# Equilibrium Residuals Expose Three Regimes of Matrix-Game Strategic Reasoning in Language Models

arXiv: https://arxiv.org/abs/2605.10410 · HTML: https://mlsanity.com/paper/2605.10410/

## 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 20 of its 26 references that are in the map.
- First posted on arXiv 11 May 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, 20 of 20)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Show Your Work: Scratchpads for Intermediate Computation with Language Models](https://mlsanity.com/paper/2112.00114/) (2021)
- [What Can Transformers Learn In-Context? A Case Study of Simple Function Classes](https://mlsanity.com/paper/2208.01066/) (2022)
- [Self-Rewarding Language Models](https://mlsanity.com/paper/2401.10020/) (2024)
- [Self-Play Fine-Tuning Converts Weak Language Models to Strong Language Models](https://mlsanity.com/paper/2401.01335/) (2024)
- [CodeRL: Mastering Code Generation through Pretrained Models and Deep Reinforcement Learning](https://mlsanity.com/paper/2207.01780/) (2022)
- [A Unified Game-Theoretic Approach to Multiagent Reinforcement Learning](https://mlsanity.com/paper/1711.00832/) (2017)
- [Self-Play Preference Optimization for Language Model Alignment](https://mlsanity.com/paper/2405.00675/) (2024)
- [Nash Learning from Human Feedback](https://mlsanity.com/paper/2312.00886/) (2023)
- [Exploring Length Generalization in Large Language Models](https://mlsanity.com/paper/2207.04901/) (2022)
- [Can Large Language Models Serve as Rational Players in Game Theory? A Systematic Analysis](https://mlsanity.com/paper/2312.05488/) (2023)
- [Strategic Reasoning with Language Models](https://mlsanity.com/paper/2305.19165/) (2023)
- [TMGBench: A Systematic Game Benchmark for Evaluating Strategic Reasoning Abilities of LLMs](https://mlsanity.com/paper/2410.10479/) (2024)
- [MARSHAL: Incentivizing Multi-Agent Reasoning via Self-Play with Strategic LLMs](https://mlsanity.com/paper/2510.15414/) (2025)
- [Large Language Models Playing Mixed Strategy Nash Equilibrium Games](https://mlsanity.com/paper/2406.10574/) (2024)

## Abstract (arXiv)

Large language models can score well on named game-theory benchmarks while failing on the same strategic computation once semantic cues are removed. We show this gap with procedurally generated zero-sum matrix games: a model that recognizes familiar games drops to 34%, 18%, and 2% success on anonymous 2×2, 3×3, and 5×5 payoff matrices. The benchmark separates semantic recall, learned approximate Nash computation, and an output-interface bottleneck that limits scale. Training only on 2×2 and 3×3 games, supervised fine-tuning raises unseen 5×5--7×7 success from 2% to 61%, while exploitability-reward training averages 37% with high seed variance. We prove that the exploitability residual is 2-Lipschitz in payoff perturbations, unlike discontinuous vertex-returning LP equilibrium selectors, explaining why residual training can transfer under payoff shifts even when formatting instability limits mean performance. A dominated-action padding experiment provides causal evidence: trained models solve 3×3 games embedded in much larger matrices, while random-padded controls fail and dense 12×12 games remain near failure. Procedural evaluation is therefore necessary for measuring strategic reasoning, and residual rewards expose a real but format-limited route to approximate equilibrium computation.