Game theoretical defensive resource allocations in the face of a partially strategic attacker considering equity constraints
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Since September 11, 2001, many game-theoretic models have been developed to study homeland security games between governments (defenders) and terrorists (attackers, adversary, enemy). Considering a partially strategic attacker, equity constraints and multi-period games would improve their connection with reality, which has not been extensively studied in the literature and will be addressed in this dissertation. First, we develop a novel hybrid model in which a centralized defender allocates defensive resources among multiple potential targets to minimize total expected loss, in the face of an attacker being either strategic or non-strategic. The attack probabilities of a strategic attacker are endogenously determined in the model, while the attack probabilities of a non-strategic attacker are exogenously provided. We study the robustness of defensive resource allocations by comparing the defender's total expected losses when: (a) the defender knows the probability that the attacker is strategic; (b) the defender falsely believes that the attacker is fully strategic, when the attacker could be non-strategic; and (c) the defender falsely believes that the attacker is fully non-strategic, when the attacker could be strategic. We find that game-theoretic models are preferred even when the probability that the terrorist is non-strategic is significantly greater than 50%. We conclude that defensive resource allocations based on game-theoretic models would not incur too much additional expected loss, as compared to non-game-theoretic models. Second, few studies consider the tradeoff between equity and efficiency in defensive resource allocation. We fill this gap by developing a novel model in which a defender allocates defensive resources among multiple potential targets, while reserving a portion (represented by the equity coefficient) for equal distribution (according to geographical areas, population, density, etc.). Such a way to model equity is one of many alternatives, but was directly inspired by homeland security resource allocation practices. The defender is faced with a strategic attacker (adaptive adversary) whose attack probabilities are endogenously determined in the model. We study the effect of the equity coefficient on the optimal defensive resource allocations and the corresponding expected loss. We find that the cost of equity (in terms of increased expected loss) increases convexly in the equity coefficient. Furthermore, such cost is lower when: (a) government uses per-valuation equity; (b) the cost-effectiveness coefficient of defense increases; and (c) the total defense budget increases. Our model, results and insights could be used to assist policy making. Third, the optimal allocation of a large amount of defensive resources over time remains a challenge. We apply game theory to study defensive resource allocations in a multi-period multi-target defender-attacker game. At each period, the defender allocates a limited amount of defensive resources over multiple targets and the attacker assigns some attack probabilities to those targets. We assume that within each period the attacker chooses targets with maximal expected loss and thus is myopic. We consider four cases: (a) cost-effectiveness coefficient is high or low, (b) there is carry-over defense or not, (c) the defender is myopic or long-sighted, and (d) total attack probability is high or low. Then the analysis characterizes the defender's optimal allocations and develops an algorithm to identify the optimal allocations. We find that at equilibrium increases in carry-over defense could lead to less allocation to defended targets from the second period on and thus lead to higher number of defended targets. Similarly, increases in cost-effectiveness of defense leads to less allocation to each defended targets and higher number of defended targets. To the best of our knowledge, no previous research has investigated the strategic interactions between attacker and defender with a multi-period multi-target game.