  ---0.4--> D gets 4. D draws at 14. \$0.25.

| 0.25

D gets 4  again and stands at 18. \$1

The value of game is -\$0.15. Optimal solution is to draw.

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

Solution.

 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*

There are 3 equilibria. All in the last row,  in columns 3,5,6. They are positions with two *.

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

Solution. Double line  indicate a strategy profiles which are equilibria.

initial position  B
(1,2,3) the payoff at both equilibria
/ /                                         \
//                                                 \
A    (1,2,3)                                       B   (0,0,2) or (-1,0,1)
/ /          \                                             /  /      \ \
B  (1,2,3)      C   (-1,-2,1)         (-1,0,1)    B       C   (0,0,2)
/      \      /         \ \                             /   /      \     /      \  \
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

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                 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*  equilibrium
3  1  2                 -1   1  -2
3  2  1               1*  -1*  0*  equilibrium
3  2  2                 -1   0   0

*maximal payoff for this player if the other players do not change.