Abstract
Two players, the Defender and the Attacker play the following game. A matroid \(M=(S,\mathcal {I})\), a weight function \(d:S\rightarrow \mathbb {R}^+\) and a cost function \(c:S\rightarrow \mathbb {R}\) are given. The Defender chooses a base B of the matroid M and the Attacker chooses an element \(s\in S\) of the ground set. In all cases, the Attacker pays the Defender the cost of attack c(s). Besides that, if \(s\in B\) then the Defender pays the Attacker the amount d(s); if, on the other hand, \(s\notin B\) then there is no further payoff. Special cases of this two-player, zero-sum game were considered and solved in various security-related applications. In this paper we show that it is also possible to compute Nash-equilibrium mixed strategies for both players in strongly polynomial time in the above general matroid setting. We also consider a further generalization where common bases of two matroids are chosen by the Defender and apply this to define and efficiently compute a new reliability metric on digraphs.
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References
Alpcan, T., Başar, T.: Network Security: A Decision and Game-Theoretic Approach. Cambridge University Press, New York (2010)
Cunningham, W.H.: Optimal attack and reinforcement of a network. J. ACM (JACM) 32(3), 549–561 (1985)
Cunningham, W.H.: Testing membership in matroid polyhedra. J. Comb. Theory Ser. B 36(2), 161–188 (1984)
Gueye, A., Walrand, J.C., Anantharam, V.: Design of network topology in an adversarial environment. In: Proceedings of the 1st International Conference on Decision and Game Theory for Security, GameSec10, Berlin, Germany, pp. 1–20 (2010)
Gueye, A., Walrand, J.C., Anantharam, V.: A network topology design game: how to choose communication links in an adversarial environment. In: Proceedings of the 2nd International ICST Conference on Game Theory for Networks, GameNets, vol. 11 (2011)
Gusfield, D.: Connectivity and edge-disjoint spanning trees. Inf. Process. Lett. 16(2), 87–89 (1983)
Han, Z.: Game Theory in Wireless and Communication Networks: Theory, Models, and Applications. Cambridge University Press, New York (2012)
Laszka, A., Szeszlér, D.: Hide and seek in digital communication: the steganography game. In: Proceedings of the 9th Hungarian–Japanese Symposium on Discrete Mathematics and its Applications, Fukuoka, Japan, pp. 126–136 (2015)
Machado, R., Tekinay, S.: A survey of game-theoretic approaches in wireless sensor networks. Comput. Netw. 52(16), 3047–3061 (2008)
Manshaei, M.H., Zhu, Q., Alpcan, T., Bacsar, T., Hubaux, J.P.: Game theory meets network security and privacy. ACM Comput Surv (CSUR). vol. 45.3, Article No. 25 (2013)
Matoušek, J., Gärtner, B.: Understanding and Using Linear Programming, 226 p. Springer, Berlin (2007)
Neumann, Jv: Zur Theorie der Gesellschaftsspiele. Math. Ann. 100(1), 295–320 (1928)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics, vol. 24, 1879 p. Springer, Berlin (2003)
Tambe, M.: Security and game theory: algorithms, deployed systems, lessons learned. Cambridge University Press, New York (2012)
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Research supported by Grant No. OTKA 185101 of the Hungarian Scientific Research Fund.
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Szeszlér, D. Security games on matroids. Math. Program. 161, 347–364 (2017). https://doi.org/10.1007/s10107-016-1011-9
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DOI: https://doi.org/10.1007/s10107-016-1011-9