The Search and Rescue Game on a Cycle

TL;DR

This paper analyzes a search and rescue game on a cycle graph, providing solutions for single and multiple targets with success probabilities, including adaptive and non-adaptive search strategies, and extends to a continuous model.

Contribution

It offers the first complete solutions for search strategies on cycle graphs in both discrete and continuous settings, considering adaptive and non-adaptive approaches.

Findings

Exact solutions for single target with equal success probabilities.

Solutions for multiple targets with adaptive search strategies.

Approximate solutions for non-adaptive strategies in the continuous model.

Abstract

We consider a search and rescue game introduced recently by the first author. An immobile target or targets (for example, injured hikers) are hidden on a graph. The terrain is assumed to dangerous, so that when any given vertex of the graph is searched, there is a certain probability that the search will come to an end, otherwise with the complementary {\em success probability} the search can continue. A Searcher searches the graph with the aim of finding all the targets with maximum probability. Here, we focus on the game in the case that the graph is a cycle. In the case that there is only one target, we solve the game for equal success probabilities, and for a class of games with unequal success probabilities. For multiple targets and equal success probabilities, we give a solution for an adaptive Searcher and a solution in a special case for a non-adaptive Searcher. We also consider a continuous version of the model, giving a full solution for an adaptive Searcher and approximately optimal solutions in the non-adaptive case.