Back to Blackwell: Closing the Loop on Intransitivity in Multi-Objective Preference Fine-Tuning

A recurring challenge in preference fine-tuning (PFT) is handling $\textit{intransitive}$ (i.e., cyclic) preferences. Intransitive preferences often stem from either $\textit{(i)}$ inconsistent rankings along a single objective or $\textit{(ii)}$ scalarizing multiple objectives into a single metric. Regardless of their source, the downstream implication of intransitive preferences is the same: there is no well-defined optimal policy, breaking a core assumption of the standard PFT pipeline. In response, we propose a novel, game-theoretic solution concept -- the $\textit{Maximum Entropy Blackwell Winner}$ ($\textit{MaxEntBW}$) -- that is well-defined under multi-objective intransitive preferences. To enable computing MaxEntBWs at scale, we derive $\texttt{PROSPER}$: a provably efficient PFT algorithm. Unlike prior self-play techniques, $\texttt{PROSPER}$ directly handles multiple objectives without requiring scalarization. We then apply $\texttt{PROSPER}$ to the problem of fine-tuning large language models (LLMs) from multi-objective LLM-as-a-Judge feedback (e.g., rubric-based judges), a setting where both sources of intransitivity arise. We find that $\texttt{PROSPER}$ outperforms all baselines considered across both instruction following and general chat benchmarks, releasing trained model checkpoints at the 7B and 3B parameter scales.