Attack Detection in Dynamic Games with Quadratic Measurements

TL;DR

The paper addresses secure state estimation in dynamic games by pairing a quadratic, attack-resistant observer with a statistical test that detects attacks on a vulnerable linear sensor. It introduces a two-observer framework (Kalman filter for linear measurements and an EKF-like updater with a projection for the quadratic measurement) and proves feasibility, prox-regularity, and monotone error reduction in the noise-free setting. A wild-bootstrap MMD test compares the trajectories of the two observers, accommodating temporal dependence to detect distributional shifts caused by attacks. Numerical experiments in a pursuit–evasion game show that the quadratic observer maintains accuracy under linear attacks, while the MMD test provides prompt, reliable detection. The approach offers a principled, data-driven method for resilient state estimation and attack detection in multi-agent dynamic games with secure utility measurements.

Abstract

This paper studies attack detection for discrete-time linear systems with stochastic process noise that produce both a vulnerable (i.e., attackable) linear measurement and a secured (i.e., unattackable) quadratic measurement. The motivating application of this model is a dynamic-game setting where the quadratic measurement is interpreted as a system-level utility or reward, and control inputs into the linear system are interpreted as control policies that, once applied, are known to all game participants and which steer the system towards a game-theoretic equilibrium (e.g., Nash equilibrium). To detect attacks on the linear channel, we develop a novel quadratic-utility-aware observer that leverages the secured quadratic output and enforces measurement consistency via a projection step. We establish three properties for this observer: feasibility of the true state, prox-regularity of the quadratic-constraint set, and a monotone error-reduction guarantee in the noise-free case. To detect adversarial manipulation, we compare linear and quadratic observer trajectories using a wild bootstrap maximum mean discrepancy (MMD) test that provides valid inference under temporal dependence. We validate our framework using numerical experiments of a pursuit-evasion game, where the quadratic observer preserves estimation accuracy under linear-sensor attacks, while the statistical test detects distributional divergence between the observers' trajectories.

Figures (3)

• Figure 1: Representative trial from the repeated experiments. Left: linear observer. Right: quadratic observer. True trajectories are shown for the evader (blue) and pursuer (red). The estimated pursuer trajectory is overlaid with hollow circles: green for pre–attack samples and orange for post–attack samples. Yellow "X" marks the attack onset. Faint lines trace the motion path and arrows indicate instantaneous velocity.
• Figure 2: Observer MSE over time aggregated across $M=100$ runs. Red: linear observer; blue: quadratic observer. Solid lines: mean MSE; shaded regions: $\pm$ SE. Vertical dashed line indicates attack onset.
• Figure 3: Attack detection via wild bootstrap MMD over $M=100$ repeated experiments. Green: mean MMD test statistic; orange: mean critical value. Solid lines: mean; shaded bands: $\pm$ SE. Vertical dotted line marks attack onset; red shaded region denotes the attack period.

Theorems & Definitions (8)

• Proposition 1: Prox-Regularity via Stacked Amenability
• proof
• Lemma 1: Feasibility of State with Adaptive Bounds
• proof
• Lemma 2: Cross-Error Term Inequality
• proof
• Theorem 1: Projection Error Bound
• proof