# Safe and Nested Subgame Solving for Imperfect-Information Games

arXiv: https://arxiv.org/abs/1705.02955 · HTML: https://mlsanity.com/paper/1705.02955/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2017.

## Built on by (most cited first, 30 of 41)

- [Learning to run a Power Network Challenge: a Retrospective Analysis](https://mlsanity.com/paper/2103.03104/) (2021)
- [ESCHER: Eschewing Importance Sampling in Games by Computing a History Value Function to Estimate Regret](https://mlsanity.com/paper/2206.04122/) (2022)
- [Student of Games: A unified learning algorithm for both perfect and imperfect information games](https://mlsanity.com/paper/2112.03178/) (2021)
- [PokerGPT: An End-to-End Lightweight Solver for Multi-Player Texas Hold'em via Large Language Model](https://mlsanity.com/paper/2401.06781/) (2024)
- [Solving Common-Payoff Games with Approximate Policy Iteration](https://mlsanity.com/paper/2101.04237/) (2021)
- [Bandit Linear Optimization for Sequential Decision Making and Extensive-Form Games](https://mlsanity.com/paper/2103.04546/) (2021)
- [RL-CFR: Improving Action Abstraction for Imperfect Information Extensive-Form Games with Reinforcement Learning](https://mlsanity.com/paper/2403.04344/) (2024)
- [Scalable Decision Making for Games of Imperfect Information](https://mlsanity.com/paper/2511.07312/) (2025)
- [A Marriage between Adversarial Team Games and 2-player Games: Enabling Abstractions, No-regret Learning, and Subgame Solving](https://mlsanity.com/paper/2206.09161/) (2022)
- [Minimizing Weighted Counterfactual Regret with Optimistic Online Mirror Descent](https://mlsanity.com/paper/2404.13891/) (2024)
- [Abstracting Imperfect Information Away from Two-Player Zero-Sum Games](https://mlsanity.com/paper/2301.09159/) (2023)
- [General search techniques without common knowledge for imperfect-information games, and application to superhuman Fog of War chess](https://mlsanity.com/paper/2506.01242/) (2025)
- [Confucius3-Math: A Lightweight High-Performance Reasoning LLM for Chinese K-12 Mathematics Learning](https://mlsanity.com/paper/2506.18330/) (2025)
- [A Theory of Inference Compute Scaling: Reasoning through Directed Stochastic Skill Search](https://mlsanity.com/paper/2507.00004/) (2025)
- [PokerSkill: LLMs Can Play Expert-Level Poker without Training or Solvers](https://mlsanity.com/paper/2605.30094/) (2026)
- [Hierarchical Deep Counterfactual Regret Minimization](https://mlsanity.com/paper/2305.17327/) (2023)
- [Agents That Certify Their Own Exploits: Confidence-Scheduled Restricted Responses for Safe Opponent Exploitation](https://mlsanity.com/paper/2607.28520/) (2026)
- [Safe Search for Stackelberg Equilibria in Extensive-Form Games](https://mlsanity.com/paper/2102.01775/) (2021)
- [A Survey of Decision Making in Adversarial Games](https://mlsanity.com/paper/2207.07971/) (2022)
- [The Update-Equivalence Framework for Decision-Time Planning](https://mlsanity.com/paper/2304.13138/) (2023)
- [GTO Wizard Benchmark](https://mlsanity.com/paper/2603.23660/) (2026)
- [Accelerate Multi-Agent Reinforcement Learning in Zero-Sum Games with Subgame Curriculum Learning](https://mlsanity.com/paper/2310.04796/) (2023)
- [Solver-Guided Reasoning for Mixed-Equilibrium Strategies](https://mlsanity.com/paper/2608.06741/) (2026)
- [D2CFR: Minimize Counterfactual Regret with Deep Dueling Neural Network](https://mlsanity.com/paper/2105.12328/) (2021)
- [AI in Human-computer Gaming: Techniques, Challenges and Opportunities](https://mlsanity.com/paper/2111.07631/) (2021)
- [Safe adaptation in multiagent competition](https://mlsanity.com/paper/2203.07562/) (2022)
- [CFR-p: Counterfactual Regret Minimization with Hierarchical Policy Abstraction, and its Application to Two-player Mahjong](https://mlsanity.com/paper/2307.12087/) (2023)
- [Beyond Outcome-Based Imperfect-Recall: Higher-Resolution Abstractions for Imperfect-Information Games](https://mlsanity.com/paper/2510.15094/) (2025)
- [AV-AIVAT: 74x Cheaper Agent Evaluation with Certified Anytime-Valid Stopping in Imperfect-Information Games](https://mlsanity.com/paper/2608.06362/) (2026)
- [Towards a Sample Efficient Reinforcement Learning Pipeline for Vision Based Robotics](https://mlsanity.com/paper/2105.09719/) (2021)

## Abstract (arXiv)

In imperfect-information games, the optimal strategy in a subgame may depend on the strategy in other, unreached subgames. Thus a subgame cannot be solved in isolation and must instead consider the strategy for the entire game as a whole, unlike perfect-information games. Nevertheless, it is possible to first approximate a solution for the whole game and then improve it by solving individual subgames. This is referred to as subgame solving. We introduce subgame-solving techniques that outperform prior methods both in theory and practice. We also show how to adapt them, and past subgame-solving techniques, to respond to opponent actions that are outside the original action abstraction; this significantly outperforms the prior state-of-the-art approach, action translation. Finally, we show that subgame solving can be repeated as the game progresses down the game tree, leading to far lower exploitability. These techniques were a key component of Libratus, the first AI to defeat top humans in heads-up no-limit Texas hold'em poker.