Module 4
Game Theory
Prepared by Lee Revere and John Large
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-1
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Learning Objectives
Students will be able to:
1. Understand the principles of
zero-sum, two-person games.
2. Analyze pure strategy games
and use dominance to reduce the
size of the game.
3. Solve mixed strategy games
when there is no saddle point.
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-2
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Module Outline
M4.1 Introduction
M4.2 Language of Games
M4.3 The Minimax Criterion
M4.4 Pure Strategy Games
M4.5 Mixed Strategy Games
M4.6 Dominance
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-3
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Introduction
Game theory is the study of how
optimal strategies are formulated
in conflict. Game theory has been
effectively used for:
 War strategies
 Union negotiators
 Competitive business strategies
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-4
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Introduction
(continued)
 Game models are classified by the
number of players, the sum of all
payoffs, and the number of
strategies employed.
 A zero sum game implies that what
is gained by one player is lost for
the other.
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-5
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Language of Games
Consider a duopoly competitive business
market in which one company is
considering advertising in hopes of luring
customers away from its competitor. The
company is considering radio and/or
newspaper advertisements.
Let’s use game theory to determine the
best strategy.
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-6
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Language of Games
(continued)
Below is the payoff matrix (as a percent of
change in market share) for Store X. A positive
number means that X wins and Y loses, while a
negative number implies Y wins and X loses.
STORE X’s
PAYOFFs
X’s strategy 1
(use radio)
X’s strategy 2
(use newspaper)
Y’s
Y’s strategy 2
strategy 1
(use
(use radio) newspaper)
3
5
1
-2
Note: Although X is considering the
advertisements (therefore the results favor X), Y
must play the game.
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-7
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Language of Games
(continued)
Store X’s
Strategy
Stores Y’s
Strategy
Outcome (% change in
market share)
X1: Radio
Y1: Radio
X wins 3
Y loses 3
X1: Radio
Y2:
Newspaper
X wins 5
Y loses 5
X2:
Newspaper
Y1: Radio
X wins 1
Y loses 1
X2:
Newspaper
Y2:
Newspaper
X loses 2
Y wins 2
Note: Although X is considering the
advertisements (therefore the results favor X), Y
must play the game.
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-8
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
The Minimax Criterion
The minimax criterion is used in
a two-person zero-sum game. Each
person should choose the strategy
that minimizes the maximum loss.
Note: This is identical to maximizing one’s
minimum gains.
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-9
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
The Minimax Criterion
(continued)
The upper value of the game is
equal to the minimum of the
maximum values in the columns.
The lower value of the game is
equal to the maximum of the
minimum values in the rows.
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-10
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
The Minimax Criterion
(continued)
Lower Value of the
Game: Maximum of
the minimums
STORE X’s
PAYOFFs
Y1
(radio)
Y2
(newspaper)
Minimum
X1
(radio)
3
5
3
X2
(newspaper)
1
-2
2
Maximum
3
5
Upper Value of the Game:
Minimum of the maximums
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-11
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
The Minimax Criterion
(continued)
A saddle point condition exists if the upper and
lower values are equal. This is called a pure
strategy because both players will follow the
same strategy.
STORE X’s
PAYOFFs
Y1
(radio)
Y2
(newspaper)
Minimum
X1
(radio)
3
5
3
X2
(newspaper)
1
-2
2
Maximum
3
5
Saddle point: Both upper and lower values are 3.
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-12
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
The Minimax Criterion
(continued)
Let’s look at a second example of a pure
strategy game.
STORE X’s
PAYOFFs
Y1
(radio)
Y2
(newspaper)
Minimum
X1
(radio)
10
6
6
X2
(newspaper)
-12
2
Maximum
10
6
Lower value
Saddle point
-12
Upper value
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-13
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Mixed Strategy Game
A mixed strategy game exists
when there is no saddle point. Each
player will then optimize their
expected gain by determining the
percent of time to use each strategy.
Note: The expected gain is determined using
an approach very similar to the expected
monetary value approach.
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-14
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Mixed Strategy Games
(continued)
Y1
(P)
Y2
(1-P)
Expected
Gain
X1
4
2
4P + 2(1-P)
(Q)
X2
1
10
1p + 10(1-P)
(1-Q)
Expected 4Q +
2Q +
Gain
1(1-Q) 10(1-Q)
Set these two equations equal to
each other and solve for Q
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-15
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Set these two equations equal to
each other and solve for P
Each player seeks to maximize his/her expected
gain by altering the percent of time (P or Q) that
he/she use each strategy.
Mixed Strategy Games
(continued)
 4P + 2(1-P) = 1P + 10(1-P)
4P – 2P – 1P + 10P = 10 – 2
P = 8/11 and 1-P = 3/11
Expected payoff: 1P + 10(1-P)
= 1(8/11) + 10(3/11)
= 3.46
 4Q + 1(1-Q) = 2Q + 10(1-Q)
4Q – 1Q – 2Q + 10Q = 10 – 1
Q = 9/11 and 1-Q = 2/11
Expected payoff: 2Q + 10(1-Q)
= 2(9/11) + 10(2/11)
= 3.46
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-16
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Dominance
Dominance is a principle that can
be used to reduce the size of games
by eliminating strategies that would
never be played.
Note: A strategy can be eliminated if all its
game’s outcomes are the same or worse than
the corresponding outcomes of another
strategy.
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-17
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Dominance
(continued)
Initial game
Y1
Y2
X1
4
3
X2
2
20
X3
1
1
X3 is a dominated strategy
Game after removal of dominated strategy
Y1
Y2
X1
4
3
X2
2
20
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-18
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Dominance
(continued)
Initial game
Y1
Y2
Y3
Y4
X1
-5
4
6
-3
X2
-2
6
2
-20
Game after removal of dominated strategies
Y1
Y4
X1
-5
-3
X2
-2
-20
To accompany Quantitative Analysis
for Management,9e
by Render/Stair/Hanna
M4-19
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458