Computation of Nash Equilibria of Attack and Defense Games on Networks

TL;DR

This work addresses computing mixed Nash equilibria for attack and defense games on networks, where an attacker traverses from node $0$ to a target and defenders invest in convex protection costs. The authors introduce network reduction by removing neutral nodes and a subset of linkers to obtain a proper graph, then solve the equilibrium attack-tree game on the reduced graph and reconstruct the NE for the original network. They prove a polynomial-time algorithm with worst-case complexity $O(n^4)$, applicable to any connected graph, and derive explicit first-order conditions that characterize the NE, yielding a unique solution under strictly increasing defender values $b_i$. The approach unifies the equilibrium structure with a tree of equilibrium attack paths, enabling efficient computation and potential extensions to broader convex costs, with practical implications for network interdiction and defense planning.

Abstract

We consider the computation of a Nash equilibrium in attack and defense games on networks (Bloch et al. [1]). We prove that a Nash Equilibrium of the game can be computed in polynomial time with respect to the number of nodes in the network. We propose an algorithm that runs in O(n4) time with respect to the number of nodes of the network, n.

Theorems & Definitions (14)

• definition thmcounterdefinition: Attack and defense game on a network
• theorem thmcountertheorem: Bloch et al. paper
• lemma thmcounterlemma: Bloch et al. paper
• lemma thmcounterlemma: Bloch et al. paper, Theorem 2
• lemma thmcounterlemma: Bloch et al. paper
• lemma thmcounterlemma: Bloch et al. paper
• lemma thmcounterlemma: Bloch et al. paper
• definition thmcounterdefinition: Equilibrium attack tree game
• theorem thmcountertheorem: Bloch et al. paper
• lemma thmcounterlemma