PART III. Two-Person General-Sum Games
1. Bimatrix Games — Safety Levels
The simplest case to consider beyond two-person zero-sum games are the two-person
non-zero-sum 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), Addison-Wesley Publishing Co. Inc.
1.1 General-Sum Strategic Form Games.
Two-person general-sum games may
be de±ned 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
real-valued functions
u
1
(
x, y
)and
u
2
(
x, y
) de±ned on
X
×
Y
, representing the payo²s to
the two players. If I chooses
x
∈
X
and II chooses
y
∈
Y
, then I receives
u
1
(
x, y
)andII
receives
u
2
(
x, y
).
A ±nite two-person game in strategic form can be represented as a matrix of ordered
pairs, sometimes called a bimatrix. The ±rst component of the pair represents Player I’s
payo² and the second component represents Player II’s payo². 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 ±nite 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
ij