Statistical Impossibility and Possibility of Aligning LLMs with Human Preferences: From Condorcet Paradox to Nash Equilibrium

Kaizhao Liu, Qi Long, Zhekun Shi, Weijie J. Su and Jiancong Xiao

Papers from arXiv.org

Abstract: Aligning large language models (LLMs) with diverse human preferences is critical for ensuring fairness and informed outcomes when deploying these models for decision-making. In this paper, we seek to uncover fundamental statistical limits concerning aligning LLMs with human preferences, with a focus on the probabilistic representation of human preferences and the preservation of diverse preferences in aligned LLMs. We first show that human preferences can be represented by a reward model if and only if the preference among LLM-generated responses is free of any Condorcet cycle. Moreover, we prove that Condorcet cycles exist with probability converging to one exponentially fast under a probabilistic preference model, thereby demonstrating the impossibility of fully aligning human preferences using reward-based approaches such as reinforcement learning from human feedback. Next, we explore the conditions under which LLMs would employ mixed strategies -- meaning they do not collapse to a single response -- when aligned in the limit using a non-reward-based approach, such as Nash learning from human feedback (NLHF). We identify a necessary and sufficient condition for mixed strategies: the absence of a response that is preferred over all others by a majority. As a blessing, we prove that this condition holds with high probability under the probabilistic preference model, thereby highlighting the statistical possibility of preserving minority preferences without explicit regularization in aligning LLMs. Finally, we leverage insights from our statistical results to design a novel, computationally efficient algorithm for finding Nash equilibria in aligning LLMs with NLHF. Our experiments show that Llama-3.2-1B, aligned with our algorithm, achieves a win rate of 60.55\% against the base model.

Date: 2025-03
References: Add references at CitEc
Citations:

Downloads: (external link)
http://arxiv.org/pdf/2503.10990 Latest version (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:2503.10990

Access Statistics for this paper

More papers in Papers from arXiv.org
Bibliographic data for series maintained by arXiv administrators ().