Scaling Inference-Time Computation via Opponent Simulation: Enabling Online Strategic Adaptation in Repeated Negotiation

A strong LLM strategic agent should reason prospectively over uncertain futures, adapt its strategy to opponents' behavioral tendencies, and continuously recalibrate its decision process from interaction experience. However, incorporating these sources in free-form reasoning could lead to unsupported strategic assumptions, inconsistent opponent estimates, and harmful interference from irrelevant historical interactions. To address these issues, we propose SAGE, a training-free inference-time framework that structures LLM strategic reasoning around three coordinated operations: anchor, adapt, and recalibrate. SAGE first anchors reasoning to an equilibrium policy that provides a strategically valid prior. It then conditions deviations from this anchor on a soft belief over opponent behavioral tendencies, enabling opponent-specific exploitation. Finally, SAGE distills strategically related interactions into counterfactual hypotheses about previously missing considerations, allowing past experience to recalibrate the model's reasoning. We evaluate SAGE on three repeated imperfect-information games: Leduc Hold'em, Liar's Dice, and Goofspiel, against various opponent types in each game. Compared with reasoning-intensive LLM agents, including Suspicion-Agent, ReTA, Agent-Pro, EMO, and Hypothetical Minds, SAGE achieves up to a 127.6% payoff improvement in Liar's Dice while reducing input and output token usage by up to 80% and 90%, respectively. In direct match-up play, it attains non-negative mean payoff against 5/10, 8/10, and 8/10 evaluated opponents in Leduc Hold'em, Liar's Dice, and Goofspiel, respectively, while using relatively fewer tokens. Code is available at https://github.com/chenzhwsysu57/SAGE.

Large language models (LLMs) are increasingly tasked with strategic decision-making under incomplete information, such as in negotiation and policymaking. While LLMs can excel at many such tasks, they also fail in ways that are poorly understood. We shed light on these failures by uncovering two fundamental gaps in the internal mechanisms underlying the decision-making of LLMs in incomplete-information games, supported by experiments with open-weight models Llama 3.1, Qwen3, and gpt-oss. First, an observation-belief gap: LLMs encode internal beliefs about latent game states that are substantially more accurate than their own verbal reports, yet these beliefs are brittle. In particular, the belief accuracy degrades with multi-hop reasoning, exhibits primacy and recency biases, and drifts away from Bayesian coherence over extended interactions. Second, a belief-action gap: The implicit conversion of internal beliefs into actions is weaker than that of the beliefs externalized in the prompt, yet neither belief-conditioning consistently achieves higher game payoffs. These results show how analyzing LLMs' internal processes can expose systematic vulnerabilities that warrant caution before deploying LLMs in strategic domains without robust guardrails.

Game-theoretic benchmarks claim that large language model (LLM) strategic decision-making varies across game types, interaction structures, and capability dimensions. However, they only evaluate strategic performance across individual fixed models. In parallel, contemporary LLM routers pick models under quality-cost trade-offs, but with no notion of a strategic structure. They do not account for an opponent that might be unknown, and that can adapt over time and shift the feedback signal. To address this, we extend the game-theoretic evaluations and define strategic selection as the meta game layered on top of the base game. In each round, a strategic selector fields an LLM from a portfolio of models to navigate a game payoff matrix that is latent and learned online. We find that the evaluation signal is game-specific, and that extracting a reliable signal requires a much higher number of seed runs $N$ compared to traditional benchmarking setups. Evaluating across 5 games and 10 LLMs, we show a model's strategic competence varies significantly depending on the game setup and the opponent faced. Building on this, we implement a suite of selector methods and find that approaches that rely on query embeddings or LLM self-reflection do not consistently capture the underlying strategic signal. A lightweight multi-armed bandit algorithm performs more reliably across games, though it still falls short of the best fixed model. Finally, we show that simple non-stationary and adversarial bandit variants fail to outperform a stationary bandit baseline, pointing to a gap in how current methods handle a shifting opponent in strategic settings.