BRAID: Learning Equilibrium Maps in Interdependent Security Games via Weight-Tied Iterative Graph Neural Networks

2026-08-14

AI Summary

This study addresses the computational inefficiency of Nash equilibrium computation in cybersecurity games and the challenge of repeated solving under parameter perturbations by proposing BRAID. This model employs a weight-tied iterative graph neural network to directly map game parameters to equilibrium solutions, bypassing traditional iterative algorithms. By simulating best-response dynamics and incorporating an internal equilibrium training strategy, BRAID enables single-pass forward prediction without requiring sensitivity labels. Experimental results demonstrate that BRAID achieves a 43-fold acceleration in inference speed while maintaining high accuracy across diverse utility functions and network scales. Furthermore, it effectively recovers parameter sensitivities, significantly enhancing both computational efficiency and robustness in equilibrium prediction for dynamic cybersecurity environments.

Abstract

Computing Nash equilibria in interdependent security (IDS) games on networks is computationally expensive: best-response dynamics may need hundreds of iterations per instance, and downstream tasks such as auditing, stress-testing, and incentive design often require repeatedly re-solving the game under parameter perturbations. We propose BRAID, a Best-Response Amortized Iterative Dynamics model that uses a weight-tied iterative graph neural network to learn a direct map from game parameters to Nash equilibrium effort profiles, replacing iterative best response computation with a single forward pass that is up to 43X faster per instance. BRAID is derived from the best-response fixed-point structure of IDS games: its SUM aggregation reflects additive neighbor coupling, and a weight-tied gated recurrent unit (GRU) mirrors a damped best-response update. The same architecture applies across IDS specifications that vary investment-cost curvature and neighborhood aggregation, including log-linear, quadratic-cost, and log constant-elasticity-of-substitution (CES) utilities. Beyond equilibrium prediction, BRAID also recovers how equilibrium efforts change under perturbations to game parameters, including costs and network edge weights. We make this sensitivity recovery an explicit evaluation target and introduce two training strategies, interior-equilibrium training and input-noise regularization, that improve the local behavior of the learned equilibrium map without using sensitivity labels. Experiments show that BRAID effectively predicts Nash equilibria and recovers equilibrium sensitivities across utility specifications and network sizes.

Problem

• Interdependent Security Games
• Nash Equilibrium
• Computational Cost
• Parameter Perturbations

Innovation

• Weight-Tied Iterative GNN
• Best-Response Amortization
• Nash Equilibrium Learning
• Sensitivity Recovery
• Interdependent Security Games