On Search Games That Include Ambush

Abstract

Abstract. We present a stochastic game that models ambush/search in a finite region Q which has area but no other structure. The searcher can search a unit area of Q in unit time or adopt an "ambush" mode for a certain period. The searcher "captures" the hider when the searched region contains the hider's location or if the hider moves while the searcher is in ambush mode. The payoff in this zero sum game is the capture time. Our game is motivated by the (still unsolved) princess and monster game on a star graph with a large number of leaves.Key words. noisy search game, ambush strategy, Poisson process AMS subject classifications. 91A05, 91A24 DOI. 10.1137/110845665Isaacs [22] introduced the princess and monster game on a graph in his book on differential games. This game on a graph has remained largely unsolved, contrary to the game in a domain, which was solved by Gal [17]. In the 1980s Alpern and Asic [3,4] solved the game on a graph with three unit length arcs connecting two nodes. This is the only nontrivial graph for which the game has been solved up to now. Gal [18] already noticed that the game on a graph can be solved if ambush strategies are not allowed. Progress on the princess and monster game on a graph has been slow, and that is why we propose to approach the game from a stochastic point of view, ignoring the geometry of the space, in order to focus on ambush strategies.In the original princess and monster game, the players have no visibility. There are versions of the game in which either one or both players have limited or full visibility, so there is an increased probability of detection. However, even if there is no visibility, there may be central locations in which the monster has a higher probability of detecting the princess, simply because she has a high probability of crossing that location once she moves to a new hiding place. In the princess and monster game on a graph, this would apply if all nodes are connected to just a few central nodes. Instead of searching the entire space as quickly as possible, the Monster may want to spend extra time in these central locations, waiting in ambush. In our paper, we study the trade-off between ambush and search in the princess and monster game. A similar type of trade-off occurs in the search of heterogeneous environments, when the searcher has a sensor with a sensitivity that depends on the location, such as studied in [11].There is a considerable literature on games that involve search, and the names of the games vary. They can be games between cops and robbers, or pursuers and evaders, or searchers and hiders, or hunters and rabbits. For instance, the game that we consider here is similar to the hunter and rabbit game on a graph that is studied in [1]. In our paper, we study the expected capture time, or search value [3]

Optimal search and ambush for a hider who can escape the search region

“…The searcher wins the game if he either …nds the hider while cruise searching or ambushes him while he is attempting to ‡ee; the hider wins if he ‡ees successfully (while the searcher is cruising) or has not been found by time T: The optimal searcher strategy involves decreasing the ambush probability over time, to a limit of zero. This surprising behaviour is opposite to that found recently by Alpern et al (2011Alpern et al ( , 2013 in a predator-prey game with similar dynamics but without the possibility of the hider escaping. Our work also complements that of Zoroa et al (2014) on searching for multiple prey and Gal and Casas (2014) for a combined model of search and pursuit.…”

“…Such search strategies were initially excluded by Gal (1979) in order to obtain certain results, but later incorporated into the theory by Alpern and Asic (1986), who showed that in the …gure eight network they had to be present in optimal search strategies. Such strategies were shown to be important in a predator-prey context for dual mode predators (who can alternate between a 'sit and wait'mode and a cruising mode) by Alpern et al (2011Alpern et al ( ,2013. Thus our game has four possible outcomes.…”

Optimal Trade-Off Between Speed and Acuity When Searching for a Small Object

“…Such a stochastic model has been studied as a stochastic game (Alpern et al, 2014) in the the context of search for a mobile hider.…”

Prey should hide more randomly when a predator attacks more persistently

“…The search game models in [6,7] had no pursuit phase, and the issue raised there concerned the optimal alternation between cruise search and ambush search. Following those papers, Zoroa et al [9,10] advanced the theory of search games to include ambush.…”