Individual versus overarching protection against strategic attacks

Abstract

This article considers a system consisting of elements that can be protected and attacked individually and collectively. To destroy the system, the attacker must always penetrate/destroy the collective (overarching) protection. In the case of the parallel system, it also must destroy all elements, whereas in the case of the series system, it must destroy at least one element. Both the attacker and the defender have limited resources and can distribute these freely between the two types of protection. The attacker chooses the resource distribution and the number of attacked elements to maximize the system destruction probability. 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 analytical solutions are presented and analysed. The asymptotical analysis of the solutions is presented. The influence of the game parameters on the optimal defence and attack strategies is discussed.

Appendix

Theorem

• Let

  

  for differentiable F and G, and (X ^*, x ^*) be a Nash saddle point of φ. Then, we have

  

  and

  

Proof

• For (X ^*, x ^*) we have

  

  and

  

  Solving (A.4) and (A.5) together we get

  

  From (A.6) it follows that F(x ^*/X ^*)=F(1), F′(x ^*/X ^*)=F′(1), G((1−x ^*)/(1−X ^*))=G(1), G′((1−x ^*)/(1−X ^*))=G′(1), which yields

  

  and

  

  which completes the proof. □

Corollary 1

• For

  
  

Proof

• Representing φ(X, x)=F(x/X)G((1−x)/(1−X)), where

  

  and

  

  we get (A.9) from (A.7). □

Corollary 2

• For

  
  

Proof

• Representing φ(X, x)=F(x/X)G((1−x)/(1−X)), where

  

  and

  

  we get (A.10) from (A.7). □