Math 486. Oct. 1, 2009. Midterm
1.
5 problems, 15 pts each.
Name____________________
m1 ..... /75. total ................./.....
Return this page. Write your name on the class list, above, and
on every
additional paper you use.
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 1,5, or 6 stones
in a move from a pile. Initial position: 2 piles, 10 and 20 stones.
2. Blackjack. Player (P) has hard 17. Dealer (D) shows
10.
Cards remaining including D's face-down are 4, 5, 7, 7,7.
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 |
| 1,6 |
5,0 |
0, 3 |
6,0 |
3,5 | 3,3 |
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
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 0 0
3 2
2
-1 0 1