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

A Searcher seeks to find a stationary Hider located at some point H (not necessarily a node) on a given network Q . The Searcher can move along the network from a given starting point at unit speed, but to actually find the Hider she must pass it while moving at a fixed slower speed (which may depend on the arc). In this “bimodal search game,” the payoff is the first time the Searcher passes the Hider while moving at her slow speed. This game models the search for a small or well hidden object (e.g., a contact lens, improvised explosive device, predator search for camouflaged prey). We define a bimodal Chinese postman tour as a tour of minimum time δ which traverses every point of every arc at least once in the slow mode. For trees and weakly Eulerian networks (networks containing a number of disjoint Eulerian cycles connected in a tree-like fashion) the value of the bimodal search game is δ /2. For trees, the optimal Hider strategy has full support on the network. This differs from traditional search games, where it is optimal for him to hide only at leaf nodes. We then consider the notion of a lucky Searcher who can also detect the Hider with a positive probability q even when passing him at her fast speed. This paper has particular importance for demining problems.