The Price of Defense and Fractional Matchings

  • Marios Mavronicolas
  • Vicky Papadopoulou
  • Giuseppe Persiano
  • Anna Philippou
  • Paul Spirakis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4308)


Consider a network vulnerable to security attacks and equipped with defense mechanisms. How much is the loss in the provided security guarantees due to the selfish nature of attacks and defenses? The Price of Defense was recently introduced in [7] as a worst-case measure, over all associated Nash equilibria, of this loss. In the particular strategic game considered in [7], there are two classes of confronting randomized players on a graph G(V, E): νattackers, each choosing vertices and wishing to minimize the probability of being caught, and a single defender, who chooses edges and gains the expected number of attackers it catches.

In this work, we continue the study of the Price of Defense. We obtain the following results:

– The Price of Defense is at least \(\frac{|V|}{2}\); this implies that the Perfect Matching Nash equilibria considered in [7] are optimal with respect to the Price of Defense, so that the lower bound is tight.

– We define Defense-Optimal graphs as those admitting a Nash equilibrium that attains the (tight) lower bound of \(\frac{|V|}{2}\). We obtain:

∙ A graph is Defense-Optimal if and only if it has a Fractional Perfect Matching. Since graphs with a Fractional Perfect Matching are recognizable in polynomial time, the same holds for Defense-Optimal graphs.

∙ We identify a very simple graph that is Defense-Optimal but has no Perfect Matching Nash equilibrium.

– Inspired by the established connection between Nash equilibria and Fractional Perfect Matchings, we transfer a known bivaluedness result about Fractional Matchings to a certain class of Nash equilibria. So, the connection to Fractional Graph Theory may be the key to revealing the combinatorial structure of Nash equilibria for our network security game.


Nash Equilibrium Perfect Match Strategic Game Uniform Probability Distribution Security Attack 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Aspnes, J., Chang, K., Yampolskiy, A.: Inoculation Strategies for Victims of Viruses and the Sum-of-Squares Problem. In: Proceedings of the 16th Annual ACMSIAM Symposium on Discrete Algorithms, pp. 43–52 (2005)Google Scholar
  2. 2.
    Bourjolly, J.-M., Pulleyblank, W.R.: König-Egerváry Graphs, 2-Bicritical Graphs and Fractional Matchings. Discrete Applied Mathematics 24, 63–82 (1989)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cheswick, E.R., Bellovin, S.M.: Firewalls and Internet Security. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  4. 4.
    Gelastou, M., Mavronicolas, M., Papadopoulou, V., Philippou, A., Spirakis, P.: The Power of the Defender. In: CD-ROM Proceedings of the 2nd International Workshop on Incentive-Based Computing (July 2006)Google Scholar
  5. 5.
    Koutsoupias, E., Papadimitriou, C.H.: Worst-Case Equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. 6.
    Markham, T., Payne, C.: Security at the Network Edge: A Distributed Firewall Architecture. In: Proceedings of the 2nd DARPA Information Survivability Conference and Exposition, vol. 1, pp. 279–286 (2001)Google Scholar
  7. 7.
    Mavronicolas, M., Michael, L., Papadopoulou, V.G., Philippou, A., Spirakis, P.G.: The Price of Defense. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 717–728. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Mavronicolas, M., Papadopoulou, V.G., Philippou, A., Spirakis, P.G.: A Network Game with Attacker and Protector Entities. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 288–297. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Mavronicolas, M., Papadopoulou, V.G., Philippou, A., Spiraki, P.G.: A Graph-Theoretic Network Security Game. In: Deng, X., Ye, Y. (eds.) WINE 2005. LNCS, vol. 3828, pp. 969–978. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Mavronicolas, M., Spirakis, P.: The Price of Selfish Routing. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 510–519 (2001)Google Scholar
  11. 11.
    Moscibroda, T., Schmid, S., Wattenhofer, R.: When Selfish Meets Evil: Byzantine Players in a Virus Inoculation Game. In: Proceedings of the 25th Annual ACM Symposium on Principles of Distributed Computing, pp. 35–44 (2006)Google Scholar
  12. 12.
    Nash, J.F.: Equilibrium Points in N-Person Games. Proceedings of National Acanemy of Sciences of the United States of America 36, 48–49 (1950)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Nash, J.F.: Non-Cooperative Games. Annals of Mathematics 54(2), 286–295 (1951)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Scheinerman, E.R., Ullman, D.H.: Fractional Graph Theory. Wiley-Interscience Series in Discrete Mathematics and Optimization (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Marios Mavronicolas
    • 1
  • Vicky Papadopoulou
    • 1
  • Giuseppe Persiano
    • 2
  • Anna Philippou
    • 1
  • Paul Spirakis
    • 3
  1. 1.Department of Computer ScienceUniversity of CyprusNicosiaCyprus
  2. 2.Dipartimento di Informatica ed Applicazioni “Renato M. Capocelli”Università di SalernoItaly
  3. 3.Research Academic Computer Technology Institute, Greece & Department of Computer Engineering and InformaticsUniversity of PatrasGreece

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