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

Abstract

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.

Who Should Pay the Cost: A Game-theoretic Model for Government Subsidized Investments to Improve National Cybersecurity

“…The first line is interdependent security game [Kunreuther and Heal, 2003;Kearns and Ortiz, 2004] where users are interdependent and can experience direct risk from internal contamination and indirect risk transferred from their neighbors and is widely applied to security settings such as airline baggage security, fire safety and computer virus. An extension of these works is interdependent defense game [Chan et al, 2012;Chan et al, 2017] where a strategic attacker is added into the game. However, they focus on the computation of Nash equilibria between users and the attacker and there is no defender who tries to optimize the global security level.…”

“…Both users and the attacker are termed as followers. The interactions between followers exactly follow the assumptions and models in interdependent security (defense) game, which is widely investigated in [Kunreuther and Heal, 2003;Kearns and Ortiz, 2004;Chan et al, 2012;Chan et al, 2017].…”

“…is the probability that there is at least one of user i's neighbors will transfer the contamination to him. The utilities (omitting the x i 's) and terms are defined the same way as in [Kearns and Ortiz, 2004;Chan et al, 2017]. The government's utility is defined to be the negative of the summation of the expected loss of users…”

“…We consider the game where the attacker can only attack one user. The setting without the government is extensively investigated in [Chan et al, 2012;Chan et al, 2017], which provides some useful results for us to develop our algorithm. The following proposition proves that, except the case where all users invest, there is no other pure NE between followers.…”

“…Then the proof can be adapted from the proof of Proposition 5 in [Chan et al, 2017] by setting the attack cost to zero. Generally speaking, the proof of the proposition is based on three facts: i) when z i = ∆ i , the users will be indifferent to the investment which is observed from Eq.…”