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Amine
No mine
Reconnaissance
r
k
α
+
v
k
−
1
1
−
n
+
v
k
−
1
No reconnaissance
r
k
(
−
β
)+
v
k
−
1
1
+
v
k
−
1
Tabl e 8. 3
Payoff matrix for
Γ
(
k
)
—two boats case
Tab le
8.3
differs from Table
8.2
in the left upper part. If the ships dispatch recon-
naissance boats, a stage contains
n
slots for the circulation of a boat and 1 slot for
the move of ships. Even if a boat missed a mine, it would be found by another boat
except for the interval while 1 slot move of the ships. Thus, the reward of the ships
is
. On the contrary, one of the ships strikes the mine if it is laid after the
boats circulate the ring (and before the ships move). Thus, the loss of the ships is
β
/
(
α
n
/
(
n
+
1
)
.
Next, the game
n
+
1
)
Γ
(
m
)
for
m
>
1 can be expressed as follows.
r
m
(
α
n
−
β
n
)+
Γ
(
−
)
−
+
Γ
(
−
)
m
1
1
n
m
1
Γ
(
m
)=
+
1
r
m
(
−
β
)+
Γ
(
m
−
1
)
1
+
Γ
(
m
−
1
)
The game value
v
m
for
Γ
(
m
)
,where
v
1
=
1
−
(
r
1
β
+
1
)
/
(
r
1
(
α
+
β
)
/
(
n
+
1
)+
1
)
,
can be solved as follows.
val
r
m
(
α
n
−
β
1
)+
v
m
−
1
1
−
n
+
v
m
−
1
=
v
m
n
+
(
−
β
)+
+
r
m
v
m
−
1
1
v
m
−
1
val
r
m
(
α
n
−
β
n
)
1
−
n
=
v
m
−
1
+
+
1
r
m
(
−
β
)
1
1
r
m
β
+
1
=
v
m
−
1
+
−
r
m
(
α
+
β
)
/
(
n
+
1
)+
1
m
k
=
1
r
k
β
+
1
=
m
−
1
.
r
k
(
α
+
β
)
/
(
n
+
1
)+
Similar to the argument for
m
=
1, the optimal strategies of player A and player
m
), denoted by
a
k
and
b
k
,are
Bforthe
k
-th stage (1
≤
k
≤
n
2
(
r
k
β
+
1
)(
n
+
1
)
r
k
(
α
n
−
β
)+
−
1
a
k
=
)
,
r
k
(
α
+
β
)
n
+
n
(
n
+
1
r
k
(
α
+
β
)
n
+
n
(
n
+
1
)
and
1
r
k
(
α
+
β
)+
n
+
r
k
(
α
+
β
)
r
k
(
α
+
β
)+
b
k
=
1
,
,
n
+
n
+
1
respectively.
If
k
is very large or
α
<
β
n
holds, we have
1
n
,
n
−
1
a
k
b
k
(
,
1
,
0
)
.
n
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