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of predation on the predator, available food (Olive [
14
]), and the necessity of either
overwhelming or not alarming the prey (Casas et al. [
7
]). The suggestion from game
theory is that such variation could represent optimal play if the interaction is mod-
elled as a zero-sum game; this follows from [
3
], and is also considered elsewhere,
such as in Zoroa et al. [
18
].
Our game is also somewhat connected with the question of timing when to
flee from an approaching predator, where the probability of successful escape is
increasing the further away the predator is and the quicker they are moving, with an
assumption that the predator's speed comes at the cost of accuracy; see Broom and
Ruxton [
6
] for more discussion of this point. This can be linked to our condition that
the hider always escapes against a searching predator, even if the predator inspects
the cell the hider started the round in: this is reasonable if we assume that a moving
hider is “too difficult a target” for a moving searcher to effectively intercept.
16.3 Constructing Payoff Matrices
We start with some additional terminology. We denote the searcher strategy set
S
,
and the hider strategy set
H
, with elements of these sets denoted by lower-case
s
and
h
. Specific hider strategies are denoted
h
i
,where
i
∈{
1
...
L
,
L
+
}
; thus
|
H
|
=
1. Given the searcher's constraint, the strategy
h
L
+
(which is intended to refer
to moving in any period greater than
L
) is equivalent to not moving at all. It is
straightforward to show that this is never optimal for the hider - at a minimum, if
period
L
is reached, it must be sensible to move, since in that period the searcher will
be obligated to search - and thus after this section such strategies will be excluded
from our payoff matrices.
Since every searcher strategy is constructed by choosing
K
periods out of a total
of
L
periods, we have
L
+
|
=
K
.When
K
L
(
L
−
1
)
2
|
S
=
2, we have
|
S
|
=
, and thus
|
S
|
=
L
2
L
K
O
.
The expected time until capture based on a pair of strategies
s
and
h
for a game
defined by
K
and
L
is denoted
T
L
,
K
(
(
)
. In general, for any fixed
K
,wehave
|
S
|
=
O
(
)
.
The
value
of a game, i.e. the expected time until capture under optimal play, is
denoted
T
L
,
K
(see [
4
] for more discussion of this concept of value in search games).
The optimal (max-min and min-max) hider and searcher strategies are denoted
s
and
h
;wethenhave
T
L
,
K
=
s
,
h
)
, which again may be abbreviated to
T
. The sub-
scripts
L
and
K
will be dropped when they are clear from the context.
We now turn to constructing game matrices, starting by describing how the
expected time until capture (less precisely referred to as the payoff) is calculated
from any pair
s
h
T
(
s
,
)
H
. Searcher strategies in general can be difficult to
denote neatly; for this section, we will use the game where
K
∈
S
and
h
∈
4asan
illustrative example, and a search strategy that involves searching in, say, the sec-
ond and fourth periods will be denoted
s
2
,
4
. In general, a searcher strategy
s
can be
=
2,
L
=
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