# Turbocharging Solution Concepts: Solving NEs, CEs and CCEs with Neural Equilibrium Solvers

arXiv: https://arxiv.org/abs/2210.09257 · HTML: https://mlsanity.com/paper/2210.09257/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 63 references that are in the map.
- First posted on arXiv 17 Oct 2022.

## Builds on (most cited first, 19 of 19)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [High-Performance Large-Scale Image Recognition Without Normalization](https://mlsanity.com/paper/2102.06171/) (2021)
- [A Unified Game-Theoretic Approach to Multiagent Reinforcement Learning](https://mlsanity.com/paper/1711.00832/) (2017)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [A Rewriting System for Convex Optimization Problems](https://mlsanity.com/paper/1709.04494/) (2017)
- [V-Learning -- A Simple, Efficient, Decentralized Algorithm for Multiagent RL](https://mlsanity.com/paper/2110.14555/) (2021)
- [Near-Optimal Reinforcement Learning with Self-Play](https://mlsanity.com/paper/2006.12007/) (2020)
- [The general theory of permutation equivarant neural networks and higher order graph variational encoders](https://mlsanity.com/paper/2004.03990/) (2020)
- [What game are we playing? End-to-end learning in normal and extensive form games](https://mlsanity.com/paper/1805.02777/) (2018)
- [Multi-Agent Training beyond Zero-Sum with Correlated Equilibrium Meta-Solvers](https://mlsanity.com/paper/2106.09435/) (2021)
- [NeuPL: Neural Population Learning](https://mlsanity.com/paper/2202.07415/) (2022)
- [Simplex Neural Population Learning: Any-Mixture Bayes-Optimality in Symmetric Zero-sum Games](https://mlsanity.com/paper/2205.15879/) (2022)
- [Computing Approximate Nash Equilibria in Polymatrix Games](https://mlsanity.com/paper/1409.3741/) (2014)
- [An Empirical Study on Computing Equilibria in Polymatrix Games](https://mlsanity.com/paper/1602.06865/) (2016)

## Built on by (most cited first, 10 of 10)

- [Re-evaluating Open-ended Evaluation of Large Language Models](https://mlsanity.com/paper/2502.20170/) (2025)
- [Neural Population Learning beyond Symmetric Zero-sum Games](https://mlsanity.com/paper/2401.05133/) (2024)
- [Computing Game Symmetries and Equilibria That Respect Them](https://mlsanity.com/paper/2501.08905/) (2025)
- [Smooth Nash Equilibria: Algorithms and Complexity](https://mlsanity.com/paper/2309.12226/) (2023)
- [WardropNet: Traffic Flow Predictions via Equilibrium-Augmented Learning](https://mlsanity.com/paper/2410.06656/) (2024)
- [Are Equivariant Equilibrium Approximators Beneficial?](https://mlsanity.com/paper/2301.11481/) (2023)
- [Deep Incentive Design with Differentiable Equilibrium Blocks](https://mlsanity.com/paper/2603.07705/) (2026)
- [Fully Distributed GNE Algorithms for Multi-Robot Placement without Consensus on Multipliers](https://mlsanity.com/paper/2608.29388/) (2026)
- [Large-Scale Contextual Market Equilibrium Computation through Deep Learning](https://mlsanity.com/paper/2406.15459/) (2024)
- [BRAID: Learning Equilibrium Maps in Interdependent Security Games via Weight-Tied Iterative Graph Neural Networks](https://mlsanity.com/paper/2608.14856/) (2026)

## Abstract (arXiv)

Solution concepts such as Nash Equilibria, Correlated Equilibria, and Coarse Correlated Equilibria are useful components for many multiagent machine learning algorithms. Unfortunately, solving a normal-form game could take prohibitive or non-deterministic time to converge, and could fail. We introduce the Neural Equilibrium Solver which utilizes a special equivariant neural network architecture to approximately solve the space of all games of fixed shape, buying speed and determinism. We define a flexible equilibrium selection framework, that is capable of uniquely selecting an equilibrium that minimizes relative entropy, or maximizes welfare. The network is trained without needing to generate any supervised training data. We show remarkable zero-shot generalization to larger games. We argue that such a network is a powerful component for many possible multiagent algorithms.