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of the points 0
represent points an
appropriately small distance to the left and right of
x
respectively. The argument
used for the particular case now show that choosing these points with probabilities
10
,
a
+
,
b
+
,
(
1
−
b
)
−,
(
1
−
a
)
−,
1where
x
+
and
x
−
/
36 respectively is an optimal Infiltrator strategy.
One might have hoped that the expression for
v
36
,
3
/
36
,
5
/
36
,
5
/
36
,
3
/
36
,
10
/
(
a
,
b
)
in Theorem
4
also covers
the case
b
≥
1
/
2
.
However this is not the case because Baston and Bostock [
2
]
(
/
,
/
)=
/
=
/
=
/
,
showed that
v
1
3
1
2
1
4 whereas, for
a
1
3and
b
1
2
Theorem
3
gives
(
/
,
/
)=
−
{
(
+
λ
)
/
(
)
,
(
+
λ
)
/
(
)
}
=
/
.
v
1
3
1
2
1
max
1
2
λ
2
2
λ
1
6
1
1
0
0
Γ
(
,
)
/
≤
<
/
Theorem
4
enables us to express the value of
a
b
when 1
3
b
1
2ina
simpler form than Lee [
8
].
Theorem 5.
Let a
≤
b
,
1
/
3
≤
b
<
1
/
2
and
λ
i
defined by
(
9.2
).
(i) If
λ
2
≥
2
,
v
(
a
,
b
)=(
2
m
−
6
)
/
3
m
where
m
=
min
{
2
λ
0
,
3
λ
1
,
6
λ
2
}.
(ii) If
λ
2
=
1
,
v
(
a
,
b
)=(
m
−
1
)
/
2
m
where
m
=
min
{
λ
0
,
2
λ
1
−
1
}.
Λ
+
=
{
Proof.
(i) Let
λ
2
≥
2
,
then
3
}
and
λ
3
=
0 so, by Theorem
4
,
3
+
λ
0
3
2
+
λ
1
3
1
+
λ
2
3
2
3
−
2
2
2
3
2
v
(
a
,
b
)=
1
−
max
{
λ
0
,
λ
1
,
λ
2
}
=
max
{
λ
0
,
λ
1
,
λ
2
}
6
and (i) follows.
(ii) Let
Λ
+
=
{
λ
2
=
1
,
then
2
,
3
}
so, by Theorem
4
,
3
+
λ
0
3
2
+
λ
1
3
1
+
λ
0
2
λ
1
v
(
a
,
b
)=
1
−
max
{
λ
0
,
λ
1
,
λ
0
,
1
}
2
λ
1
−
Now
(
3
+
λ
0
)
/
(
3
λ
0
)
≤
(
1
+
λ
0
)
/
(
2
λ
0
)
because
λ
0
≥
3and
(
λ
1
)
/
(
2
λ
1
−
1
)
≥
(
2
+
λ
1
)
/
(
3
λ
1
)
because
λ
1
≥
2
.
Thus
1
1
v
(
a
,
b
)=
1
/
2
−
max
{
0
,
)
}
2
λ
2
(
2
λ
−
1
1
and (ii) follows.
We now give examples to show that every case in the theorem arises:
=
/
,
=
/
≥
<
<
,
a
1
8
b
3
8gives
λ
2and6
λ
3
λ
2
λ
2
2
1
0
a
=
3
/
39
,
b
=
13
/
39 gives
λ
2
≥
2and2
λ
0
<
3
λ
1
<
6
λ
2
,
a
=
2
/
15
,
b
=
5
/
15 gives
λ
2
≥
2and3
λ
1
<
2
λ
0
<
6
λ
2
,
a
=
3
/
10
,
b
=
4
/
10 gives
λ
2
=
1and2
λ
1
−
1
<
λ
0
,
a
=
2
/
20
,
b
=
9
/
20 gives
λ
2
=
1and
λ
0
<
2
λ
1
−
1
.
Acknowledgements
The first author was supported, in part, by NATO Grant PST.CLG.976391.
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