# SAGE: Structured Strategic Reasoning for Efficient LLM Game Playing

arXiv: https://arxiv.org/abs/2609.34342 · HTML: https://mlsanity.com/paper/2609.34342/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reasoning for agents, planning and code, one of 17 directions in LLM reasoning.
- Builds on 11 of its 48 references that are in the map.
- First posted on arXiv 28 Sep 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, 11 of 11)

- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Theory of Mind for Multi-Agent Collaboration via Large Language Models](https://mlsanity.com/paper/2310.10701/) (2023)
- [Agent-Pro: Learning to Evolve via Policy-Level Reflection and Optimization](https://mlsanity.com/paper/2402.17574/) (2024)
- [Suspicion-Agent: Playing Imperfect Information Games with Theory of Mind Aware GPT-4](https://mlsanity.com/paper/2309.17277/) (2023)
- [Hypothetical Minds: Scaffolding Theory of Mind for Multi-Agent Tasks with Large Language Models](https://mlsanity.com/paper/2407.07086/) (2024)
- [Mastering Board Games by External and Internal Planning with Language Models](https://mlsanity.com/paper/2412.12119/) (2024)
- [Strategist: Self-improvement of LLM Decision Making via Bi-Level Tree Search](https://mlsanity.com/paper/2408.10635/) (2024)
- [Can Large Language Models Master Complex Card Games?](https://mlsanity.com/paper/2509.01328/) (2025)
- [Scaling Inference-Time Computation via Opponent Simulation: Enabling Online Strategic Adaptation in Repeated Negotiation](https://mlsanity.com/paper/2602.19309/) (2026)
- [StratFormer: Adaptive Opponent Modeling and Exploitation in Imperfect-Information Games](https://mlsanity.com/paper/2604.25796/) (2026)
- [PolicyEvol-Agent: Evolving Policy via Environment Perception and Self-Awareness with Theory of Mind](https://mlsanity.com/paper/2504.15313/) (2025)

## Abstract (arXiv)

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.