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L
where column
i
corresponds to computer
C
i
and
row
j
to the one-hour period beginning at
j
o'clock. The pure strategies for the
hacker are the subsets of
L
of cardinality equal to
s
; the pure strategies for the com-
pany are the subsets of
L
with just one point in each column; if the hacker is not
detected, he receives a quantity
c
i
for every computer
C
i
where he has made at least
k
incursions, and zero otherwise. This payoff can be formalized as follows, when
the hacker uses his strategy
A
, and the company its strategy
B
,
M
=
{
1
,
2
,...,
20
}×{
1
,
2
,...,
24
}
(
,
)
A
B
is defined by
⎧
⎨
j
∈
H
c
j
if
A
∩
B
= ∅
,
M
(
A
,
B
)=
⎩
0if
A
∩
B
= ∅
,
where
H
is the set of the columns of
A
with
k
elements at least. The study of this
problem can be simplified with the method developed in this chapter and it is studied
in [
11
].
A two-person zero-sum game will be expressed by
G
=(
,
,
)
X
Y
M
where
X
,
Y
are
the sets of pure strategies for players I and II, respectively, and
M
:
X
×
Y
→
R
(3.1)
is the payoff function which represents the winnings of player I and the losses of
player II. Player I chooses a strategy
A
∈
X
, player II chooses a strategy
B
∈
Y
and
these choices determine the payoff
M
(
A
,
B
)
to player I and
−
M
(
A
,
B
)
to player II.
Throughout this chapter
X
and
Y
are finite sets, therefore a probability distribu-
tion on
X
, that is to say, a mixed strategy for player I, can be written as a function
x
:
X
→
R
C
∈
X
x
(
C
)=
1. Similarly, a mixed strategy for
such that
x
(
C
)
≥
0forall
C
∈
X
and
player II will be given by a function
y
:
Y
→
R
C
∈
Y
y
(
C
)=
1. When the players use their mixed
such that
y
(
C
)
≥
0forall
C
∈
Y
and
strategies
x
and
y
, the payoff
M
(
x
,
y
)
is the expected value of
M
(
A
,
B
)
.
3.2 Transformations on the Strategy Space
In this section we develop a method to facilitate the study and resolution of games
where the strategies for the players are subsets of a finite set and these and the
payoff function satisfy certain conditions. This method is based on a well known
invariance property [
1
,
8
]: we show how, given a game
G
, we can build a game
G
,
which is easier to study than
G
and has the same value as
G
. This game has fewer
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