Security Games with Decision and Observation Errors

arXiv:1003.2767v1 [cs.GT] 14 Mar 2010 Security Games with Decision and Observation Errors Kien C. Nguyen, Tansu Alpcan, and Tamer Bas¸ar Abstract— We study two-player security games which can be viewed as sequences of nonzero-sum matrix games played by an Attacker and a Defender. The evolution of the game is based on a stochastic fictitious play process. Players do not have access to each other’s payoff matrix. Each has to observe the other’s actions up to present and plays the action generated based on the best response to these observations. However, when the game is played over a communication network, there are several practical issues that need to be taken into account: First, the players may make random decision errors from time to time. Second, the players’ observations of each other’s previous actions may be incorrect. The players will try to compensate for these errors based on the information they have. We examine convergence property of the game in such scenarios, and establish convergence to the equilibrium point under some mild assumptions when both players are restricted to two actions. I. INTRODUCTION Game theory has recently been used as an effective tool to model and solve many security problems in computer and communication networks. In a noncooperative matrix game between an Attacker and a Defender, if the payoff matrices are assumed to be known to both players, each player can compute the set of Nash equilibria of the game and play one of these strategies to maximize her expected gain (or minimize its expected loss)1. However, in practice, the players do not necessarily have full knowledge of each other’s payoff function. If the game is repeated, a mechanism called fictitious play (FP) can be used for each player to learn her opponent’s motivations. In a FP process, each player observes all the actions and makes estimates of the mixed strategy of her opponent. At each stage, she updates this estimate and plays the pure strategy that is the best response (or generated based on the best response) to the current estimate of the other’s mixed strategy. It can be seen that in a FP process, if one person plays a fixed strategy (either of the pure or mixed type), the other person’s strategy will converge to the best response to this fixed strategy. Furthermore, it has been shown that, for many classes of games, such a FP process will finally render both players playing the Nash equilibrium. This work was supported by Deutsche Telekom Laboratories and the Boeing Company. Tamer Bas¸ar and Kien C. Nguyen are with the Department of Electrical and Computer Engineering and the Coordinated Sci- ence Laboratory, University of Illinois at Urbana-Champaign, USA basar1@illinois.edu, knguyen4@illinois.edu Tansu Alpcan is with Deutsche Telekom Laboratories, Technical Univer- sity of Berlin, Berlin, Germany tansu.alpcan@telekom.de 1The problem of each player choosing a Nash equilibrium out of multiple Nash equilibria is not discussed within the scope of this paper. In this paper, we examine a two-player game, where an Attacker (denoted as player 1 or P1) and a Defender (denoted as player 2 or P2) participate in a discrete-time repeated nonzero-sum matrix game. In a general setting, the Attacker has m possible actions and the Defender has n posssible actions to choose from. When such a security game is played between two automated systems over a network, in order to have a good model, we have to take into account several practical issues. First, the players may make random decision errors from time to time. Instead of playing an action aj i that is the output of the best-response computation, player i may play another action ak i with some probability (which is typically small for functional systems). Second, the observation that each player makes on her opponent’s actions may also be incorrect, which will definitely affect her own responding actions. There are many factors giving rise to these problems: The non-idealiality of electronic and software systems, the uncertain and noisy characteristic of observation data, and the erroneous nature of the channels on which commands and observations are communicated, to name a few. It is these scenarios that we aim to address in this paper. We examine convergence of players’ strategies in the FP process with decision and observation errors. If these strategies do converge, we quantify the new Nash equilibrium and thus estimate how these decision and observation errors affect the learning process and the equilibrium of the game. Security games have been examined extensively in a number of papers, see for example, [1]–[4]. The work in [5] employs the framework of Bayesian games to address the intrusion detection problem in wireless ad hoc networks. In [6], the author examines the intrusion detection problem in heterogenous networks as a nonzero-sum static game. The work in [7] addresses this problem using the framework of zero-sum stochastic games [8]. In [9], we develop

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