Solver-Guided Reasoning for Mixed-Equilibrium Strategies

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 \textbf{250 million mixed-strategy decisions}; MDT together with other techniques \textbf{reduces the $\ell_1$ 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.

GTO Wizard Benchmark

We introduce GTO Wizard Benchmark, a public API and standardized evaluation framework for benchmarking algorithms in Heads-Up No-Limit Texas Hold'em (HUNL). The benchmark evaluates agents against GTO Wizard AI, a state-of-the-art superhuman poker agent that approximates Nash Equilibria, and defeated Slumbot, the 2018 Annual Computer Poker Competition champion and previous strongest publicly accessible HUNL benchmark, by $19.4$ $\pm$ $4.1$ bb/100. Variance is a fundamental challenge in poker evaluation; we address this by integrating AIVAT, a provably unbiased variance reduction technique that achieves equivalent statistical significance with ten times fewer hands than naive Monte Carlo evaluation. We conduct a comprehensive benchmarking study of state-of-the-art large language models under zero-shot conditions, including GPT-5.4, Claude Opus 4.6, Gemini 3.1 Pro, Grok 4, and others. Initial results and analysis reveal dramatic progress in LLM reasoning over recent years, yet all models remain far below the baseline established by our benchmark. Qualitative analysis reveals clear opportunities for improvement, including representation and the ability to reason over hidden states. This benchmark provides researchers with a precise and quantifiable setting to evaluate advances in planning and reasoning in multi-agent systems with partial observability.