# Solver-Guided Reasoning for Mixed-Equilibrium Strategies

arXiv: https://arxiv.org/abs/2608.06741 · HTML: https://mlsanity.com/paper/2608.06741/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning for agents, planning and code, one of 17 directions in LLM reasoning.
- Builds on 25 of its 49 references that are in the map.
- First posted on arXiv 7 Aug 2026.

## Placed in

- LLM reasoning / [Reasoning for agents, planning and code](https://mlsanity.com/lr/reasoning-for-agents-planning-and-code/)

## Builds on (most cited first, 25 of 25)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [Concept Bottleneck Models](https://mlsanity.com/paper/2007.04612/) (2020)
- [STaR: Bootstrapping Reasoning With Reasoning](https://mlsanity.com/paper/2203.14465/) (2022)
- [Textbooks Are All You Need](https://mlsanity.com/paper/2306.11644/) (2023)
- [Policy Distillation](https://mlsanity.com/paper/1511.06295/) (2015)
- [Neural Additive Models: Interpretable Machine Learning with Neural Nets](https://mlsanity.com/paper/2004.13912/) (2020)
- [Distilling a Neural Network Into a Soft Decision Tree](https://mlsanity.com/paper/1711.09784/) (2017)
- [DeepStack: Expert-Level Artificial Intelligence in No-Limit Poker](https://mlsanity.com/paper/1701.01724/) (2017)
- [Programmatically Interpretable Reinforcement Learning](https://mlsanity.com/paper/1804.02477/) (2018)
- [Verifiable Reinforcement Learning via Policy Extraction](https://mlsanity.com/paper/1805.08328/) (2018)
- [Acquisition of Chess Knowledge in AlphaZero](https://mlsanity.com/paper/2111.09259/) (2021)
- [Combining Deep Reinforcement Learning and Search for Imperfect-Information Games](https://mlsanity.com/paper/2007.13544/) (2020)
- [Deep Counterfactual Regret Minimization](https://mlsanity.com/paper/1811.00164/) (2018)
- [Solving Imperfect-Information Games via Discounted Regret Minimization](https://mlsanity.com/paper/1809.04040/) (2018)
- [Suspicion-Agent: Playing Imperfect Information Games with Theory of Mind Aware GPT-4](https://mlsanity.com/paper/2309.17277/) (2023)
- [Safe and Nested Subgame Solving for Imperfect-Information Games](https://mlsanity.com/paper/1705.02955/) (2017)
- [Are ChatGPT and GPT-4 Good Poker Players? -- A Pre-Flop Analysis](https://mlsanity.com/paper/2308.12466/) (2023)
- [PokerGPT: An End-to-End Lightweight Solver for Multi-Player Texas Hold'em via Large Language Model](https://mlsanity.com/paper/2401.06781/) (2024)
- [PokerBench: Training Large Language Models to become Professional Poker Players](https://mlsanity.com/paper/2501.08328/) (2025)
- [How Far Are LLMs from Professional Poker Players? Revisiting Game-Theoretic Reasoning with Agentic Tool Use](https://mlsanity.com/paper/2602.00528/) (2026)
- [GTO Wizard Benchmark](https://mlsanity.com/paper/2603.23660/) (2026)

## Built on by (most cited first, 2 of 2)

- [Abstraction Agent](https://mlsanity.com/paper/2609.04303/) (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)

## Abstract (arXiv)

Reasoning in large language models (LLMs) is often grounded in human text, human demonstrations, and human-generated rationales. For equilibrium reasoning in complex games, however, relying on human data can be suboptimal. In fact, human play is often guided by intuition and heuristics and can deviate substantially from game equilibrium. This discrepancy is amplified in games with mixed-strategy equilibria, where human data is heavily biased toward pure strategies. Consequently, conditioning LLMs on this data yields weak game strategies. To grant LLMs the reasoning capacity in games, in this work, we study how to elicit equilibrium play using solver output. We propose Mixed-Strategy Decision Tree (MDT), which articulates the silent optimality of the equilibrium into sparse strategic rules that both humans and LLMs could understand. Using solver output rather than human annotation allows us to extend the input to arbitrarily new states and continuations. We instantiate this study on No-Limit Texas Hold'em by querying a solver oracle for over 250 million mixed-strategy decisions; MDT together with other techniques reduces the ℓ1 distance to the equilibrium by 52.6% across 8 different LLM configurations. A Route-only ablation tests the incremental contribution of the shadow-based contrast, while complete River-endgame and Liar's Dice experiments evaluate strategic fidelity and portability beyond the original NLH communication setting.