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For instance, suppose the searcher inspects in the first period (that is, plays
s
1
),
while the hider moves (plays
h
1
). No capture occurs in that period; instead the hider
escapes but, at the end of the period, returns to the only hiding place available, while
the searcher returns to their ambush position. The searcher is informed of the hider's
escape and a new round starts, so the payoff is 1
T
.
This game can be solved using basic game theory techniques, with the only
complicating factor being that the value of the game is unknown and is included
in the payoffs. Given the interpretation of the game, any payoff must be strictly
positive, and we therefore have
T
+
>
0. As would be expected, there is no equilib-
rium in pure strategies. To solve in mixed strategies: let
q
s
i
represent the prob-
ability attached by the searcher to searching in the
i
-th period; for simplicity, in
this section we will shorten this to simply
q
i
. For the searcher to make the hider
indifferent between their two remaining strategies, we require:
(
)
q
1
(
1
+
T
)+(
1
−
q
1
)=
q
1
+(
2
+
T
)(
1
−
q
1
)
,
(16.7)
1
+
T
⇒
q
1
=
1
,
(16.8)
2
T
+
By symmetry, an equivalent equation will hold for the probability of the hider mov-
ing in the first period. To solve for
T
, we can substitute (
16.8
) into:
q
1
(
1
+
T
)+(
1
−
q
1
)=
T
,
(16.9)
which represents the payoff the hider will receive in equilibrium. This provides us
with an expression solely in terms of
T
:
2
(
1
+
T
)
1
+
T
1
+(
1
−
1
)=
T
,
(16.10)
2
T
+
2
T
+
+
√
2
T
2
,
1
(
h
⇒
,
s
)=
1
,
(16.11)
where we have eliminated the negative root of the quadratic due to
T
being positive.
Substituting into (
16.8
), we conclude that the searcher inspects in the first period
with probability:
√
2
√
2
√
2
2
+
q
1
=
2
√
2
=
+
√
2
=
2
−
,
(16.12)
3
+
1
which by the symmetry of the game is likewise the equilibrium probability that the
hider moves in the first period; and thus the probability of the searcher searching or
the hider moving in the second period is
2
:
+
√
2
=
√
2
1
(
1
−
q
1
)=
−
1
,
(16.13)
1
2
For incidental interest, 1
/
(
1
+
√
2
)
has been known as the silver ratio, which, in conjunction
with the results of Sect.
16.7
, seems worth mentioning if only to identify a theme of metallurgical
nomenclature through this chapter.
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