Succession of hide–seek and pursuit–evasion at heterogeneous locations

Abstract

Many interactions between searching agents and their elusive targets are composed of a succession of steps, whether in the context of immune systems, predation or counterterrorism. In the simplest case, a two-step process starts with a search-and-hide phase, also called a hide-and-seek phase, followed by a round of pursuit-escape. Our aim is to link these two processes, usually analysed separately and with different models, in a single game theory context. We define a matrix game in which a searcher looks at a fixed number of discrete locations only once each searching for a hider, which can escape with varying probabilities according to its location. The value of the game is the overall probability of capture after k looks. The optimal search and hide strategies are described. If a searcher looks only once into any of the locations, an optimal hider chooses it's hiding place so as to make all locations equally attractive. This optimal strategy remains true as long as the number of looks is below an easily calculated threshold; however, above this threshold, the optimal position for the hider is where it has the highest probability of escaping once spotted.

Sit-and-wait versus active-search hunting: A behavioral ecological model of optimal search mode

“…As in our model, Alpern et al (2011) allow for the possibility that prey detection declines with increased predator velocity in their "ambush search" game. In a second model that consists of search or hide-and-seek coupled to pursuit-evasion, Gal and Casas (2014) show that prey will respond to a predator that visits each possible hiding location for a probabilistic capture below a threshold number of times by randomizing their positions. However, if the predator revisits particular sites above that threshold, then the prey elect the fixed position with the greatest probability of successfully evading a pursuit once they are spotted.…”

Prey should hide more randomly when a predator attacks more persistently

“…These boxes may be heterogeneous in the cost of searching and in the probability that a hider can be overlooked even if his location is searched. The literature on this aspect of our model has been discussed in [4]. A new aspect of search games that appears in this paper is that of persistence of attacks-if the prey escapes, the predator may attempt to find it again.…”

“…In fact, the number of predator-prey interactions with a rapid sequence of repeated attacks on the same prey by the same predator abound in nature, and older literature provides lengthy descriptions of such interactions, from pompilid wasps pursuing spiders to falcons attacking passerine prey (see Fabre's description in [4] for the first case and [15] for the second case). These descriptions often lack crucial information to formalize them as repeated games, and are not quantified.…”

“…Also, these studies do not provide a map of the environment and hence an appreciation of the number of hiding places available to the prey. This aspect does matter as it defines the number of possible hiding locations [4].…”

“…reduced the likelihood of secondary pursuits, after initial escape of the prey, to nearly zero'. With the inclusion of a persistence probability, this paper can be viewed as a repeated game extension of the single stage game of Gal & Casas [4], where an unsuccessful pursuit ended the game. Our main result is that when the persistence probability is high, the prey should hide more randomly (distributed over more potential locations) rather than more concentrated on the best sites for fleeing.…”