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

Wednesday 18th June 2014, 4.00pm

32L LG.08, LSE

Shmuel Gal

University of Haifa

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Abstract:

A predator (searcher) looks for a prey (hider) in a search space consisting of n locations. The hider chooses a location and the searcher inspects k different locations, where k is a parameter of the game (the `giving-up time’ for the continuous version). If the predator visits a location i at which the prey hides, then the game moves into a pursuit-evasion phase. In this phase capture is not certain but occurs with probability p_i.

We show that for all k smaller than an easily calculated threshold, it is optimal to hide with probability proportional to 1/p_i for each location i: If k exceeds the threshold, then the optimal hiding strategy is always to stay at the location with the smallest p_i.

We extend this game to a repeated game. During the k looks among the different locations within a single patch, there can be any of three events. First, if the searcher does not find the hider, then the game ends with zero payoff for the searcher and a payoff of one to the hider. Second, if the searcher finds the hider and catches it, then the game ends with a payoff of one to the searcher and zero to the hider. Finally, if the searcher finds the hider but does not catch it then the hider escapes to another patch and the process restarts. We show that in this game the optimal hiding strategy is to always make all the locations equally "attractive" for the searcher, no matter how large is k: This situation is quite different from the one stage game in which solutions of this type occur only if k is below the threshold.

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