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Hide-and-Seek Game with Capacitated Locations and Imperfect Detection

Bastián Bahamondes, Mathieu Dahan

TL;DR

The paper addresses a strategic hide-and-seek problem with multiple, capacitated locations where a seeker may inspect up to $r_S$ sites and a hider may conceal up to $r_H$ items, with location-dependent detection probabilities $p_i$. It introduces a two-step solution that first solves a lower-dimensional continuous zero-sum game $\tilde{\Gamma}$ to obtain equilibrium marginals, then cooperatively constructs mixed strategies for the original game $\Gamma$ consistent with those marginals. The authors prove an exact equivalence between NE of $\Gamma$ and NE of $\tilde{\Gamma}$, analytically solve $\tilde{\Gamma}$ revealing three regime patterns that determine optimal marginal allocations, and develop a quadratic-time combinatorial algorithm to realize NE in $\Gamma$ with linear-support strategies. This approach yields NE in $O(n^2)$ time and provides structural insights into how capacities, detection rates, and budgets influence location criticality and defender-resource coordination, with clear applicability to security and auditing contexts.

Abstract

We consider a variant of the hide-and-seek game in which a seeker inspects multiple hiding locations to find multiple items hidden by a hider. Each hiding location has a maximum hiding capacity and a probability of detecting its hidden items when an inspection by the seeker takes place. The objective of the seeker (resp. hider) is to minimize (resp. maximize) the expected number of undetected items. This model is motivated by strategic inspection problems, where a security agency is tasked with coordinating multiple inspection resources to detect and seize illegal commodities hidden by a criminal organization. To solve this large-scale zero-sum game, we leverage its structure and show that its mixed strategies Nash equilibria can be characterized using their unidimensional marginal distributions, which are Nash equilibria of a lower dimensional continuous zero-sum game. This leads to a two-step approach for efficiently solving our hide-and-seek game: First, we analytically solve the continuous game and compute the equilibrium marginal distributions. Second, we derive a combinatorial algorithm to coordinate the players' resources and compute equilibrium mixed strategies that satisfy the marginal distributions. We show that this solution approach computes a Nash equilibrium of the hide-and-seek game in quadratic time with linear support. Our analysis reveals a complex interplay between the game parameters and allows us to evaluate their impact on the players' behaviors in equilibrium and the criticality of each location.

Hide-and-Seek Game with Capacitated Locations and Imperfect Detection

TL;DR

The paper addresses a strategic hide-and-seek problem with multiple, capacitated locations where a seeker may inspect up to sites and a hider may conceal up to items, with location-dependent detection probabilities . It introduces a two-step solution that first solves a lower-dimensional continuous zero-sum game to obtain equilibrium marginals, then cooperatively constructs mixed strategies for the original game consistent with those marginals. The authors prove an exact equivalence between NE of and NE of , analytically solve revealing three regime patterns that determine optimal marginal allocations, and develop a quadratic-time combinatorial algorithm to realize NE in with linear-support strategies. This approach yields NE in time and provides structural insights into how capacities, detection rates, and budgets influence location criticality and defender-resource coordination, with clear applicability to security and auditing contexts.

Abstract

We consider a variant of the hide-and-seek game in which a seeker inspects multiple hiding locations to find multiple items hidden by a hider. Each hiding location has a maximum hiding capacity and a probability of detecting its hidden items when an inspection by the seeker takes place. The objective of the seeker (resp. hider) is to minimize (resp. maximize) the expected number of undetected items. This model is motivated by strategic inspection problems, where a security agency is tasked with coordinating multiple inspection resources to detect and seize illegal commodities hidden by a criminal organization. To solve this large-scale zero-sum game, we leverage its structure and show that its mixed strategies Nash equilibria can be characterized using their unidimensional marginal distributions, which are Nash equilibria of a lower dimensional continuous zero-sum game. This leads to a two-step approach for efficiently solving our hide-and-seek game: First, we analytically solve the continuous game and compute the equilibrium marginal distributions. Second, we derive a combinatorial algorithm to coordinate the players' resources and compute equilibrium mixed strategies that satisfy the marginal distributions. We show that this solution approach computes a Nash equilibrium of the hide-and-seek game in quadratic time with linear support. Our analysis reveals a complex interplay between the game parameters and allows us to evaluate their impact on the players' behaviors in equilibrium and the criticality of each location.
Paper Structure (13 sections, 14 theorems, 44 equations, 5 figures, 1 algorithm)

This paper contains 13 sections, 14 theorems, 44 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Problem Description
  5. Analytical Characterization of Equilibrium Strategies
  6. Continuous Equivalence
  7. Preliminary Analysis
  8. Analytical Characterization
  9. Parametric Analysis
  10. Equilibrium Computation
  11. Conclusion
  12. Proofs of Section \ref{['sec: analytical characterization of equilibrium strategies']}
  13. Proofs of Section \ref{['sec: equilibrium computation']}

Key Result

Lemma 1

Consider a vector of capacities $b \in \mathbb{Z}^n_{>0}$, a budget of resources $r \in \mathbb{Z}_{>0}$, and a vector $\rho^\prime \in \mathbb{R}^n$. Then, $\rho^\prime \in \widetilde{\mathcal{A}}(b,r)$ if and only if there exists a probability distribution $\sigma \in \Delta(b,r)$ that satisfies $

Figures (5)

  • Figure 1: Illustration of a NE for Regime Pattern 1 when $r_\textnormal{S} = 3$ and $r_\textnormal{H}=6$. The hiding capacity of each location is represented by the corresponding number of squares. Marginal inspection probabilities (resp. expected numbers of hidden items) in equilibrium are represented by the blue (resp. red) colors.
  • Figure 2: Illustration of a NE for Regime Pattern 2 when $r_\textnormal{S} = 6$ and $r_\textnormal{H}=6$. The hiding capacity of each location is represented by the corresponding number of squares. Marginal inspection probabilities (resp. expected numbers of hidden items) in equilibrium are represented by the blue (resp. red) colors.
  • Figure 3: Illustration of a NE for Regime Pattern 3 when $r_\textnormal{S} = 4$ and $r_\textnormal{H}=10$. The hiding capacity of each location is represented by the corresponding number of squares. Marginal inspection probabilities (resp. expected numbers of hidden items) in equilibrium are represented by the blue (resp. red) colors.
  • Figure 4: Illustration of equilibrium regions as a function of the number of resources $r_\textnormal{S}$ and $r_\textnormal{H}$ for the hide-and-seek instance from Figures \ref{['fig: regime pattern 1']}-\ref{['fig: regime pattern 3']}.
  • Figure 5: Fraction of undetected items in equilibrium as a function of $r_\textnormal{S}$, for different values of $r_\textnormal{H}$.

Theorems & Definitions (30)

  • Lemma 1
  • Proposition 1
  • Lemma 2
  • Theorem 1
  • Proposition 2
  • Example 1
  • Example 2
  • Example 3
  • Theorem 2
  • Corollary 1
  • ...and 20 more