Game-Theoretic Optimization on School Safety: Resource Allocation Against Strategic Attacks

School security remains a critical concern due to the increasing frequency of violent incidents, requiring a strategic balance between physical security measures and mental health programs. This study develops a game-theoretic framework to model the interaction between a school as the defender and a potential attacker, aiming to identify optimal investment decisions across two complementary layers of defense. Numerical illustrations calibrated with data from U.S. school shootings provide empirical support for the analysis. Through one-way and two-way sensitivity analyses, robustness tests, and scenario-based indifference curve analysis, we explore how attacker and defender valuations, intervention effectiveness, and defensive costs influence equilibrium strategies. The findings show that physical security measures have the strongest deterrent effect, but lasting protection depends on balanced investment in both security and mental health once their effectiveness exceeds critical thresholds. While physical security offers immediate deterrence, mental health interventions are essential for addressing underlying risk factors, emphasizing the complementary nature of the two approaches. The framework contributes to evidence-based decision-making for educational institutions and suggests future extensions to include external threats, incomplete information, and dynamic investment strategies.