A stochastic game framework for patrolling a border

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 a Nash equilibrium to gain insight into 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 of our non zero-sum game 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.