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

TL;DR

This paper introduces a novel strategic hide-and-seek game model with capacitated locations and imperfect detection, providing an efficient two-step solution approach to compute Nash equilibria in large-scale security inspection scenarios.

Contribution

It develops a structured method to solve a complex zero-sum game by characterizing equilibria through marginal distributions and designing a quadratic-time algorithm.

Findings

Equilibrium strategies can be derived from unidimensional marginal distributions.

The solution approach efficiently computes Nash equilibria in quadratic time.

Game parameters significantly influence players' behaviors and location criticality.

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.