Information Technology Reference
In-Depth Information
fact that any specific solution would almost certainly be calculated numerically in
any case, it seems preferable to examine instead the comparatively tidy infinite case.
In general, the matrix for the two-cell game,
A
∞
,
2
, for some suitably large but
finite value of
L
, has the following form:
s
1
,
2
s
1
,
3
s
1
,
4
s
2
,
3
s
2
,
4
...
...
⎛
⎝
⎞
⎠
h
1
1
+
T
1
+
T
1
+
T
...
1
1
...
h
2
(
3
+
T
)
/
2
3
3
/
2
/
2
...
2
+
T
2
+
T
...
h
3
3
(
4
+
T
)
/
2
(
5
+
T
)
/
2
5
/
2
2
...
/
2
...
h
4
3
(
5
+
T
)
/
2
5
6
+
T
/
2
2
...
/
2
/
2
...
A
∞
,
2
=
h
5
3
5
5
/
2
2
/
2
...
/
2
3
...
h
6
3
5
5
/
2
2
/
2
...
/
2
3
...
.
.
.
.
.
.
It can be shown that there is an equilibrium where the searcher gives positive
weight only to what we term
wait-then-exhaustive
strategies. These are those such
that, once the searcher starts searching, they do not stop until the entire space has
been inspected; i.e., if they search for the first time in period
n
, then they also search
in period
n
+
1.
Given this behaviour by the searcher, the payoffs faced by the hider are then as
follows:
⎧
⎨
m
+
T
if
m
=
n
h
m
s
n
,
n
+
1
1
T
2
(
,
)=
m
+
2
+
if
m
=
n
+
1
,
T
∞
,
2
(16.29)
⎩
1
2
}
min
{
m
,
n
+
otherwise
Denote by
A
2
,
∞
the reduced matrix for infinite L that excludes any strategies other
than those that are wait-then-exhaustive. Thus:
s
1
,
2
s
2
,
3
s
3
,
4
s
4
,
5
...
⎛
⎝
⎞
⎠
h
1
1
+
T
1
1
1
...
h
2
(
3
+
T
)
/
2
+
T
2
2
...
2
h
3
3
(
5
+
T
)
/
2
/
2
3
+
T
3
...
h
4
3
5
(
7
+
T
)
/
2
/
2
/
2
4
+
T
...
A
2
,
∞
=
h
5
3
5
7
(
9
+
T
)
/
2
/
2
/
2
/
2
...
h
6
3
5
7
9
/
2
/
2
/
2
/
2
...
.
.
.
.
.
.
.
.
This game can be solved using a very similar approach to that which was
employed for the case where
K
1, but, as the resulting equations are substan-
tially more complex and space-consuming, we will omit the details and state only
that it is possible to prove the following, where
=
ϕ
is thegoldenratio (1.618...),and
Φ
is the inverseof the goldenratio (0.618...):
Theorem 5.
T
∞
,
2
, the value of the silent search-ambush game when K
=
2
,L
=
∞
,
1
2
,
that is, approximately
2
is equal to
, or equivalently to
ϕ
.
618
.
1
−
Φ
Search WWH ::
Custom Search