Continuous Patrolling and Hiding Games

TL;DR

This paper introduces two zero-sum games modeling attack and hide scenarios, one dynamic and one static, providing a mathematical framework for optimal patrolling and hiding strategies in finite sets.

Contribution

It formulates and analyzes continuous patrolling and hiding games, offering new models for strategic search and concealment in finite environments.

Findings

The patrolling game models dynamic attack detection with optimal trajectories.

The hiding game captures static concealment strategies with probabilistic success.

Mathematical analysis provides equilibrium strategies for both games.

Abstract

We present two zero-sum games modeling situations where one player attacks (or hides in) a finite dimensional nonempty compact set, and the other tries to prevent the attack (or find him). The first game, called patrolling game, corresponds to a dynamic formulation of this situation in the sense that the attacker chooses a time and a point to attack and the patroller chooses a continuous trajectory to maximize the probability of finding the attack point in a given time. Whereas the second game, called hiding game, corresponds to a static formulation in which both the searcher and the hider choose simultaneously a point and the searcher maximizes the probability of being at distance less than a given threshold of the hider.