# The Consensus Game: Language Model Generation via Equilibrium Search

arXiv: https://arxiv.org/abs/2310.09139 · HTML: https://mlsanity.com/paper/2310.09139/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 52 references that are in the map.
- First posted on arXiv 13 Oct 2023.

## Builds on (most cited first, 30 of 40)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [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)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Reflexion: Language Agents with Verbal Reinforcement Learning](https://mlsanity.com/paper/2303.11366/) (2023)
- [Self-Refine: Iterative Refinement with Self-Feedback](https://mlsanity.com/paper/2303.17651/) (2023)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [The Curious Case of Neural Text Degeneration](https://mlsanity.com/paper/1904.09751/) (2019)
- [Beyond the Imitation Game: Quantifying and extrapolating the capabilities of language models](https://mlsanity.com/paper/2206.04615/) (2022)
- [Improving Factuality and Reasoning in Language Models through Multiagent Debate](https://mlsanity.com/paper/2305.14325/) (2023)
- [LaMDA: Language Models for Dialog Applications](https://mlsanity.com/paper/2201.08239/) (2022)
- [Hierarchical Neural Story Generation](https://mlsanity.com/paper/1805.04833/) (2018)
- [A General Language Assistant as a Laboratory for Alignment](https://mlsanity.com/paper/2112.00861/) (2021)
- [Teaching Large Language Models to Self-Debug](https://mlsanity.com/paper/2304.05128/) (2023)
- [How Can We Know What Language Models Know?](https://mlsanity.com/paper/1911.12543/) (2019)
- [RACE: Large-scale ReAding Comprehension Dataset From Examinations](https://mlsanity.com/paper/1704.04683/) (2017)
- [Contrastive Decoding: Open-ended Text Generation as Optimization](https://mlsanity.com/paper/2210.15097/) (2022)
- [Improving alignment of dialogue agents via targeted human judgements](https://mlsanity.com/paper/2209.14375/) (2022)
- [ReConcile: Round-Table Conference Improves Reasoning via Consensus among Diverse LLMs](https://mlsanity.com/paper/2309.13007/) (2023)
- [Inverse Scaling: When Bigger Isn't Better](https://mlsanity.com/paper/2306.09479/) (2023)
- [The Capacity for Moral Self-Correction in Large Language Models](https://mlsanity.com/paper/2302.07459/) (2023)
- [Mastering the Game of Stratego with Model-Free Multiagent Reinforcement Learning](https://mlsanity.com/paper/2206.15378/) (2022)
- [Locally Typical Sampling](https://mlsanity.com/paper/2202.00666/) (2022)
- [Language Model Cascades](https://mlsanity.com/paper/2207.10342/) (2022)

## Built on by (most cited first, 30 of 36)

- [Nash Learning from Human Feedback](https://mlsanity.com/paper/2312.00886/) (2023)
- [PrivacyLens: Evaluating Privacy Norm Awareness of Language Models in Action](https://mlsanity.com/paper/2409.00138/) (2024)
- [Towards Guaranteed Safe AI: A Framework for Ensuring Robust and Reliable AI Systems](https://mlsanity.com/paper/2405.06624/) (2024)
- [Multiagent Finetuning: Self Improvement with Diverse Reasoning Chains](https://mlsanity.com/paper/2501.05707/) (2025)
- [Learning How Hard to Think: Input-Adaptive Allocation of LM Computation](https://mlsanity.com/paper/2410.04707/) (2024)
- [Internal Consistency and Self-Feedback in Large Language Models: A Survey](https://mlsanity.com/paper/2407.14507/) (2024)
- [A Language Model's Guide Through Latent Space](https://mlsanity.com/paper/2402.14433/) (2024)
- [Privacy in Action: Towards Realistic Privacy Mitigation and Evaluation for LLM-Powered Agents](https://mlsanity.com/paper/2509.17488/) (2025)
- [Amulet: ReAlignment During Test Time for Personalized Preference Adaptation of LLMs](https://mlsanity.com/paper/2502.19148/) (2025)
- [Steering Language Models with Game-Theoretic Solvers](https://mlsanity.com/paper/2402.01704/) (2024)
- [A Survey on LLM Test-Time Compute via Search: Tasks, LLM Profiling, Search Algorithms, and Relevant Frameworks](https://mlsanity.com/paper/2501.10069/) (2025)
- [Game Theory Meets Large Language Models: A Systematic Survey with Taxonomy and New Frontiers](https://mlsanity.com/paper/2502.09053/) (2025)
- [Instance-Dependent Regret Bounds for Learning Two-Player Zero-Sum Games with Bandit Feedback](https://mlsanity.com/paper/2502.17625/) (2025)
- [Incentivizing Truthful Language Models via Peer Elicitation Games](https://mlsanity.com/paper/2505.13636/) (2025)
- [Position: Towards a Responsible LLM-empowered Multi-Agent Systems](https://mlsanity.com/paper/2502.01714/) (2025)
- [Decoding Game: On Minimax Optimality of Heuristic Text Generation Strategies](https://mlsanity.com/paper/2410.03968/) (2024)
- [Self-Consistency Falls Short! The Adverse Effects of Positional Bias on Long-Context Problems](https://mlsanity.com/paper/2411.01101/) (2024)
- [Stop Reducing Responsibility in LLM-Powered Multi-Agent Systems to Local Alignment](https://mlsanity.com/paper/2510.14008/) (2025)
- [Engagement-Driven Content Generation with Large Language Models](https://mlsanity.com/paper/2411.13187/) (2024)
- [Multiplayer Federated Learning: Reaching Equilibrium with Less Communication](https://mlsanity.com/paper/2501.08263/) (2025)
- [Regularized Conventions: Equilibrium Computation as a Model of Pragmatic Reasoning](https://mlsanity.com/paper/2311.09712/) (2023)
- [Truth or Deceit? A Bayesian Decoding Game Enhances Consistency and Reliability](https://mlsanity.com/paper/2410.01064/) (2024)
- [Utility-inspired Reward Transformations Improve Reinforcement Learning Training of Language Models](https://mlsanity.com/paper/2501.06248/) (2025)
- [On Separation Between Best-Iterate, Random-Iterate, and Last-Iterate Convergence of Learning in Games](https://mlsanity.com/paper/2503.02825/) (2025)
- [A Game-Theoretic Negotiation Framework for Cross-Cultural Consensus in LLMs](https://mlsanity.com/paper/2506.13245/) (2025)
- [Reranking-based Generation for Unbiased Perspective Summarization](https://mlsanity.com/paper/2506.15925/) (2025)
- [ALIGN: Aligned Delegation with Performance Guarantees for Multi-Agent LLM Reasoning](https://mlsanity.com/paper/2602.00127/) (2026)
- [LinguaGame: A Linguistically Grounded Game-Theoretic Paradigm for Multi-Agent Dialogue Generation](https://mlsanity.com/paper/2601.04516/) (2026)
- [Power and Limitations of Aggregation in Compound AI Systems](https://mlsanity.com/paper/2602.21556/) (2026)
- [HCAG: Hierarchical Abstraction and Retrieval-Augmented Generation on Theoretical Repositories with LLMs](https://mlsanity.com/paper/2603.20299/) (2026)

## Abstract (arXiv)

When applied to question answering and other text generation tasks, language models (LMs) may be queried generatively (by sampling answers from their output distribution) or discriminatively (by using them to score or rank a set of candidate outputs). These procedures sometimes yield very different predictions. How do we reconcile mutually incompatible scoring procedures to obtain coherent LM predictions? We introduce a new, a training-free, game-theoretic procedure for language model decoding. Our approach casts language model decoding as a regularized imperfect-information sequential signaling game - which we term the CONSENSUS GAME - in which a GENERATOR seeks to communicate an abstract correctness parameter using natural language sentences to a DISCRIMINATOR. We develop computational procedures for finding approximate equilibria of this game, resulting in a decoding algorithm we call EQUILIBRIUM-RANKING. Applied to a large number of tasks (including reading comprehension, commonsense reasoning, mathematical problem-solving, and dialog), EQUILIBRIUM-RANKING consistently, and sometimes substantially, improves performance over existing LM decoding procedures - on multiple benchmarks, we observe that applying EQUILIBRIUM-RANKING to LLaMA-7B outperforms the much larger LLaMA-65B and PaLM-540B models. These results highlight the promise of game-theoretic tools for addressing fundamental challenges of truthfulness and consistency in LMs.