Answers to Odd Exercises for Section 9.5
1. A3 dominates A1, B3 dominates B1
3. A1 dominates A2 B3 dominates B4 , B2 dominates B4
5. B2 dominates B3, then A1 dominates A2 and A3, then B2 dominates B1. A will play A1
and B will play B2 with a payoff of -2 to A.
7. A1 dominates A2 and B1 dominates B3. No final solution.
9. A Nash equilibrium exists. A would choose A1 and B would choose B2, with a payoff
of -2 to A.
11. A Nash equilibrium exists. A will play A2 and B will play B2 for a payoff of 1 to A.
1 1 2 1
13. P( X  5)      0.333...  33%
6 6 6 3
15. The expected value of X  E ( X )  x1  P( X  x1 )  x2  P( X  x2 )  ...xn  P( X  xn )
n
1
1
1
1
1
1 1 2 3 4 5 6 21
1
  xi P( X  xi )  1  2   3   4   5   6        
3
6
6
6
6
6
6 6 6 6 6 6 6 6
2
i 1
17. Max v
Subject to: v  4 x1  1x2  1x3
v  3x1  0 x2  2 x3
v  1x1  3x2  1x1
x1  x2  x3  1
x1 , x2 , x3  0
19.
Max v   v 
Subject to: 2 x1  1x2  4 x3  v   v   0
 2 x1  3 x2  2 x3  v   v   0
4 x1  5 x2  5 x3  v   v   0
x1  x2  x3  1
x1 , x2 , x3 , v  , v   0
21. Max v
Subject to: v  -2 x1  4 x2  2 x3
v  1x1  1x2  3 x3
v  5 x1  3 x2  4 x3
x1  x2  x3  1
x1 , x2 , x3  0
A should play A2 63.6% of the time and A3 36.4% of the time for an expected
payoff of 0.455 .
23. Max v   v 
Subject to: 1x1  1x2  3 x3  v   v   0
2 x1  1x2  0 x3  v   v   0
0 x1  0 x2  2 x3  v   v   0
x1  x2  x3  1
x1 , x2 , x3 , v  , v   0
A should play A1 and A2 each 50% of the time for an estimated payoff to A of 0.
25. a) Yes; A1 is dominated by A2 .
b) No; there is no Nash equilibrium.
c) Max v
Subject to: v  1x1  6 x2  5 x3
v  2 x1  8x2  15x3
v  6 x1  2 x2  7 x1
x1  x2  x3  1
x1 , x2 , x3  0
You should play man-to-man about 68% (roughly 2/3) of the time, and double-team
their star player about 32% (roughly 1/3) of the time, for an expected savings of about
0.84 points (around one point).
27. a) B1 is dominated by B2 , A1 is dominated by A2 .
b) Yes; A plays A2 and B plays B2 .
c) Yes; A plays A2 and B plays B2 .
29. a) Sbtract 0.50 from all of the entries in the matrix to get the zero-sum payoff matrix,
so that they correspond to the amount that the final Pay increase is over $0.50 .
b) No; there are no dominated strategies.
c) No; there are no pure strategy Nash equilibria.
d) Max v
Subject to: v  0.50 x1  0.25 x2
v  0.10 x1  0.25 x3
v  0.05 x2  0.50 x1
x1  x2  x3  1
x1 , x2 , x3  0
The Union should propose $0.00 with a probability of 29.4% , $0.50 with a
probability of 62.5% , and $1.00 with a probability of 8.1% . The shadow prices
indicate that Management should propose $0.00 with a probability of 9.9% , $0.50
with a probability of 58.8% , and $1.00 with a probability of 31.3% .
e) Min v   v 
Subject to:  0.50 y1  0.10 y2  v   v   0
0.25y1  0.05 y3  v   v   0
 0.25 y2  0.50 y3  v   v   0
y1  y2  y3  1
y1 , y2 , y1 , y4 , v  , v   0
The solutions come out the same as in part (d).
31. a) y1 = 0.5 , y2 = 0.5 , ν = -1
b) Min v   v 
Subject to: y1  3 y2  v   v   0
-4y1  2 y2  v   v   0
y1  y2  1
y1 , y2 , v  , v   0
The solution comes out the same as in part (a), with ν+ = 0 and ν- = 1 , so that
ν = ν+ - ν- = 0 – 1 = -1 .