When Sally Found Harry: A Stochastic Search Game
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Harry hides on an edge of a graph and does not move from there. Sally, starting from a known origin, tries to find him as soon as she can. Harry's goal is to be found as late as possible. At any given time, each edge of the graph is either active or inactive, independently of the other edges, with a known probability of being active. This situation can be modeled as a zero-sum two-person stochastic game. We show that the game has a value and we provide upper and lower bounds for this value. We show that, as the probability of each edge being active goes to 1, the value of the game converges to the value of the deterministic game, where all edges are always active. Finally, by generalizing optimal strategies of the deterministic case, we provide more refined results for trees and Eulerian graphs.
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