  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.
