A stochastic game framework for patrolling a border

TL;DR

This paper models border patrolling as a stochastic game with smugglers and a patroller, introducing a continuous contraband choice and developing efficient algorithms to find Nash equilibria, providing insights into real-world border security strategies.

Contribution

It extends existing models by incorporating continuous contraband quantities and offers new algorithms for computing Nash equilibria more efficiently.

Findings

Nash equilibria properties depend on smugglers' aggregation and discount factors.

The model can be reduced to a zero-sum game under certain conditions.

New algorithms improve computational efficiency for equilibrium analysis.

Abstract

In this paper we consider a stochastic game for modelling the interactions between smugglers and a patroller along a border. The problem we examine involves a group of cooperating smugglers making regular attempts to bring small amounts of illicit goods across a border. A single patroller has the goal of preventing the smugglers from doing so, but must pay a cost to travel from one location to another. We model the problem as a two-player stochastic game and look to find the Nash equilibrium to gain insight to real world problems. Our framework extends the literature by assuming that the smugglers choose a continuous quantity of contraband, complicating the analysis of the game. We discuss a number of properties of Nash equilibria, including the aggregation of smugglers, the discount factors of the players, and the equivalence to a zero-sum game. Additionally, we present algorithms to find Nash equilibria that are more computationally efficient than existing methods. We also consider certain assumptions on the parameters of the model that give interesting equilibrium strategies for the players.