Supplement 1
Game Theory
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
S-1
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Learning Objectives
After completing this supplement,
students will be able to:
• Understand the principles of zerosum, two person games.
• Analyze pure strategy games and use
dominance to reduce the size of the
game.
• Solved mixed strategy games when
there is no saddle point.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
S-2
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Supplement Outline
S1.1 Introduction
S1.2 Language of games
S1.3 Minimax Criterion
S1.4 Pure strategy games
S1.5 Mixed strategy games
S1.6 Dominance
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
S-3
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Introduction
• Game: a contest involving two or
more decision makers, each of
whom wants to win.
• Game theory: the study of how
optimal strategies are formed in
conflict
• Games classified by:
• Number of players
• Sum of all payoffs
• Number of strategies employed
• Zero-sum game: the sum of the
losses must equal the sum of the
gains
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
S-4
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Payoff Table
Game Player
X’s Strategies
Game player Y’s
strategies
X1
Use radio
X2
Use
newspaper
Y1
Use
radio
3
Y2
Use
newspaper
5
1
-2
+ entry, s X wins and Y loses
- entry, Y wins and X loses
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
S-5
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Outcomes
Store X’s
Strategy
X1 Radio
X1 Radio
X2
Newspaper
X2
Newspaper
Store Y’s
Strategy
Outcome %
change in
market share)
Y1 Radio
X wins 3
Y loses 3
Y2
X wins 5
Newspaper
Y loses 5
Y1 Radio
X wins 1
Y loses 1
Y2
X loses 2
Newspaper
Y wins 2
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
S-6
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Minimax Criterion
• In a zero-sum game, each
person can choose the strategy
that minimizes the maximum
loss
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
S-7
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Minimax Criterion
Maximums of
minimums
Saddle point
Y1
Y2 Minimum
X1
3
5
3
X2
1
-2
-2
Maximum
3
5
Minimum of maximums
Note: an equilibrium or saddle point exists if the
upper value of the game is equal to the lower
value of the game. This is called the value of
the game.
This is a “pure strategy” game
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
S-8
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Whenever a saddle point is
present, the strategy a player
should follow will always be the
same, regardless of the strategy
of the other player.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
S-9
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Pure Strategy
X’s pure
strategy
Y’s pure
strategy
Second
Player’s (Y)
Strategies
Y1
Y2
First
Player’s
(X)
Strategies
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
X1
3
5
X2
1
-2
S-10
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Pure Strategy
Minimax Criterion
Player Y’s
Strategies
Player X’s
strategies
Minimum
Row
Number
Y1
Y2
X1
10
6
6
X2
-12
2
-12
10
6
Maximum
Column
Number
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
S-11
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Mixed Strategy Game
X1
Q
X2
1-Q
Y1
P
4
Y2
1-P
2
Expected
Gain
4P+2(1-P)
1
10
1P+10(1-p)
4Q+1(1-Q) 2Q+10(1-q)
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
S-12
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Solving for P & Q
4P+2(1-P) = 1P+10(1-P)
or: P = 8/11 and 1-p = 3/11
Expected payoff:
EPX=1P+10(1-P)
=1(8/11)+10(3/11) = 3.46
4Q+1(1-Q)=2Q+10(1-q)
or: Q=9/11 and 1-Q = 2/11
Expected payoff:
EPY=3.46
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
S-13
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Dominance
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, 8e
by Render/Stair/Hanna
S-14
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Domination
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, 8e
by Render/Stair/Hanna
S-15
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Domination
Initial Game
Y1
Y2
Y3
Y4
X1
-5
4
6
-3
X2
-2
6
2
-20
Game after dominated strategies are removed
Y1
Y4
X1
-5
-3
X2
-2
-20
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
S-16
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458