# DICE: Entropy-Regularized Equilibrium Selection for Stable Multi-Agent LLM Coordination

arXiv: https://arxiv.org/abs/2606.08068 · HTML: https://mlsanity.com/paper/2606.08068/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Social simulation and multi-agent debate, one of 16 directions in LLM agents and tool use.
- Builds on 21 of its 56 references that are in the map.
- First posted on arXiv 6 Jun 2026.

## Placed in

- LLM agents and tool use / [Social simulation and multi-agent debate](https://mlsanity.com/agents/social-simulation-and-multi-agent-debate/)

## Builds on (most cited first, 21 of 21)

- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Improving Factuality and Reasoning in Language Models through Multiagent Debate](https://mlsanity.com/paper/2305.14325/) (2023)
- [Least-to-Most Prompting Enables Complex Reasoning in Large Language Models](https://mlsanity.com/paper/2205.10625/) (2022)
- [Why Do Multi-Agent LLM Systems Fail?](https://mlsanity.com/paper/2503.13657/) (2025)
- [Value-Decomposition Networks For Cooperative Multi-Agent Learning](https://mlsanity.com/paper/1706.05296/) (2017)
- [Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review](https://mlsanity.com/paper/1805.00909/) (2018)
- [Complexity-Based Prompting for Multi-Step Reasoning](https://mlsanity.com/paper/2210.00720/) (2022)
- [DeepInception: Hypnotize Large Language Model to Be Jailbreaker](https://mlsanity.com/paper/2311.03191/) (2023)
- [QPLEX: Duplex Dueling Multi-Agent Q-Learning](https://mlsanity.com/paper/2008.01062/) (2020)
- [ZebraLogic: On the Scaling Limits of LLMs for Logical Reasoning](https://mlsanity.com/paper/2502.01100/) (2025)
- [Towards a Science of Scaling Agent Systems](https://mlsanity.com/paper/2512.08296/) (2025)
- [EvoAgent: Towards Automatic Multi-Agent Generation via Evolutionary Algorithms](https://mlsanity.com/paper/2406.14228/) (2024)
- [GroupDebate: Enhancing the Efficiency of Multi-Agent Debate Using Group Discussion](https://mlsanity.com/paper/2409.14051/) (2024)
- [Contrastive Chain-of-Thought Prompting](https://mlsanity.com/paper/2311.09277/) (2023)
- [AutoLogi: Automated Generation of Logic Puzzles for Evaluating Reasoning Abilities of Large Language Models](https://mlsanity.com/paper/2502.16906/) (2025)
- [Statistical Impossibility and Possibility of Aligning LLMs with Human Preferences: From Condorcet Paradox to Nash Equilibrium](https://mlsanity.com/paper/2503.10990/) (2025)
- [From Passive to Active Reasoning: Can Large Language Models Ask the Right Questions under Incomplete Information?](https://mlsanity.com/paper/2506.08295/) (2025)
- [Fundamental Limits of Game-Theoretic LLM Alignment: Smith Consistency and Preference Matching](https://mlsanity.com/paper/2505.20627/) (2025)
- [AlphaApollo: A System for Deep Agentic Reasoning](https://mlsanity.com/paper/2510.06261/) (2025)
- [SAT: Sequential Agent Tuning for Coordinator Free Plug and Play Multi-LLM Training with Monotonic Improvement Guarantees](https://mlsanity.com/paper/2605.05216/) (2026)
- [TeamTR: Trust-Region Fine-Tuning for Multi-Agent LLM Coordination](https://mlsanity.com/paper/2605.15207/) (2026)

## Abstract (arXiv)

Multi-agent large language model (LLM) systems often fail to reliably outperform a single strong model equipped with best-of-N sampling. We argue that a core source of this instability is ill-posed equilibrium selection: current systems specify what information agents share, but not which coordination convention should be selected. We formalize a broad class of such systems as discounted incomplete-information Markov games and show that two common pathologies, oscillation between competing conventions and drift across them, can both induce unstable learning and linear Bayesian regret. To obtain a well-posed target, we introduce the Heterogeneous Quantal Response Equilibrium (HQRE), an entropy-regularized equilibrium concept with agent- and state-dependent temperatures. Under a monotonicity condition, HQRE is unique, admits linearly convergent mirror updates, and yields bounded Bayesian regret; the same condition yields rollout-measurable stability diagnostics. We instantiate this objective in two algorithms: DICE-PC, which coordinates frozen models through prompt-control actions, and DICE-FT, which performs parameter-efficient mirror fine-tuning. Across eleven benchmarks in four domains, DICE improves accuracy-cost trade-offs over strong within-class baselines; on reasoning and planning tasks, DICE-PC improves by 4.3 percentage points on average and DICE-FT by 8.5 points.