Similar Models, Broken Odds: Where Luce's Axiom Fails in Chatbot Arena

We propose a method for evaluating the robustness of widely used LLM ranking systems -- variants of a Bradley--Terry model -- to dropping a worst-case very small fraction of preference data. Our approach is computationally fast and easy to adopt. When we apply our method to matchups from popular LLM ranking platforms, including Chatbot Arena and derivatives, we find that the rankings of top-performing models can be remarkably sensitive to the removal of a small fraction of preferences; for instance, dropping just 0.003% of human preferences can change the top-ranked model on Chatbot Arena. Our robustness check identifies the specific preferences most responsible for such ranking flips, allowing for inspection of these influential preferences. We observe that the rankings derived from MT-bench preferences are notably more robust than those from Chatbot Arena, likely due to MT-bench's use of expert annotators and carefully constructed prompts. Finally, we find that neither rankings based on crowdsourced human evaluations nor those based on LLM-as-a-judge preferences are systematically more sensitive than the other.

Large language models are often used as judges to score candidate responses, then validated with a single global metric such as correlation with reference labels. This can be misleading when the real deployment task is best-of-n selection within a prompt. In a 5,000-prompt best-of-4 benchmark from Chatbot Arena, a judge with moderate global correlation (r = 0.47) captures only 21.0% of the improvement that perfect selection would achieve over random choice. The gap arises because global agreement is driven largely by prompt-level baseline effects, while selection depends on within-prompt ranking: within-prompt correlation is only r_within = 0.27, and coarse pointwise scoring creates ties in 67% of pairwise comparisons. In a matched-pair best-of-2 audit, explicit pairwise judging recovers much of this lost signal, raising recovery from 21.1% to 61.2%. For judge-based selection, the relevant audit should report within-prompt signal, tie rates, and recovery/top-1 accuracy, not global agreement alone.

With the rise in capabilities of large language models (LLMs) and their deployment in real-world tasks, evaluating LLM alignment with human preferences has become an important challenge. Current benchmarks average preferences across all users to compute aggregate ratings, overlooking individual user preferences when establishing model rankings. Since users have varying preferences in different contexts, we call for personalized LLM benchmarks that rank models according to individual needs. We compute personalized model rankings using ELO ratings and Bradley-Terry coefficients for 115 active Chatbot Arena users and analyze how user query characteristics (topics and writing style) relate to LLM ranking variations. We demonstrate that individual rankings of LLM models diverge dramatically from aggregate LLM rankings, with Bradley-Terry correlations averaging only $ρ= 0.04$ (57% of users show near-zero or negative correlation) and ELO ratings showing moderate correlation ($ρ= 0.43$). Through topic modeling and style analysis, we find users exhibit substantial heterogeneity in topical interests and communication styles, influencing their model preferences. We further show that a compact combination of topic and style features provides a useful feature space for predicting user-specific model rankings. Our results provide strong quantitative evidence that aggregate benchmarks fail to capture individual preferences for most users, and highlight the importance of developing personalized benchmarks that rank LLM models according to individual user preferences.