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where
x
A
means the probability distribution on
X
uniformly concentrated on
A
.That
is to say,
1
A
,
x
A
(
C
)=
if
C
∈
A
,
x
A
(
C
)=
0
,
if
C
∈
X
−
A
.
The last member of (
3.
2
)
does not depend on
B
∈
B
,
in other words, it takes the
same value for any
B
∈
B
. In fact, bearing in mind properties 3 and 4, we have that
T
s
n
B
, from which we
can easily obtain the independence. The expression (
3.2
) can also be c
om
puted by
M
M
x
A
,
T
s
1
B
=
M
x
A
,
T
s
1
T
s
2
M
(
x
A
,
B
)=
M
(
x
A
,
T
1
B
)=
...
2
(
A
,
y
B
)
,w
he
r
e
y
B
is the distribution on
Y
uniformly concentrated on
B
⊂
Y
.The
game
G
=(
X
,
Y
,
M
)
, constructed above, will be called the associated (averaged)
game of
G
=(
X
,
Y
,
M
)
.
Theorem 1.
Given the game G
,letx, y be the
o
ptim
al m
i
xe
d strategies
for players I and II respectively in the associated game G
=(
X
,
Y
,
M
)
=(
X
,
Y
,
M
)
. Then the
mixed strategies of game G defined by
x
)
A
,
(
A
x
(
A
)=
A
∈
X
,
y
)
B
,
(
B
y
(
B
)=
B
∈
Y
are optimal in the game G and both games have the same value.
Proof.
By definition (
3.2
), we easily obtain that
A
∈
X
x
A
M
A
,
B
=
A
∈
X
x
A
M
(
x
A
,
B
)
=
A
∈
X
x
A
1
A
C
∈
A
M
(
C
,
B
)
,
B
∈
Y
.
And therefore we have
M
x
B
=
,
M
(
x
,
B
)
.
A similar reasoning leads to
M
A
y
,
=
(
,
)
.
M
A
y
Hence the inequalities
M
A
y
≤
M
x
B
,
,
M
(
x
,
y
)
≤
,
A
∈
X
,
B
∈
Y
become
M
(
A
,
y
)
≤
M
(
x
,
y
)
≤
M
(
x
,
B
)
,
A
∈
X
,
B
∈
Y
which completes the proof.
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