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

arXiv: https://arxiv.org/abs/2608.14856 · HTML: https://mlsanity.com/paper/2608.14856/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 37 references that are in the map.
- First posted on arXiv 14 Aug 2026.

## Builds on (most cited first, 12 of 12)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Gated Graph Sequence Neural Networks](https://mlsanity.com/paper/1511.05493/) (2015)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Turbocharging Solution Concepts: Solving NEs, CEs and CCEs with Neural Equilibrium Solvers](https://mlsanity.com/paper/2210.09257/) (2022)
- [Learning to Infer Structures of Network Games](https://mlsanity.com/paper/2206.08119/) (2022)
- [Reinforcement Nash Equilibrium Solver](https://mlsanity.com/paper/2405.03518/) (2024)
- [Interdependent Security Games on Networks under Behavioral Probability Weighting](https://mlsanity.com/paper/1510.09106/) (2015)
- [Finite-Agent Stochastic Differential Games on Large Graphs: II. Graph-Based Architectures](https://mlsanity.com/paper/2509.12484/) (2025)

## Abstract (arXiv)

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.