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Theorem 3.
Let
(
X
,
Y
,
M
)
be a game on a lattice satisfying
=
F
T
s
X
Y
=
,
X
(3.8)
(
,
)=
(
,
)
M
T
s
A
T
s
B
M
A
B
(
A
∈
X
,
B
∈
Y
,
s
=
1
,
2
,...
n
)
.
An optimal strategy for player II is the uniform distribution on Y
=
F
,
1
|F|
=
1
m
n
,
y
(
B
)=
y
F
(
B
)=
B
∈ F.
(3.9)
If we call A
0
∈
X a strategy which fulfills
(
,
y
F
)=
(
,
y
F
)
,
max
A
∈
X
M
A
M
A
0
(3.10)
the
n
an optimal strategy for player I is the uniformly concentrated distribution
on A
0
:
1
|
,
if
A
∈
A
0
|
A
0
x
A
0
(
)=
A
(3.11)
0 f
A
∈
A
0
(
,
y
F
)
and the value of the game is M
A
0
given by (
3.10
).
Proof.
Similar to the proof of the Theorem
2
.
These theorems give a general method for solving those games satisfying (
3.4
)
or (
3.8
). In each case it will be sufficient to determine either
B
0
∈
Y
satisfying (
3.6
)
or
A
0
∈
X
satisfying (
3.10
).
Example 1.
In a conflict situation an intruder (the hider) has to carry out a sab-
otage on the perimeter of a protected zone. He has to perform the action over
n
consecutive days, and has to position himself each day at one of
m
strategic points
placed on this border in order to set a device on it. These points are represented
by 1, 2, 3, ...,
m
, considering point 1 as next to point
m
. The first day the hider
can take his place at any of the
m
points, on successive days he can either stay,
move one step to the right or move one step to the left. This constraint on the
movements of the hider can be considered as a limit on his maximum speed, and
expresses that his movements are difficult, e.g. because he has to use safe ways
to go from one point to another. Furthermore, the perimeter is protected by a pa-
troller (the searcher) who every day selects one of the
m
strategic points to in-
spect. This selection has to be done satisfying different constraints depending on
the situation. If it is assumed
n
=
6and
m
=
9, Fig.
3.3
shows a representation
of strategy
{
(
1
,
7
)
,
(
2
,
7
)
,
(
3
,
8
)
,
(
4
,
9
)
,
(
5
,
9
)
,
(
6
,
1
)
}
of the intruder on the lattice
L
=
{
1
,
2
,...,
6
,}×{
1
,
2
,...,
9
}
and on the cyclic set
{
1
,
2
,...,
9
}
.
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