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Amine
No mine
Reconnaissance
α
1
−
n
No reconnaissance
−
β
1
Tabl e 8. 1
Payoff matrix for a stage
Tab le
8.1
shows the payoff matrix of our game. In the left column, where a ter-
rorist lays a mine, if a ship dispatches a reconnaissance boat, the reward of the ship
is
. In the right column, where a
terrorist does not lay a mine, if a ship dispatches a reconnaissance boat, the reward
of the ship is 1
α
>
0. Otherwise, the reward of the ship is
−
β
n
because it takes
n
steps for the boat to circulate a ring and then
the ship proceeds to the next port.
Let
E
−
be the expected payoff of player A when player B lays a naval
mine. On the contrary, let
E
(
a
,
Mine
)
(
a
,¬
Mine
)
be the expected payoff of player A when
player B does not lay a naval mine.
E
(
a
,
Mine
)=
p
·
α
+(
1
−
p
)
·
(
−
β
)=
p
(
α
+
β
)
−
β
E
(
a
,¬
Mine
)=
p
·
(
1
−
n
)+(
1
−
p
)
·
1
=
1
−
pn
1, player A's optimal strategy
a
∗
is
Since two straight lines intersect in 0
≤
p
≤
1
α
+
β
+
β
+
n
,
α
+
n
−
1
a
∗
=
.
α
+
β
+
n
The value of the game is
α
+
β
(
1
−
n
)
n
.
(8.1)
α
+
β
+
be the expected payoff of player B when player A dispatches a
reconnaissance boat. On the contrary, let
E
Let
E
(
Recon
,
b
)
be the expected payoff of
player B when player A does not dispatch a reconnaissance boat.
(
¬
Recon
,
b
)
E
(
Recon
,
b
)=(
1
−
q
)
·
(
1
−
n
)+
q
·
α
=
q
(
n
+
α
−
1
)
−
n
+
1
E
(
¬
Recon
,
b
)=(
1
−
q
)
·
1
+
q
·
(
−
β
)=
1
−
q
(
β
+
1
)
1, player B's optimal strategy
b
∗
is
Since two straight lines intersect in 0
≤
q
≤
n
α
+
β
+
α
+
β
α
+
β
+
b
∗
=
n
,
.
n
Notice that if we use the player A's optimal strategy as it is, it means that the ad-
versary always aims to bring about a failure. So, an unlikely failure may frequently
occur against our intuition. To avoid it, we consider the empirical probability of a
terrorist from now. Let
s
be a mine-preparing probability for each stage. We assume
that the terrorist succeeds in preparing the mine by geometric distribution. Then, he
can prepare it with probability
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