Modeling Resources Allocation in Attacker-Defender Games with 'Warm Up' CSF

Like many other engineering investments, the attacker's and defender's investments may have limited impact without initial capital to "warm up" the systems. This article studies such "warm up" effects on both the attack and defense equilibrium strategies in a sequential-move game model by developing a class of novel and more realistic contest success functions. We first solve a single-target attacker-defender game analytically and provide numerical solutions to a multiple-target case. We compare the results of the models with and without consideration of the investment "warm up" effects, and find that the defender would suffer higher expected damage, and either underestimate the attacker effort or waste defense investment if the defender falsely believes that no investment "warm up" effects exist. We illustrate the model results with real data, and compare the results of the models with and without consideration of the correlation between the "warm up" threshold and the investment effectiveness. Interestingly, we find that the defender is suggested to give up defending all the targets when the attack or the defense "warm up" thresholds are sufficiently high. This article provides new insights and suggestions on policy implications for homeland security resource allocation.