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have properties 1, 2, 5 and 6. Figure
3.2
shows the effect of transformations
T
1
,
T
2
,
T
3
and
T
4
on the subset
A
=
(
1
,
4
)
,
(
2
,
2
)
,
(
2
,
3
)
,
(
2
,
4
)
,
(
3
,
3
)
,
(
3
,
4
)
,
(
3
,
7
)
,
(
4
,
2
)
,
(
4
,
4
)
}
of the lattice
{
1
,
2
,
3
,
4
}×{
1
,
2
,...,
7
}
. It is also easy to see that
F
=
F,
=
,
,...,
.
T
s
s
1
2
n
Therefore we can state the following theorems.
(
,
,
)
Theorem 2.
Let
X
Y
M
be a game on a lattice satisfying
=
F
T
s
Y
X
=
Y
,
(3.4)
M
(
T
s
A
,
T
s
B
)=
M
(
A
,
B
)
(
A
∈
X
,
B
∈
Y
,
s
=
1
,
2
,...
n
)
.
An optimal strategy for player I is the uniform distribution on X .
1
|F|
=
1
m
n
,
x
(
A
)=
x
F
(
A
)=
A
∈
X
.
(3.5)
Let B
0
∈
Y such that
1
m
n
A
∈F
(
x
F
,
)=
(
,
)
min
B
M
B
min
M
A
B
∈
Y
1
m
n
A
∈F
=
M
(
A
,
B
0
)=
M
(
x
F
,
B
0
)
.
(3.6)
Th
us
an optimal strategy for player II is the distribution on Y uniformly concentrated
in B
0
:
1
|
,
if
B
∈
B
0
|
B
0
y
B
0
(
B
)=
(3.7)
0 f
B
∈
B
0
and the value of the game is M
(
x
F
,
B
0
)
given by (
3.6
).
Proof.
Since properties 1-6 of tr
an
sfor
m
a
tions
T
s
work
h
ere, we can apply
=
X
M
the set
X
Theorem
1
. In the associated game
G
,
Y
,
=
{F}
contains the
only element
which will be the o
pt
imal strategy for player I. The optimal strategy
for player II will be a pure strategy
B
0
such that
F
M
F,
B
=
M
F,
B
0
.
m
i
B
The two strategies
x
F
,
y
B
0
are obtained from the two optimal strategies
F
,
B
0
,of
the associated game
G
, completing the proof.
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