Search for an Immobile Hider on a Binary Tree with Unreliable Locational Information
Steve Alpern, Thomas Lidbetter
TL;DR
This paper introduces G(Q,O,p), a binary-tree search game where a Searcher using unit-speed, depth-first tours receives noisy branch signals about the Hider's location with accuracy p. It derives a closed-form value V(Q) = 2qμ + (p−q)D_Q and shows the Hider's optimal leaf distribution is defined recursively, while the Searcher adopts a favored-branch strategy with bias β = ((p−q)(D_{Q1}−D_{Q2}))/(2(pμ1+qμ2)); as p→1/2 the model recovers Gal79's Equal Branch Density dynamics, and as p→1 the strategy concentrates on the farthest leaf. The results are demonstrated via a concrete example and extended through recursion to show behavior for constant-depth trees, with implications for sensor-assisted search in security and ecological contexts and potential generalization to variable-speed networks.
Abstract
Adversarial search of a network for an immobile Hider (or target) was introduced and solved for rooted trees by Gal (1979). In this zero-sum game, a Hider picks a point to hide on the tree and a Searcher picks a unit speed trajectory starting at the root. The payoff (to the Hider) is the search time. In Gal's model (and many subsequent investigations), the Searcher receives no additional information after the Hider chooses his location. In reality, the Searcher will often receive such locational information. For homeland security, mobile sensors on vehicles have been used to locate radioactive material stashed in an urban environment. In a military setting, mobile sensors can detect chemical signatures from land mines. In predator-prey search, the predator often has specially attuned senses (hearing for wolves, vision for eagles, smell for dogs, sonar for bats, pressure sensors for sharks) that may help it locate the prey. How can such noisy locational information be used by the Searcher to modify her route? We model such information as signals which indicate which of two branches of a binary tree should be searched first, where the signal has a known accuracy p<1. Our solution calculates which branch (at every branch node) is favored, meaning it should always be searched first when the signal is in that direction. When the signal is in the other direction, we calculate the probability the signal should be followed. Compared to the optimal Hider strategy in the classic search game of Gal, the Hider's optimal distribution for this model is more skewed towards leaf nodes that are further from the root.