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an
d an optimal strategy for player II is the uniformly concentrated distribution on
B
0
,where
B
0
is
B
0
=
{
(
i
,
j
)
:1
≤
j
≤
3
,
for
i
=
1
,
2
,
3
,
1
≤
j
≤
2
,
for
i
=
4
}.
A represent
at
ion of this set can be seen in Fig.
3.5
. Figure
3.6
shows some of the
elements of
B
0
.
3.4.2 Weighted Inspection Game
An inspection game is a mathematical model of a situation where an inspec-
tor verifies that another part, the inspectee, adheres to certain rules. Typically,
the inspector's resources are limited, so the verification can only be partial. The
weighted inspection game (WIG) is a game
(
X
,
Y
,
M
)
on the lattice
L
satisfying
X
=
F
Y
=
{
B
:
|
B
|
=
s
}
n
i
=
1
c
i
|
B
∩
L
i
||
A
∩
B
∩
L
i
|
M
(
A
,
B
)=
where
c
i
are constants such that
0
<
c
1
≤
c
2
≤ ...≤
c
n
.
(3.31)
Therefore, in this game the inspector, player I, makes a single inspection on each
column of
L
(each column can represent a different day, zone, product, etc.), player
II, the inspectee, chooses a subset of
L
of cardinality equal to
s
to hide one object
at each one of its points. If player I finds one of the objects hidden by player II in
column
L
i
, then he receives a quantity
c
i
for every object hidden by player II in this
column. This game has been studied in [
11
], where the case
c
1
=
c
2
=
...
=
c
n
=
c
is completely solved.
Example 3.
A farmer has 800 cows distributed in 8 cowsheds, C
1
,C
2
, ..., C
8
, 100
in every cowshed. He has decided to administer an illegal substance to 80 of his
cows. The public health agency will test one cow from each cowshed every month.
If the illegal substance is detected, then the sanitary inspector will test all the cows
of that cowshed and the farmer has to pay a fine of
c
i
times the number of positive
results. The coefficient
c
i
depends on the village where the cowshed is allocated.
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