ICM Out! Better Tournament Strategy from Computed Continuations, vs. Solvers and LLMs

The Independent Chip Model (ICM) converts tournament chips into reference prize equity, and policies are routinely constructed against those values. Because ICM reads only stack sizes, it omits action order, blind obligations, and seat rotation, and it does not price the elimination pressure a big stack puts on the short stacks it can bust. Those omissions can alter the successor-state contrasts that determine a move. We introduce Strategic-Continuation Optimization (SCO), a policy-construction method that enumerates current-hand outcomes, maps them to successor states, prices those states with continuation values computed from the finite tournament model, and optimizes and freezes the resulting current-hand policy. The fixed-ICM comparison policy changes one thing only: the same optimizer solves the same game with successor states priced by analytic ICM, so the two policies differ only through that pricing. We evaluate the resulting policies in a three-player jam/fold tournament with a \$1M prize pool. Relative to the frozen strategic-continuation benchmark, analytic ICM has \$9{,}036 mean absolute value error across all 2,838 state--seat entries. That value error rewrites the ranges it prices: measured against each decision point's own fixed-ICM jam range, SCO moves the jam frequency by an average of 14.08\%. To price those different moves, we compare all 946 states and three policy owners while changing only the focal policy and holding both opponents and the continuation evaluator fixed. The policy produced by SCO earns \$214.33 more prize equity per hand on average and is favored in 2,433 of 2,838 matched units. The ordering survives replacing the solver-built opponent with two LLMs and with a family of non-modeling threshold players. This value-to-policy-to-cost chain shows directly when ICM becomes an inadequate objective for tournament strategy construction.

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.

Towards Learning Representations of Policies in Two-Player Zero-Sum Imperfect-Information Games

We investigate the problem of learning useful policy representations (embeddings) in two-player zero-sum imperfect-information games. We make three contributions: First, we introduce methods of creating datasets of policies for a given game. Second, we propose methods to learn policy representations. Third, we introduce downstream tasks to evaluate the effectiveness of such representations. We evaluate each dataset method, embedding method, and downstream task on Kuhn and Leduc Poker. Although our methods are very basic, we demonstrate that useful behavioral representations are present in the learned embeddings. To our knowledge, this work is among the first to systematically compare self-supervised learning techniques for learning policy representations in games. Our code is available at https://github.com/VitamintK/ssl-project for others to extend.

Beyond Tracking or Shortcut: Composition-Bounded Predictive States in Poker Autoregressive Models

Hidden-state probes often recover latent labels in imperfect-information sequence models, but this alone does not establish that a model maintains a posterior belief distribution over hidden states. This paper studies this ambiguity in a no-range Limit Hold'em autoregressive model trained only on action and value targets, not on an opponent's hand or range. Opponent-range probes are positive after action/value controls in two of three seeds, and the behavior head predicts held-out actions about five percentage points above a baseline using only observable public history. However, visible public betting composition explains more opponent-range signal than residual hidden states, suggesting that most recoverable information comes from betting summaries. Action/value+composition baselines reach 16.5-16.7% top-10 accuracy while composition-residual hidden probes fall to 11.4-12.2%, and matched-composition comparisons are negative in every seed. We call this evidence pattern composition-bounded predictive support: hidden states remain behavior-predictive and opponent-range correlated, but most recoverable range information is explained by visible betting composition rather than residual hidden-state structure. This is a case-study claim about opponent-range representational evidence, not exact Bayesian posterior tracking or a causal belief mechanism. Synthetic control and oracle validations show that the same diagnostics accept posterior-sensitive states and reject raw composition states under matched controls. Thus positive belief probes should be interpreted through targeted alternatives before being treated as evidence of belief tracking.