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MODULE 4
Game Theory
TEACHING SUGGESTIONS
Teaching Suggestion M4.1: Game Theory and Conflict.
This module covers zero-sum, two-person games. Conflict is a part of our world. Students can be
Teaching Suggestion M4.2: Use of Pure Strategy Games and Dominance.
The use of pure strategy games and dominance shows students that some strategies or alterna-
ALTERNATIVE EXAMPLE
Alternative Example M4.1: Melinda (person A) and Stanley (person B) are involved in a com-
petitive situation. Both have two strategies (1 and 2) that they can play. A table showing the
winnings is presented below.
Stanley (B)
Melinda (A)
Strategy 1
Strategy 2
Strategy 1
10
1
Strategy 2
2
7
To solve this game, we determine the strategies for both players. We begin with Melinda (player
A). The equations are as follows:
For player A—Melinda:
10Q + 2(1 − Q) = 1Q + 7(1 − Q)
10Q + 2 − 2Q = 1Q + 7 − 7Q
SOLUTIONS TO DISCUSSION QUESTIONS AND PROBLEMS
M4-1. A two-person game is one in which only two players can participate. These players could
M4-2. The value of the game can be computed by multiplying the percentage that each player
plays a given strategy by the game outcomes embodied in the table of the game. Since the opti-
M4-3. A pure strategy is one in which a player will always play one strategy in the game. Dom-
M4-4. A strategy is dominated if there is another strategy that has outcomes at least as good as
M4-5. A saddle point is found by finding the largest number in each column and the smallest
number in each row. If a number in the table has both of these characteristics, it is a saddle point.
M4-6. If a game has a saddle point, it will be a pure strategy game. If there is no saddle point,
the game is a mixed strategy game.
M4-7. A mixed game is one in which each player would play every strategy a given percent of
M4-8. Strategy for X = X2
M4-9. A’s strategy = A1
M4-10. X’s strategy:
86Q + 36(1 − Q) = 42Q + 106(1 − Q),
Q =
35
57
1 − Q =
22
M4-11. 21Q + 89(1 − Q) = 116Q + 3(1 − Q)
Q =
86
181
, 1 − Q =
95
181
M4-12. A1: A selects $5 bill
A2: A selects $10 bill
B1: B selects $1 bill
B2: A selects $20 bill
a. B1 B2
b. Strategy for A:
−6Q + 11(1 − Q) = 25Q − 30(1 − Q),
72
c. Value of game = −6
41
72
+ 11
31
72
= 1.32
Since game value is positive, I’d rather be A.
M4-13. a. B1 B2
A1
6
−25
A2
−11
30
b.A’s strategy:
6Q − 11(1 − Q) = −25Q + 30(1 − Q),
So strategies remain identical
c.Value of game = 6
41
72
− 11
31
72
= −1.32
Since game value is negative, I’d rather be B.
M4-14. The game can be reduced to a 2 2 game, since X would never play X1 or X4 since X
stands to lose in every eventuality under those two strategies. Thus, the game is
Y1 Y2
X2
12
8
M4-15. A1: Shoe Town does no advertising.
A2: Shoe Town invests $15,000 in advertising.
B1: Fancy Foot does nothing.
This particular problem has a saddle point with strategies A2 and B3 and game value of −1.
M4-16. a. B1 B2 B3
M4-17. The value of the game is 3.17. The optimal strategies for A and B can be computed
along with the value of the game using QM for Windows. The results are presented below.
Mixed Strategy
For player A:
Probability of strategy 1 0.390
Probability of strategy 2 0.244
Probability of strategy 3 0.366
M4-18. Strategy X2 is dominated by both X1 and X3, so we may eliminate X2. When this is elim-
M4-19.
Y1
Y2
P
1 − P
Expected gain
X1
Q
2
−4
2P − 4(1 − P)
X2
1 − Q
6
10
6P + 10(1 − P)
Expected gain 2Q + 6(1 − Q) −4Q + 10(1 − Q)
To solve this as a mixed strategy game, the expected gain for each decision should be the same.
Taking the expected gain for player X, we have
Also, 1 − P = 1 − 1.4 = −0.4
Thus, for a mixed strategy game, the probability that X would play strategy 1 must be 1.4 which
M4-20. The best strategy for Petroleum Research is to play strategy 14 all of the time. Petrole-
um Research can expect to get a return of $3 million from this approach. These results are sum-
marized below.
Mixed Strategy
For player A:
Probability of strategy 1 0.000
Probability of strategy 7 0.000
Probability of strategy 8 0.000
Probability of strategy 9 0.000
Probability of strategy 10 0.000
Probability of strategy 11 0.000
Probability of strategy 12 0.000
Probability of strategy 13 0.000
