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Game Theory
In previous chapters, we have encountered many situations in which a
single
decision maker
chooses an optimal decision without reference to the effect that the decision has on other deci
sion makers (and without reference to the effect that the decisions of others have on him or her).
In many business situations, however, two or more decision makers simultaneously choose an ac
tion, and the action chosen by each player affects the rewards earned by the other players. For ex
ample, each fastfood company must determine an advertising and pricing policy for its product,
and each company’s decision will affect the revenues and pro±ts of other fastfood companies.
Game theory is useful for making decisions in cases where two or more decision makers
have con²icting interests. Most of our study of game theory deals with situations where there
are only two decision makers (or players), but we brie²y study
n
person (where
n
±
2) game
theory also. We begin our study of game theory with a discussion of twoplayer games in which
the players have no common interest.
14.1
TwoPerson ZeroSum and ConstantSum Games: Saddle Points
Characteristics of TwoPerson ZeroSum Games
1
There are two players (called the
row
player and the
column
player).
2
The row player must choose 1 of
m
strategies. Simultaneously, the column player must
choose 1 of
n
strategies.
3
If the row player chooses his
i
th strategy and the column player chooses his
j
th strat
egy, then the row player receives a reward of
a
ij
and the column player loses an amount
a
ij
. Thus, we may think of the row player’s reward of
a
ij
as coming from the column player.
Such a game is called a
twoperson zerosum game,
which is represented by the ma
trix in Table 1 (the game’s
reward matrix
). As previously stated,
a
ij
is the row player’s
TABLE
1
Example of TwoPerson ZeroSum Game
Row Player’s
Column Player’s Strategy
Strategy
Column 1
Column 2
±±±
Column
n
Row 1
a
11
a
12
²²²
a
1
n
Row 2
a
21
a
22
²²²
a
2
n
²
²²
²
Row
m
a
m
1
a
m
2
²²²
a
mn
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CHAPTER
14
Game Theory
reward (and the column player’s loss) if the row player chooses his
i
th strategy and the
column player chooses his
j
th column strategy.
For example, in the twoperson zerosum game in Table 2, the row player would re
ceive two units (and the column player would lose two units) if the row player chose his
second strategy and the column player chose his ﬁrst strategy.
A twoperson zerosum game has the property that for any choice of strategies, the sum
of the rewards to the players is zero. In a zerosum game, every dollar that one player
wins comes out of the other player’s pocket, so the two players have totally conﬂicting in
terests. Thus, cooperation between the two players would not occur.
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