Bayesian hypergame approach to equilibrium stability and robustness in moving target defense

TL;DR

The work addresses equilibrium stability and robustness in moving target defense under incomplete information and asymmetric cognition by formulating a hyper Bayesian Stackelberg framework. It combines Bayesian Stackelberg games for the attacker’s incomplete information with a hypergame representation that captures the defender’s exact knowledge, defining the hyper Bayesian Nash equilibrium (HBNE) and a linear-equation stability condition that guarantees strategic and cognitive stability. The authors prove existence of BSSE and HBNE via a Harsanyi transformation and provide corollaries for common type distributions, while introducing robust HBNE to address perturbations in the attacker’s perceived distribution, showing that small distributional changes preserve stability. Numerical experiments validate the theoretical results, demonstrating high incidence of stability conditions and practical robustness against perturbations such as malicious jamming in multi-type settings, reinforcing the framework’s applicability to CPS cybersecurity and related MTD deployments.

Abstract

We investigate the equilibrium stability and robustness in a class of moving target defense problems, in which players have both incomplete information and asymmetric cognition. We first establish a Bayesian Stackelberg game model for incomplete information and then employ a hypergame reformulation to address asymmetric cognition. With the core concept of the hyper Bayesian Nash equilibrium (HBNE), a condition for achieving both the strategic and cognitive stability in equilibria can be realized by solving linear equations. Moreover, to deal with players' underlying perturbed knowledge, we study the equilibrium robustness by presenting a condition of robust HBNE under the given configuration. Experiments evaluate our theoretical results.

Figures (2)

• Figure 1: The equivalence of BSSE and HBNE. The x-axis is for the target amount in (a), and for the type amount in (b). The y-axis is for ratios where Case 1 and Case 2 happen in 100 instances, depicted in blue and red, respectively.
• Figure 2: Robust HBNE in two settings. Any point $(P(\theta_1),P(\theta_2))$ in this figure refers to a distribution $P$ over $\theta_1$, $\theta_2$, and $\theta_3$ with $P(\theta_3)=1-P(\theta_1)-P(\theta_2)$. If a point has a same-color neighborhood, then the corresponding HBNE is robust. Figures (a) and (b) are two different settings with randomly generated utilities.

Theorems & Definitions (11)

• Definition 1
• Definition 2
• Definition 3
• Remark 1
• Lemma 1
• Theorem 1
• Corollary 1
• Corollary 2
• Corollary 3
• Definition 4