Individual vs. overarching protection for minimizing the expected damage caused by an attack

The article considers a system consisting of identical elements which can be protected and attacked individually and collectively. The system is aimed at supplying a demand. If, following the attack, the cumulative performance of the elements becomes less than the demand the damage proportional to the unsupplied demand is inflicted. Additional damage is associated with the destruction of the equipment. To destroy any system element the attacker always must penetrate/destroy the collective (overarching) protection. Both the attacker and the defender have limited resources and can distribute them freely between the two types of attack and protection. The attacker chooses the resource distribution and the number of attacked elements to maximize the expected damage associated with equipment losses and unsupplied demand. The defender chooses the resource distribution and the number of protected elements to minimize the system destruction probability. The bi-contest minmax game is formulated and its solutions are presented and analyzed. The influence of the game parameters on the optimal defense and attack strategies is discussed.