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A review of minimum cost box searching games

Thomas Lidbetter

TL;DR

This paper surveys minimum-cost box searching games, a class of zero-sum search problems where a Hider hides in one or more boxes with detection that may fail and with heterogeneous search times. It covers equal and unequal search times as well as equal and non-equal detection probabilities, derives closed-form values in several regimes, and presents strategy constructions such as equalizing distributions, randomized orderings, and best-response frontiers. In the multiple-target setting, it provides a neat equilibrium for perfect detection and analyzes the impact of adaptivity, while highlighting algorithmic approaches (e.g., Clarkson 2024) and open questions for broader cases. The discussion points to rich extensions, including non-equal detection, higher target counts, and compact strategy representations, inviting further theoretical and computational advances.

Abstract

We consider a class of zero-sum search games in which a Hider hides one or more target among a set of $n$ boxes. The boxes may require differing amount of time to search, and detection may be imperfect, so that there is a certain probability that a target may not be found when a box is searched, even when it is there. A Searcher must choose how to search the boxes sequentially, and wishes to minimize the expected time to find the target(s), whereas the Hider wishes to maximize this payoff. We review some known solutions to different cases of this game.

A review of minimum cost box searching games

TL;DR

This paper surveys minimum-cost box searching games, a class of zero-sum search problems where a Hider hides in one or more boxes with detection that may fail and with heterogeneous search times. It covers equal and unequal search times as well as equal and non-equal detection probabilities, derives closed-form values in several regimes, and presents strategy constructions such as equalizing distributions, randomized orderings, and best-response frontiers. In the multiple-target setting, it provides a neat equilibrium for perfect detection and analyzes the impact of adaptivity, while highlighting algorithmic approaches (e.g., Clarkson 2024) and open questions for broader cases. The discussion points to rich extensions, including non-equal detection, higher target counts, and compact strategy representations, inviting further theoretical and computational advances.

Abstract

We consider a class of zero-sum search games in which a Hider hides one or more target among a set of boxes. The boxes may require differing amount of time to search, and detection may be imperfect, so that there is a certain probability that a target may not be found when a box is searched, even when it is there. A Searcher must choose how to search the boxes sequentially, and wishes to minimize the expected time to find the target(s), whereas the Hider wishes to maximize this payoff. We review some known solutions to different cases of this game.

Paper Structure

This paper contains 10 sections, 1 theorem, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Game Definition and Preliminaries
  3. Equal Search Times and Detection Probabilities
  4. Equal Detection Probabilities
  5. Non-Equal Detection Probabilities
  6. Proof:
  7. Example
  8. Remarks
  9. Multiple Targets
  10. Concluding Remarks

Key Result

Lemma 2

Let $\mathbf p$ and $\theta$ be optimal strategies for the Hider and Searcher, respectively.

Figures (3)

  • Figure 1: Graphical representation of the expected search time of a search $\xi$ against a fixed Hider strategy $\mathbf p$.
  • Figure 2: The set of performance vectors of strategies $\xi^1$ and $\xi^2$.
  • Figure 3: A subset of the best response frontier.

Theorems & Definitions (2)

  • Definition 1
  • Lemma 2