Modeling a Multitarget Attacker–Defender Game with Budget Constraints

Though the choices of terrorists’ attack targets are vast, their resources are limited. In this paper, a game-theoretical model is proposed to study both the defender’s (government) and the attacker’s (terrorist) expenditures among multiple targets under budget constraints to guide investment in defense. We study how the defender’s and the attacker’s equilibrium allocations depend on the budget constraints, target valuations, cost effectiveness of their investments, and inherent defense levels of targets in both sequential-move and simultaneous-move games. The equilibrium solutions are provided using the Karush–Kuhn–Tucker conditions. At the subgame-perfect Nash equilibrium, the defender’s total marginal effects are the same among targets. Moreover, the defender’s total marginal effects can be decomposed into direct and indirect effects. We also use the multiple-infrastructure and multiple-urban-area data sets to demonstrate the model results. The regression analysis shows that both the attacker’s and the defender’s equilibrium investments increase with their own target valuations, because a higher valuation for themselves means a higher attractiveness. Interestingly, the attacker’s equilibrium investment is negatively correlated with the defender’s target valuations, since a higher defender’s valuation would make it more difficult for the attacker to successfully attack the target. By contrast, the defender’s equilibrium investment is positively correlated with the attacker’s target valuations, as a higher attacker’s valuations would increase the urgency for the defender to protect the target. To show the utility of the new model, we compare the results of this model and the model in which the defender assumes that only a single target will be attacked when there could actually be multiple targets attacked. Our results show that the defender will suffer higher expected losses if she assumes that the attacker will attack only one target. The analysis of the attacker’s and the defender’s budget constraints show that (a) the higher the budget the defender has, the less likely it is that her most valuable target will be attacked; (b) a higher proportion of defense resources should be allocated to the most valuable target if the defender’s budget is low; and (c) the attacker is less concentrated on attacking the most valuable target and spreads the resources to attack more targets as his budget increases.