Darlington, Matthew and Glazebrook, Kevin and Leslie, David and Shone, Robert (2024) The border patrol game. PhD thesis, Lancaster University.
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Abstract
The issue of how best to patrol a border can be found in many settings. Important examples include protecting important infrastructure such as airports, preventing the smuggling of illicit items, or defending computers in a network. This thesis contributes to the existing literature by developing two unique models for border patrol. We begin by introducing a model where a group of smugglers play a game against a single patroller. We investigate how communication and cooperation between the smugglers affect the equilibria in the game. Smugglers are located at different locations along a border and, for each smuggler, a decision is made about whether they will attack. Simultaneously, the patroller chooses one of the locations to defend. Smugglers obtain rewards for making successful attacks, but incur penalty costs if they are caught by the patroller. The reward to an individual smuggler for making a successful attack decreases with the total number of successful attacks made, so that the smugglers obtain diminishing marginal returns as they smuggle larger quantities of items. We define equilibria in three different cases: selfish smugglers without communication, selfish smugglers with communication, and cooperative smugglers. We study the equilibria in each case and establish properties of the associated smuggler and patroller strategies. We show that communication and cooperation both tend to improve the smugglers’ expected returns, while (perhaps counter-intuitively) the smugglers attack less often when they are cooperating than when they are communicating but acting selfishly. Our second model considers a similar problem to the first, but we make some important changes to the game. We simplify the payoff structure of the model, however, we are then able to study a repeated game with inter-temporal dependence. The application of the model is examining a group of cooperating smugglers who make 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. Furthermore, we introduce a multiple patroller extension to the second model. The addition of multiple patrollers increases the complexity of the game, and exactly finding Nash equilibria becomes an even more complex task. We describe how we model the multiple patroller extension, and detail why previous methods for the single patroller game no longer apply. Three different techniques to study the game are then discussed. Firstly, we look at a method using subgradient descent to try to find the exact Nash equilibrium in the game. Secondly, we look at two different heuristics for the patroller’s strategy. We first consider the myopic strategy for the patrollers, and then introduce a method of partitioning the border into multiple segments each defended by one patroller. The performance of the heuristics is numerically investigated and used to evaluate the convergence of the subgradient method. Finally, we discuss how reinforcement learning can be applied to our model. We consider two different reinforcement learning approaches, fictitious play and Q-learning, and then provide a numerical analysis of the resulting player behaviours. We conclude with several directions of further work for the multiple patroller problem.