How Many Attackers Can Selfish Defenders Catch?

In a distributed system with attacks and defenses, an economic investment in defense mechanisms aims at increasing the degree of system protection against the attacks. We study such investments in the selfish setting, where both attackers and defenders are self-interested entities. In particular, we assume a reward-sharing scheme among interdependent defenders; each defender wishes to maximize its own fair share of the attackers caught due to him (and possibly due to the involvement of others). Addressed in this work is the fundamental question of determining the maximum amount of protection achievable by a number of such defenders against a number of attackers if the system is in a Nash equilibrium. As a measure of system protection, we adapt the defense-ratio, which describes the expected proportion of attackers caught by defenders. In a defense-optimal Nash equilibrium, the defense-ratio is optimized. We discover that the answer to this question depends in a quantitatively subtle way on the invested number of defenders. We identify graph-theoretic thresholds for the number of defenders that determine the possibility of optimizing a defense-ratio. In this vein, we obtain, through an extensive combinatorial analysis of Nash equilibria, a comprehensive collection of trade-off results.