Interdependent Defense Games with Applications to Internet Security at the Level of Autonomous Systems

Hau Chan,Luis Ortiz,Michael Ceyko

doi:10.3390/g8010013

Abstract

We propose interdependent defense (IDD) games, a computational game-theoretic framework to study aspects of the interdependence of risk and security in multi-agent systems under deliberate external attacks. Our model builds upon interdependent security (IDS) games, a model by Heal and Kunreuther that considers the source of the risk to be the result of a fixed randomized-strategy. We adapt IDS games to model the attacker’s deliberate behavior. We define the attacker’s pure-strategy space and utility function and derive appropriate cost functions for the defenders. We provide a complete characterization of mixed-strategy Nash equilibria (MSNE), and design a simple polynomial-time algorithm for computing all of them for an important subclass of IDD games. We also show that an efficient algorithm to determine whether some attacker’s strategy can be a part of an MSNE in an instance of IDD games is unlikely to exist. Yet, we provide a dynamic programming (DP) algorithm to compute an approximate MSNE when the graph/network structure of the game is a directed tree with a single source. We also show that the DP algorithm is a fully polynomial-time approximation scheme. In addition, we propose a generator of random instances of IDD games based on the real-world Internet-derived graph at the level of autonomous systems (≈27 K nodes and ≈100 K edges as measured in March 2010 by the DIMES project). We call such games Internet games. We introduce and empirically evaluate two heuristics from the literature on learning-in-games, best-response gradient dynamics (BRGD) and smooth best-response dynamics (SBRD), to compute an approximate MSNE in IDD games with arbitrary graph structures, such as randomly-generated instances of Internet games. In general, preliminary experiments applying our proposed heuristics are promising. Our experiments show that, while BRGD is a useful technique for the case of Internet games up to certain approximation level, SBRD is more efficient and provides better approximations than BRGD. Finally, we discuss several extensions, future work, and open problems.

Highlights

Attacks carried out by hackers and terrorists in recent decades have led to increased efforts by both government and the private sector to create and adopt mechanisms to prevent future attacks
For the purpose of studying the computational complexity of single-attack interdependent defense (IDD) games, it is natural to view the computation of an mixed-strategy Nash equilibria (MSNE) as a two-part process
The first question we address is, what is the relation between the constant λ and the actual approximation quality achieved in practice? Table 2 shows the impact λ has on, for the smallest -MSNE we can obtain for an instance of the Internet games (IGs)

Summary

Introduction

Attacks carried out by hackers and terrorists in recent decades have led to increased efforts by both government and the private sector to create and adopt mechanisms to prevent future attacks. This effort has yielded a more focused research attention to models, computational and otherwise, that facilitate and help to improve (homeland) security for both physical infrastructure and cyberspace. Introduced and studied by economists Kunreuther and Heal [3], IDS games model general abstract security problems In those problems, an individual within a population considers whether to voluntarily invest in some protection mechanisms or security against a risk they may face. This is because of transfer risks; that is, the “bad event” may be transferable from a compromised individual to another

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Results

Conclusion