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

Abstract

Chatbot Arena ranks language models with the Bradley-Terry model, which gives each model one score and so assumes Luce's choice axiom. Choice theory predicts where this fails: near-copies are told apart more decisively than their scores imply. On 1.8 million Arena votes, votes between versions of one model are 1.37 times as decisive as Bradley-Terry predicts, between sizes of one generation 1.26 times and between models of one lab 1.12 times; a release a year later gives consistent multipliers, and the effect survives alternative tie treatments, prompt subsets and topics, and timing checks. How alike two models'answers are worded and formatted tracks the effect. A correction with four to six parameters predicts held-out votes better than Bradley-Terry and than models with hundreds of parameters, mostly by modelling ties; frozen and carried to new releases in a test fixed in advance, its gains are concentrated in ties, and better head-to-head odds are not established. The leaderboard order barely moves. With language-model judges and three-answer menus, a near-copy takes votes mostly from its twin when choices are pooled over prompts (log-odds shift +0.10, 95% interval 0.05 to 0.16); within a prompt the shift is smaller and clear for two of four judges.