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1
m
n
M
(
x
F
,
B
0
)=
min
B
Y
M
(
x
F
,
B
)=
min
B
M
(
A
,
B
)
∑
∈
∈
Y
A
∈F
1
m
n
n
∑
1
m
n
n
∑
=
|
∩
||
∩
∩
|
=
|
∩
∩
|
min
B
∑
A
∈F
c
i
B
L
i
A
B
L
i
min
B
∑
A
∈F
c
i
s
i
A
B
L
i
i
=
1
i
=
1
n
i
=
1
c
i
s
i
s
i
m
n
−
1
n
i
=
1
c
i
s
i
1
m
n
1
m
=
min
B
=
min
s
1
,
s
2
,...,
s
n
n
∑
where
s
i
=
|
B
∩
L
i
|
,andso
s
i
=
s
. Therefore to obtain the solution of this game
i
=
1
we need to solve the following problem: Let
c
1
,
c
2
, ...,
c
n
, be positive real numbers
satisfying (
3.31
), and
s
a positive integer. Minimize the quadratic function
n
i
=
1
c
i
x
i
.
(
,
,...,
)=
f
x
1
x
2
x
n
(3.32)
subject to
n
i
=
1
x
i
=
s
and
x
i
i
=
1
,
2
,...,
n
non negative integers
(3.33)
,
α
,...,
α
≤
,
=
,
,...,
If
n
,
thenan
optimal strategy for player II is the uniformly concentrated distribution in
B
0
,where
α
n
is a solution for this problem and
α
m
i
1
2
i
1
2
B
0
=
{
(
i
,
j
)
:1
≤
j
≤
α
i
for
i
=
1
,
2
,..,
n
}
and the value of the game is
n
i
=
1
c
i
α
1
m
i
v
=
.
To solve the game of Example
3
we need to minimize (
3.32
) subject to (
3.33
),
which we do by taking advantage of the particular properties of this problem.
Lemma 5.
The minimum of the function f , given by (
3.32
), for real values x
i
satis-
fying
i
1
x
i
=
s, is achieved when x
i
takes the following values r
i
∑
=
s
r
i
=
,
i
=
1
,
2
,...,
n
(3.34)
n
j
=
1
c
−
1
c
i
j
and for any other x
1
,x
2
, ..., x
n
the following equality is satisfied
n
i
=
1
c
i
x
i
=
n
i
=
1
c
i
r
i
+
n
i
=
1
c
i
(
x
i
−
r
i
)
2
.
(3.35)
Proof.
It is sufficient to prove (
3.35
) because from this it follows that the minimum
condition is satisfied. Write
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