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k
−
1
s
r
k
=(
1
−
s
)
(
1
≤
k
≤
m
)
at the
k
-th stage. It is assumed that the terrorist has material only to prepare a mine,
therefore the terrorist can prepare a mine on stage
k
only if he has not been able
to prepare one on the previous stages
(
,
,...,
−
)
. After preparing the mine,
he makes a decision to lay it for each stage. The following table shows the payoff
matrix where the mine-preparing probability is considered. It is essentially the same
one as Table
8.1
except for the probability and the value, where
v
k
−
1
is the value
of the game
1
2
k
1
. In Sects.
8.3.1
and
8.3.2
, we consider two cases, a game
continuation case and a game termination case, after the ship is broken by a mine.
Γ
(
k
−
1
)
Amine
No mine
β
(
α
n
n
Reconnaissance
r
k
−
)+
v
k
−
1
1
−
n
+
v
k
−
1
+
1
n
+
1
No reconnaissance
r
k
(
−
β
)+
v
k
−
1
1
+
v
k
−
1
Tabl e 8. 2
Payoff matrix for
Γ
(
k
)
with mine-preparation
8.3.1 Mine-Preparing Probability: Game Continuation Case
In this section, we consider that a fleet of ships can continue to sail and thus the
game continues if one of them is broken by a mine. Then, the game
Γ
(
m
)
for
m
>
1
can be expressed as follows :
r
m
α
+
Γ
(
m
−
1
)
1
−
n
+
Γ
(
m
−
1
)
Γ
(
)=
,
m
r
m
(
−
β
)+
Γ
(
m
−
1
)
1
+
Γ
(
m
−
1
)
where
Γ
(
k
)(
k
<
m
)
is the first
k
stages of the game
Γ
(
m
)
. The game value
v
m
for
the game
Γ
(
m
)
,where
v
1
=(
r
1
(
α
+
β
(
1
−
n
)))
/
(
r
1
(
α
+
β
)+
n
)
, can be solved as
follows.
val
r
m
α
+
v
m
−
1
1
−
n
+
v
m
−
1
v
m
=
r
m
(
−
β
)+
v
m
−
1
1
+
v
m
−
1
val
r
m
α
1
−
n
=
v
m
−
1
+
(
−
β
)
r
m
1
r
m
(
α
+
β
(
1
−
n
))
=
v
m
−
1
+
r
m
(
α
+
β
)+
n
m
h
=
1
r
h
(
α
+
β
(
1
−
n
))
=
n
.
r
h
(
α
+
β
)+
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