Fundamental Limits of Game-Theoretic LLM Alignment: Smith Consistency and Preference Matching
Zhekun Shi, Kaizhao Liu, Qi Long, Weijie J. Su, Jiancong Xiao
TL;DR
This work analyzes game-theoretic LLM alignment through a generalized payoff that applies a mapping $\Psi$ to pairwise human preferences. It derives necessary and sufficient conditions on $\Psi$ for Condorcet consistency and Smith consistency, and shows that Smith-consistent methods preserve diversity via mixed Nash strategies when no Condorcet winner exists. The authors also prove an impossibility result: exact preference matching cannot be guaranteed by any smooth payoff mapping under general assumptions, highlighting fundamental limits of game-theoretic alignment. The results provide a robustness theory for NLHF-like approaches and guide design choices for payoff mappings beyond raw preferences, including a natural RLHF-generalizing payoff that retains the desirable properties.
Abstract
Nash Learning from Human Feedback is a game-theoretic framework for aligning large language models (LLMs) with human preferences by modeling learning as a two-player zero-sum game. However, using raw preference as the payoff in the game highly limits the potential of the game-theoretic LLM alignment framework. In this paper, we systematically study using what choices of payoff based on the pairwise human preferences can yield desirable alignment properties. We establish necessary and sufficient conditions for Condorcet consistency, diversity through mixed strategies, and Smith consistency. These results provide a theoretical foundation for the robustness of game-theoretic LLM alignment. Further, we show the impossibility of preference matching -- i.e., no smooth and learnable mappings of pairwise preferences can guarantee a unique Nash equilibrium that matches a target policy, even under standard assumptions like the Bradley-Terry-Luce model. This result highlights the fundamental limitation of game-theoretic LLM alignment.