# Cross-Entropy Games for Language Models: From Implicit Knowledge to General Capability Measures

arXiv: https://arxiv.org/abs/2506.06832 · HTML: https://mlsanity.com/paper/2506.06832/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 50 references that are in the map.
- First posted on arXiv 7 Jun 2025.

## Builds on (most cited first, 28 of 28)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [A Survey on Evaluation of Large Language Models](https://mlsanity.com/paper/2307.03109/) (2023)
- [Chatbot Arena: An Open Platform for Evaluating LLMs by Human Preference](https://mlsanity.com/paper/2403.04132/) (2024)
- [Contrastive Decoding: Open-ended Text Generation as Optimization](https://mlsanity.com/paper/2210.15097/) (2022)
- [Darwin Godel Machine: Open-Ended Evolution of Self-Improving Agents](https://mlsanity.com/paper/2505.22954/) (2025)
- [TextWorld: A Learning Environment for Text-based Games](https://mlsanity.com/paper/1806.11532/) (2018)
- [ARC-AGI-2: A New Challenge for Frontier AI Reasoning Systems](https://mlsanity.com/paper/2505.11831/) (2025)
- [Security and Privacy Challenges of Large Language Models: A Survey](https://mlsanity.com/paper/2402.00888/) (2024)
- [Actor-Mimic: Deep Multitask and Transfer Reinforcement Learning](https://mlsanity.com/paper/1511.06342/) (2015)
- [REASONING GYM: Reasoning Environments for Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2505.24760/) (2025)
- [AgentGym: Evolving Large Language Model-based Agents across Diverse Environments](https://mlsanity.com/paper/2406.04151/) (2024)
- [If beam search is the answer, what was the question?](https://mlsanity.com/paper/2010.02650/) (2020)
- [The Good, The Bad, and The Greedy: Evaluation of LLMs Should Not Ignore Non-Determinism](https://mlsanity.com/paper/2407.10457/) (2024)
- [On NMT Search Errors and Model Errors: Cat Got Your Tongue?](https://mlsanity.com/paper/1908.10090/) (2019)
- [A Thorough Examination of Decoding Methods in the Era of LLMs](https://mlsanity.com/paper/2402.06925/) (2024)
- [A Tutorial on Meta-Reinforcement Learning](https://mlsanity.com/paper/2301.08028/) (2023)
- [AI-GAs: AI-generating algorithms, an alternate paradigm for producing general artificial intelligence](https://mlsanity.com/paper/1905.10985/) (2019)
- [Open-Endedness is Essential for Artificial Superhuman Intelligence](https://mlsanity.com/paper/2406.04268/) (2024)
- [A Comprehensive Survey on Data Augmentation](https://mlsanity.com/paper/2405.09591/) (2024)
- [A Survey of Useful LLM Evaluation](https://mlsanity.com/paper/2406.00936/) (2024)
- [Arrows of Time for Large Language Models](https://mlsanity.com/paper/2401.17505/) (2024)
- [A Definition of Open-Ended Learning Problems for Goal-Conditioned Agents](https://mlsanity.com/paper/2311.00344/) (2023)
- [Compositional game theory](https://mlsanity.com/paper/1603.04641/) (2016)
- [When does a predictor know its own loss?](https://mlsanity.com/paper/2502.20375/) (2025)
- [Smart Proofs via Smart Contracts: Succinct and Informative Mathematical Derivations via Decentralized Markets](https://mlsanity.com/paper/2102.03044/) (2021)
- [Meta-Diversity Search in Complex Systems, A Recipe for Artificial Open-Endedness ?](https://mlsanity.com/paper/2312.00455/) (2023)

## Built on by (most cited first, 2 of 2)

- [Cognitive Training for Language Models: Towards General Capabilities via Cross-Entropy Games](https://mlsanity.com/paper/2603.22479/) (2026)
- [Cross-Entropy Games and Frost Training](https://mlsanity.com/paper/2605.27701/) (2026)

## Abstract (arXiv)

Large Language Models (LLMs) define probability measures on text. By considering the implicit knowledge question of what it means for an LLM to know such a measure and what it entails algorithmically, we are naturally led to formulate a series of tasks that go beyond generative sampling, involving forms of summarization, counterfactual thinking, anomaly detection, originality search, reverse prompting, debating, creative solving, etc. These tasks can be formulated as games based on LLM measures, which we call Cross-Entropy (Xent) Games. Xent Games can be single-player or multi-player. They involve cross-entropy scores and cross-entropy constraints, and can be expressed as simple computational graphs and programs. We show the Xent Game space is large enough to contain a wealth of interesting examples, while being constructible from basic game-theoretic consistency axioms. We then discuss how the Xent Game space can be used to measure the abilities of LLMs. This leads to the construction of Xent Game measures: finite families of Xent Games that can be used as capability benchmarks, built from a given scope, by extracting a covering measure. To address the unbounded scope problem associated with the challenge of measuring general abilities, we propose to explore the space of Xent Games in a coherent fashion, using ideas inspired by evolutionary dynamics.