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.

Behavioural Signatures of Risk-Sensitive Decision-Making in Large Language Models

As large language models (LLMs) are increasingly used in decision support, it is important to understand whether their choices under uncertainty exhibit stable and interpretable behavioural regularities. Human decision-making combines relatively persistent risk preferences with context-dependent adjustment, yet it remains unclear whether analogous behavioural structure can be observed in LLM-based decision systems. Here we examine this question using a controlled multi-model framework based on no-limit Texas Hold'em, where behaviour is quantified by Participation, measuring voluntary engagement in uncertain opportunities, and Proactiveness, measuring pre-flop risk escalation. Across homogeneous self-play and heterogeneous mixed-model interactions, frontier LLMs exhibit stable, model-specific risk profiles, forming a spectrum from conservative to aggressive decision styles. These profiles remain largely robust under changing opponent composition, while the most conservative and most aggressive models diverge further in mixed settings. Under global risk pressure and personal resource constraint, models adapt in structured but heterogeneous ways, ranging from broad behavioural contraction to selective de-escalation and near-invariant behaviour. These findings suggest that LLMs differ not only in baseline risk disposition, but also in the risk signals they respond to and the flexibility with which they adjust, providing a behavioural basis for auditing risk-sensitive decision-making in interactive settings. Our code is publicly available at: https://github.com/XuankunRong/AgentTexasPoker.