# Depth-Limited Solving for Imperfect-Information Games

arXiv: https://arxiv.org/abs/1805.08195 · HTML: https://mlsanity.com/paper/1805.08195/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 May 2018.

## Built on by (most cited first, 27 of 27)

- [ESCHER: Eschewing Importance Sampling in Games by Computing a History Value Function to Estimate Regret](https://mlsanity.com/paper/2206.04122/) (2022)
- [PokerGPT: An End-to-End Lightweight Solver for Multi-Player Texas Hold'em via Large Language Model](https://mlsanity.com/paper/2401.06781/) (2024)
- [Neural Recursive Belief States in Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2102.02274/) (2021)
- [Policy Space Response Oracles: A Survey](https://mlsanity.com/paper/2403.02227/) (2024)
- [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)
- [Real-Time Parallel Counterfactual Regret Minimization](https://mlsanity.com/paper/2605.19928/) (2026)
- [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)
- [A Survey of Decision Making in Adversarial Games](https://mlsanity.com/paper/2207.07971/) (2022)
- [Look-ahead Reasoning with a Learned Model in Imperfect Information Games](https://mlsanity.com/paper/2510.05048/) (2025)
- [AV-AIVAT: 74x Cheaper Agent Evaluation with Certified Anytime-Valid Stopping in Imperfect-Information Games](https://mlsanity.com/paper/2608.06362/) (2026)
- [DecisionHoldem: Safe Depth-Limited Solving With Diverse Opponents for Imperfect-Information Games](https://mlsanity.com/paper/2201.11580/) (2022)
- [Methods and Mechanisms for Interactive Novelty Handling in Adversarial Environments](https://mlsanity.com/paper/2302.14208/) (2023)
- [Game-Theoretic Robust Reinforcement Learning Handles Temporally-Coupled Perturbations](https://mlsanity.com/paper/2307.12062/) (2023)
- [AI planning in the imagination: High-level planning on learned abstract search spaces](https://mlsanity.com/paper/2308.08693/) (2023)
- [Accelerating Nash Equilibrium Convergence in Monte Carlo Settings Through Counterfactual Value Based Fictitious Play](https://mlsanity.com/paper/2309.03084/) (2023)
- [AlphaZeroES: Direct score maximization outperforms planning loss minimization](https://mlsanity.com/paper/2406.08687/) (2024)
- [Evolutionary Optimization of Deep Learning Agents for Sparrow Mahjong](https://mlsanity.com/paper/2508.07522/) (2025)
- [Test-time Reinforcement Learning in Imperfect Information Games](https://mlsanity.com/paper/2608.30635/) (2026)
- [GPU-CFR: 80x Faster Counterfactual Regret Minimization by Compiling the Game to Static Dataflow and CUDA Graph Replay](https://mlsanity.com/paper/2609.11923/) (2026)
- [Learning to Play: Reinforcement Learning and Games](https://mlsanity.com/paper/2610.17391/) (2026)

## Abstract (arXiv)

A fundamental challenge in imperfect-information games is that states do not have well-defined values. As a result, depth-limited search algorithms used in single-agent settings and perfect-information games do not apply. This paper introduces a principled way to conduct depth-limited solving in imperfect-information games by allowing the opponent to choose among a number of strategies for the remainder of the game at the depth limit. Each one of these strategies results in a different set of values for leaf nodes. This forces an agent to be robust to the different strategies an opponent may employ. We demonstrate the effectiveness of this approach by building a master-level heads-up no-limit Texas hold'em poker AI that defeats two prior top agents using only a 4-core CPU and 16 GB of memory. Developing such a powerful agent would have previously required a supercomputer.