The Price of Defense

Abstract

We consider a strategic game with two classes of confronting randomized players on a graph G(V, E): ν attackers, each choosing vertices and wishing to minimize the probability of being caught, and a defender, who chooses edges and gains the expected number of attackers it catches. The Price of Defense is the worst-case ratio, over all Nash equilibria, of the optimal gain of the defender over its gain at a Nash equilibrium. We provide a comprehensive collection of trade-offs between the Price of Defense and the computational efficiency of Nash equilibria.

– Through reduction to a Two-Players, Constant-Sum Game, we prove that a Nash equilibrium can be computed in polynomial time. The reduction does not provide any apparent guarantees on the Price of Defense.

– To obtain such, we analyze several structured Nash equilibria:

– In a Matching Nash equilibrium, the support of the defender is an Edge Cover. We prove that they can be computed in polynomial time, and they incur a Price of Defense of α(G), the Independence Number of G.

– In a Perfect Matching Nash equilibrium, the support of the defender is a Perfect Matching. We prove that they can be computed in polynomial time, and they incur a Price of Defense of $\frac{|V|}{2}$.

– In a Defender Uniform Nash equilibrium, the defender chooses uniformly each edge in its support. We prove that they incur a Price of Defense falling between those for Matching and Perfect Matching Nash Equilibria; however, it is ${\cal NP}$-complete to decide their existence.

– In an Attacker Symmetric and Uniform Nash equilibrium, all attackers have a common support on which each uses a uniform distribution. We prove that they can be computed in polynomial time and incur a Price of Defense of either $\frac{|V|}{2}$ or α(G).

This work was partially supported by the IST Program of the European Union under contract number IST-2004-001907 (DELIS).