GAME THEORY
Thomas S. Ferguson
Part III. TwoPerson GeneralSum Games
1. Bimatrix Games — Safety Levels.
1.1 GeneralSum Strategic Form Games.
1.2 GeneralSum Extensive Form Games.
1.3 Reducing Extensive Form to Strategic Form.
1.4 Overview.
1.5 Safety Levels.
1.6 Exercises.
2. Noncooperative Games.
2.1 Strategic Equilibria.
2.2 Examples.
2.3 Finding All PSE’s.
2.4 Iterated Elimination of Strictly Dominated Strategies.
2.5 Exercises.
3. Models of Duopoly.
3.1 The Cournot Model of Duopoly.
3.2 The Bertrand Model of Duopoly.
3.3 The Stackelberg Model of Duopoly.
3.4 Entry Deterrence.
3.5 Exercises.
4. Cooperative Games.
4.1 Feasible Sets of Payoff Vectors.
4.2 Cooperative Games with Transferable Utility.
4.3 Cooperative Games with NonTransferable Utility.
4.4 EndGame with an AllIn Player.
4.5 Exercises.
III – 1
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PART III. TwoPerson GeneralSum Games
1. Bimatrix Games — Safety Levels
The simplest case to consider beyond twoperson zerosum games are the twoperson
nonzerosum games.
Examples abound in Economics: the struggle between labor and
management, the competition between two producers of a single good, the negotiations
between buyer and seller, and so on. Good reference material may be found in books of
Owen and Straﬃn already cited. The material treated in Part III is much more oriented
to economic theory.
For a couple of good references with emphasis on applications in
economics, consult the books,
Game Theory for Applied Economists
by Robert Gibbons
(1992), Princeton University Press, and
Game Theory with Economic Applications
by H.
Scott Bierman and Luis Fernandez (1993), AddisonWesley Publishing Co. Inc.
1.1 GeneralSum Strategic Form Games.
Twoperson generalsum games may
be defined in extensive form or in strategic form. The
normal
or
strategic
form of a two
person game is given by two sets
X
and
Y
of pure strategies of the players, and two
realvalued functions
u
1
(
x, y
) and
u
2
(
x, y
) defined on
X
×
Y
, representing the payoffs to
the two players. If I chooses
x
∈
X
and II chooses
y
∈
Y
, then I receives
u
1
(
x, y
) and II
receives
u
2
(
x, y
).
A finite twoperson game in strategic form can be represented as a matrix of ordered
pairs, sometimes called a bimatrix. The first component of the pair represents Player I’s
payoff and the second component represents Player II’s payoff. The matrix has as many
rows as Player I has pure strategies and as many columns as Player II has pure strategies.
For example, the bimatrix
⎛
⎝
(1
,
4)
(2
,
0)
(
−
1
,
1)
(0
,
0)
(3
,
1)
(5
,
3)
(3
,
−
2)
(4
,
4)
(0
,
5)
(
−
2
,
3)
(4
,
1)
(2
,
2)
⎞
⎠
(1)
represents the game in which Player I has three pure strategies, the rows, and Player II
has four pure strategies, the columns. If Player I chooses row 3 and Player II column 2,
then I receives
−
2 (i.e. he loses 2) and Player II receives 3.
An alternative way of describing a finite two person game is as a pair of matrices. If
m
and
n
representing the number of pure strategies of the two players, the game may be
represented by two
m
×
n
matrices
A
and
B
. The interpretation here is that if Player I
chooses row
i
and Player II chooses column
j
, then I wins
a
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 Spring '10
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