A Discrete Search-Ambush Game with a Silent Predator - Search Theory: A Game Theoretic Perspective - page 252

Fig. 16.2 Conditional beliefs of hider ( K = 2, L = ∞ )

1
2

and K , respectively, ensuring consistency with our

conclusions in Sects. 16.5 and 16.6 . For interest, the polynomial that generates
σ K only ever has at most two real solutions, and always exactly one real solution
between 0 and 1.
Tab le 16.1 gives some detail on specific values of

= 2, where σ K is and Φ

σ K and the corresponding

conjectured game lower bound up to K = 6, comparing them to the numerical esti-

mates when L 18.

In the limit, as K

=

→ ∞ , σ K → 1, and in turn the conjectured lower bound on the

game also approaches . This is intuitively sensible: in an infinite search space, the

hider should be able to evade capture indefinitely. In addition, for large K the bound
on the value of the game appears to converge to a linear function, increasing by
around 0

∞

62749 for every unit increase in K .
To convert the discrete search-ambush game into its continuous equivalent, we
can assume that each cell, rather than taking one time period to search, takes time
1

.

/ K ; this is equivalent to dividing up a continuous search space of area unity that
takes one time period to search into K subsections. We re-obtain the continuous
search space by taking K

→ ∞

, assuming continuity in the game values in the limit.
If the conjecture is true, this suggests the value of the continuous search-ambush
game with a silent predator is greater than or equal to 0

.

62749, while Alpern et al.
[ 3 ] have previously determined that the value of the game with a noisy predator is
0

.
We also make a further conjecture regarding hider and searcher behaviour in
equilibrium.

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