Information Technology Reference
In-Depth Information
which a value has been found (
), these RED strategies have been
derived from coverings of
I
so it is difficult to see how they can be modified to
improve the payoff for other values of
β
=
2
/
(
n
−
2
)
. On the other hand optimal alternatives to
the uniform distribution for BLUE abound. For instance, when
β
β
=
1
/
2
,
the value is
1/6 so BLUE can afford to ignore all points in
[
0
,
1
/
6
)
and
(
5
/
6
,
1
]
and concentrate
the distribution in an appropriate way in
[
1
/
6
,
5
/
6
]
.
(
−
β
)
/
To illustrate the point we show that the value of the game is
2
9when
n
2
equals 1/5 for
n
/
≤
β
≤
/
.
(
−
−
β
)
/
=
β
=
/
1
5
5
7
Notice that
n
1
2 and 3 when
1
5
and equals 1/7 for
n
=
3and4when
β
=
5
/
7
.
Thus 1
/
5
≤
β
≤
5
/
7islikelytobethe
maximum range of values of
β
giving the game value
(
2
−
β
)
/
9 because not only
does Lemma
1
tell us that RED can guarantee more (namely
(
1
−
β
)
/
4) if
β
<
1
/
5
but it also suggests that RED cannot guarantee as much if
β
>
5
/
7
.
Let
⎧
⎨
0
if
x
<
(
2
−
β
)
/
9
,
−
β
)
/
9
x
+
β
)
if
F
(
x
)=
1
/
(
1
+
β
)
−
(
2
(
1
(
2
−
β
)
/
9
≤
x
<
1
/
2
,
⎩
18
(
19
β
+
7
)
/
(
1
+
β
)
if
x
=
1
/
2
and
F
(
x
)=
1
−
F
(
1
−
x
)
for 1
/
2
<
x
≤
1
.
As
x
→
1
/
2
−,
F
(
x
)
→
(
5
+
2
β
)
/
(
9
+
9
β
)
≤
1
/
2when
β
≥
1
/
5
.
Thus
F
(
x
)
is a
probability distribution over
I
whichhasajumpat1/2when
β
>
1
/
5 and is strictly
concave in the interval
)
.
Suppose BLUE employs the strategy
F
[(
2
−
β
)
/
9
,
1
/
2
We first show that the properties of
F
mean that we only need to find the payoff of certain RED intervals in detail in order
to find RED's best reply to
F
(
x
)
.
.
If
[
a
,
a
+
x
]
and
[
b
,
b
+
x
]
are two RED intervals with
F
(
a
+
x
)
−
F
(
a
)
<
F
(
b
+
x
)
−
F
(
b
)
,
then
[
a
,
a
+
x
]
gives a better payoff than
[
b
,
b
+
x
]
so we need only consider
[
a
,
a
+
x
]
.
Therefore
any RED interval of the form
[
a
,
a
+
x
]
with 0
<
a
≤
(
2
−
β
)
/
9 gives an inferior payoff than
[
0
,
x
]
and so can be ignored.
Furthermore
F
(
a
+
x
)
−
F
(
a
)
<
F
(
b
+
x
)
−
F
(
b
)
if
(
2
−
β
)
/
9
≤
b
<
a
<
a
+
x
<
1
/
2so
intervals
[
a
,
a
+
x
]
⊆
[
2
−
β
)
/
9
,
1
/
2
)
have an inferior payoff to
[
1
/
2
−
x
,
1
/
2
)
.
For
a
2so,
for intervals having 1/2 as an interior point, it is only necessary to consider those symmetric
about 1/2.
<
1
/
2
<
a
+
x
,
F
(
a
+
x
)
−
F
(
a
)
has a minimum in
a
for fixed
x
at
a
=(
1
−
x
)
/
Finally the symmetry of
F
means that we can assume a RED interval starts in
[
0
,
1
)
.
Thus, in finding RED's best reply to
F
/
2
,
the analysis is reduced to investigating
three types of RED interval, namely (i)
[
0
,
x
]
,
(ii)
[
x
,
1
/
2
)
where
x
>
(
2
−
β
)
/
9and
(iii)
[
1
/
2
−
x
,
1
/
2
+
x
]
.
(i) For
x
<
1
/
2
,
the payoff for
[
0
,
x
]
is
(
1
−
F
(
x
))
x
−
β
F
(
x
)
x
=
x
(
1
−
(
1
+
β
)
F
(
x
)) = (
2
−
β
)
/
9
.
Search WWH ::
Custom Search