A Discrete Search-Ambush Game with a Silent Predator - Search Theory: A Game Theoretic Perspective - page 249

16.6.1 Optimal Searcher and Hider Behaviour

Optimal hider and searcher strategies that support this game value (we do not
address the question of whether the same value can be obtained through alternate
strategies; that is, whether the equilibrium described in the following is unique) are
as follows:

h 1

= − Φ ,

1 (16.30)

h j h j − 1

= Φ , ∀ j > 1 , (16.31)

s n , n + 1 represents the probability of the searcher playing the wait-then-

exhaustive strategy that commences in period n, which for this section we will
simply shorten to q n . Then, to 3 decimal places:

Again, q ( )

n n

q n = − 0 . 555 ( − 0 . 521 ) + 0 . 555 ( 0 . 594 ) , (16.32)

n term in the formula for q n results in the
probabilities attached to the searcher's strategies oscillating: q n >

Thepresenceofthe ( − 0 . 521 )

q n − 1 for odd n ,

and q n < q n .

These raw probabilities on their own, however, are not particularly informative.
More revealing questions concern the equilibrium beliefs of the hider: to start with,
in each period the hider must have a certain belief, conditional on that period being
reached, regarding the likelihood that the searcher will search. In the first period
of any round, this is easily calculated: the probability of search is equal to the sum
of the probabilities attached to all searcher strategies that search in the first period.
Where K

q n − 1 for even n , while we always have q n − 2 >

2, the only such strategy with a positive weight is s 1 , 2 .
If a round progresses to the second period, however, the hider should discount
their belief in the probability that the searcher is playing s 1 , 2 , as the fact of their
survival is evidence against that proposition (and if the third period is reached, the
hider should naturally dismiss any possibility of s 1 , 2 ). 4

=

An application of Bayes'

Theorem allows us to calculate P h ( , that is, the belief held by the hider (de-

noted P h ) regarding the probability of any particular searcher strategy (denoted s )
conditional on period z being reached (denoted R z ). The a priori probabilities that
the searcher actually attaches to their strategies s

s | R z )

(remembering
that in equilibrium the hider can be treated as if they were aware of these probabil-
ities). Reiterating that s j is either 0 or 1, representing whether ambush or search
occurs in the j -th period, for general K the appropriate formula is the following:

∈ S are denoted q ( s )

1 q

z 1

j = 1

−

s j

∑

− ( s )

K

∑ s ∈ S 1

q

P h ( s | R z )= , (16.33)

z 1

j = 1 s j
K

−

∑

− ( s )

4 Under the predator-prey interpretation, the prey is thus employing a variant of the anthropic
principle: that they have not yet been eaten allows them to draw certain conclusions regarding the
likely states of the world (specifically the predator's choice of strategy).