A Network Game with Attackers and a Defender: A Survey

We survey a research line recently initiated by Mavronicolas et al. [14, 15, 16], concerning a strategic game on a graph G(V,E) with two confronting classes of randomized players: ν attackers who choose vertices and wish to minimize the probability of being caught by the defender, who chooses edges and gains the expected number of attackers it catches. So, the defender captures system rationality. In a Nash equilibrium, no single player has an incentive to unilaterally deviate from its randomized strategy. The Price of Defense is the worst-case ratio, over all Nash equilibria, of the optimal gain of the defender (which is ν) over the gain of the defender at a Nash equilibrium. We present a comprehensive collection of trade-offs between the Price of Defense and the computational efficiency of Nash equilibria proved in [14, 15, 16]. • We present an algebraic characterization of (mixed) Nash equilibria. • No (non-trivial) instance of the graph-theoretic game has a pure Nash equilibrium. This is an immediate consequence of some covering properties proved for the supports of the players in all (mixed) Nash equilibria. • We present a reduction of the game to a Zero-Sum Two-Players Game that proves that a general Nash equilibrium can be computed via Linear Programming in polynomial time. However, the reduction does not provide any apparent guarantees on the Price of Defense. • To obtain guarantees on Price of Defense, we present an analysis of several structured Nash equilibria: – In a Matching Nash equilibrium, the support of the defender is an Edge Cover of the graph. Matching Nash equilibria are shown to still be computable in polynomial time, and that 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 of the graph. Perfect Matching Nash equilibria are shown to be computable in computed in polynomial time, and that they incur a Price of Defense of |V | 2 . • We consider a generalization of the basic model with an increased power for the defender: it is able to scan a simple path of the network instead of a single edge. Deciding existence of a pure Nash equilibrium is shown to be an NP-complete problem for this model. This work was partially supported by the IST Program of the European Union under contract number IST-2004001907 (DELIS), and by research funds at University of Cyprus. Department of Computer Science, University of Cyprus, P.O. Box 20537, Nicosia CY-1678, Cyprus. Email: {mavronic,viki,annap}@ucy.ac.cy Research Academic Computer Technology Institute (RACTI), Rion, Patras 26500, Greece & Department of Computer Engineering and Informatics, University of Patras, Rion, Patras 26500, Greece. Email: [email protected]