A Discrete Search-Ambush Game with a Silent Predator - Search Theory: A Game Theoretic Perspective - page 239

16.2 Literature Review
The silent search-ambush game was invented by Steve Alpern, and is an outgrowth
of recent work by Alpern et al. [ 3 ] on the solution to a very similar game, with
two distinguishing characteristics: first, their search space is continuous, with total
area equal to unity; and second, more crucially, their hider is always aware of what
proportion of the search space has been inspected (i.e., the searcher is “noisy”).
A key result in their game is the “square root law of predation search”: that the
searcher's optimal probability of searching is equal to the square root of the fraction
of the search space that remains uninspected. Qualitatively, this implies the pace of
search should be rapid at the start of a game, slowing in favour of ambush as the
game progresses.
The issue of the potential usefulness of ambush (in the sense of the searcher
remaining motionless in the hope that the hider will move into them) extends further
back into the search game literature, arising naturally in questions of optimal search
for a mobile hider on a network, especially where the network in question has the
form of a figure-eight or n -leaf clover, or star network: here, it is reasonable to
suspect that the searcher may wish to occasionally remain still at the central node.
This issue was considered in detail by Alpern and Asic [ 1 , 2 ], finding that such
“loitering” strategies are indeed sometimes advantageous (a useful overview of this
and related issues is provided in Alpern and Gal [ 4 ]).
Our game is similar but distinct to games that model the problem of
pursuit-evasion in a continuous space with differential equations or games that con-
sider the geometric problem of optimal ambush locations. Example of this latter
type include a variety of games based on an agent attempting to defend some dis-
crete or continuous search space (see Ruckle et al. [ 15 ]; Joseph [ 12 ]; Washburn [ 17 ]
or Baston and Kikuta [ 5 ]). A case which has some interesting parallels to ours is the
one-dimensional evasion game first described by Gal [ 8 ] and solved by Lalley [ 13 ]
(with several generalisations of the game investigated by other authors); here, an in-
filtrator starts in a safe spot but then must progress incrementally through M discrete
sites within N periods of time without being detected by a defender. One result is
that a min-max strategy for the infiltrator is to play what Lalley terms “Admiral Far-
ragut” 1 strategies: waiting for some period of time before progressing through all
sites as rapidly as possible (in Sect. 16.6 , we will, less evocatively, term our equiva-
lent of these wait-then-exhaustive strategies).
Trade-offs between searching and ambushing behaviour can be seen in a variety
of real-world contexts. The issue is particularly relevant in biology, with much mod-
ern research following from the observations of different foraging modes made by
Schoener [ 16 ]. Some of many possible examples of such behaviour include brook
trout (Grant and Noakes [ 9 ]), mantids (Inoue and Marsura [ 11 ]), and between differ-
ent species of desert-dwelling lizards (Huey and Pianka [ 10 ]). Possible explanations
offered for such behavioural variations include environmental conditions, the threat
1 The reference is to the US Navy flag officer who, among other things, is remembered for the
paraphrased exclamation “Damn the torpedoes, full steam ahead!”.