New insights into Elo algorithm for practitioners and statisticians

Evaluating new large language models typically requires costly human annotation campaigns at scale. LLM-as-a-judge offers a cheaper alternative, but judge scores carry systematic errors - such as position bias, self-preference, or intransitivity - that can strongly miscalibrate the resulting rankings. We quantify the resulting judge-human disagreement at two complementary levels. At the local level, we estimate per-battle uncertainty from the judge's own score differences by propagating calibrated win probabilities rather than hard labels into the Bradley-Terry procedure. This alone provides a drastic improvement to Elo estimation accuracy, bringing LLM-derived ratings within 17.9 Elo MAE of human-derived ones when averaged over 55 held-out models on LMArena. At the global level, we apply split conformal prediction to the residual gap between LLM-derived and human-derived Elo ratings across held-out models, producing prediction intervals with distribution-free marginal coverage guarantees that account for irreducible LLM-human disagreement. Together, these two layers yield a low-cost evaluation tool that provides developers with calibrated Elo estimates and honest uncertainty bounds, without access to large-scale human annotations. To facilitate reproducibility, we release our code at https://github.com/kargibora/SoftElo .

Leaderboards rank models by their average scores on benchmark items, and they are consulted repeatedly while the evaluation is still running. Existing confidence intervals for a model's rank control their error rate only if they are computed once, after a number of items chosen in advance. If they are recomputed as results arrive, and the evaluation stops once they look decisive, their error rate exceeds its nominal level. Anytime-valid methods keep their guarantees at all sample sizes simultaneously and hence under any stopping rule. They exist for the accuracy of one model, for one pair of models and for the set of models that may be best. For pairwise battles they also give ranks. None gives ranks when all models are scored on the same items, which makes their scores dependent. We construct rank confidence sequences: for every model, a set of ranks that contains its true rank, simultaneously for all models and at all times, at a chosen error level $α$, in finite samples. The construction combines betting e-processes, one for each ordered pair of models, with closed testing over the possible orderings of the models. It allows any dependence between the models' scores on an item. The method has two advantages. A leaderboard can be inspected after every item without inflating its error rate. The evaluation of each model can stop as soon as the question asked about it is answered, which saves compute. When results are examined only once, halfway through or later, little power is lost relative to fixed-sample methods. The paper quantifies these advantages in simulations and on public leaderboard data.

Rating systems such as Elo serve as the gold standard for matchmaking in competitive chess. However, they inherently suffer from response lag due to their exclusive reliance on match outcomes, neglecting the granular quality of gameplay. Nevertheless, incorporating move-by-move information into rating adjustments presents a significant challenge given the substantial noise and the vastness of the game-state space. To address this, we propose the Drift-Diffusion-Enhanced Elo Rating System (DD-Elo), a novel skill assessment framework inspired by the drift diffusion model (DDM) from cognitive neuroscience. By modeling skill expression as a decision-making process, our model integrates move-level data to capture rapid skill fluctuations. We provide a rigorous mathematical derivation proving that DD-Elo maintains a bounded deviation from the traditional Elo system, ensuring theoretical alignment. Extensive experiments demonstrate that DD-Elo adapts to skill changes faster than Elo. Our findings suggest that DD-Elo offers an explainable, highly responsive, and backward-compatible solution for chess rating ecosystems. The implementation code is publicly available at https://github.com/Aquila-zhou1/DD-Elo .