arenakit

Bradley-Terry leaderboards with simultaneous confidence control.

Arena leaderboards turn pairwise votes into a ranked list, and the ranked list is what everyone reads. But a rank is a claim, and the votes behind it support some of those claims and not others. Two models a hair apart may be genuinely tied. Being shown first may be worth more than the gap between neighbours. A board can look decisive because a handful of people voted a great many times. arenakit takes the votes, fits the same Bradley-Terry model the leaderboards use, and reports which parts of the published order the evidence actually carries — with an interval on every number and a flag only when the evidence clears a stated bar.

The audit in one table

Nine synthetic boards, one planted defect each. Every probe in the package has a board here that makes it fire, and the clean board fires nothing.

The two figures are the same thin board seen two ways: on the left of each, six models whose intervals overlap into two tiers; on the right, the gap between each pair of neighbours measured against the evidence needed to call it real — once on its own, and once with the whole board's worth of comparisons taken into account.

make results regenerates the table, the full report and the figures from fixed seeds. CI rebuilds them from pinned dependencies and fails if a committed number differs from what the code produces.

106,134 real arena votes, audited

examples/lmarena/ audits the public LMArena preference sample: 106,134 human votes over 55 models. Committed results: examples/lmarena/results/lmarena.md.

• 1 of 54 neighbouring pairs is separated once the 54 comparisons are controlled for together; 5 are separated judged one at a time. Only the top model is distinguishable from its neighbour. The other 54 models form a single block the votes cannot order — which is not the same as saying the order is wrong, only that these votes do not establish it.
• Closing the median unresolved gap would take roughly 8.9 million votes at this design, against the 106 thousand collected. For most neighbouring pairs the honest reading is that they are tied.
• The median model's 95% rank interval spans 7 ranks, and gemma-2-9b-it-simpo — 468 battles — spans 16.
• Position matters, and it does not matter. The model shown first wins 49.53% of 65,418 decisive battles. The interval on the fitted advantage, [-0.021, -0.001] in log-odds, excludes zero, so at this sample size the effect is real. The whole interval also sits inside the +/-0.05 band this package treats as immaterial, and fitting the advantage out reorders nothing. Detectable and worth acting on are different questions, and a leaderboard needs an answer to the second one.
• Clustering the bootstrap on voters widens standard errors by 1.03x. The busiest 1% of the 49,382 voters cast 16% of all votes, so the concern is well founded, but on this data the usual battle-level bootstrap is close enough. That is a finding, not an assumption — and it is the kind of thing worth checking rather than asserting either way.
• 38.4% of votes are ties, and the ranking survives dropping them or modelling them (Rao-Kupper theta 2.35) instead of splitting them.
• Preference cycles land about where Bradley-Terry expects: 3.5% of the 14,164 decided triads are cyclic against 4.3% predicted. One strength per model is enough to describe who beats whom on this board.

The same six models, judged by people and by GPT-4

examples/mtbench/ audits both halves of MT-Bench's judgments: 3,355 votes from 65 people, and 2,400 verdicts from GPT-4 acting as a judge, on the same models and the same questions. Committed results: examples/mtbench/results/mtbench.md.

• GPT-4 changes its mind on 15.8% of pairs when the two answers swap places. That figure is not estimated here; the split records those cases explicitly, because it was collected by asking both ways round.
• GPT-4 matches the human majority on 71.3% of the 1,612 comparisons where the humans reached one.
• And it produces exactly the human leaderboard. Kendall tau between the two orders is +1.00. Judge noise that looks disqualifying one comparison at a time averages out of a six-model ranking — which is a reason to be careful about what a leaderboard proves, in both directions.
• The judge's verdicts are correlated within question: resampling questions rather than verdicts widens its intervals by 1.48x. The human votes, spread over 65 annotators, show no annotator carrying the board.

Install

pip install arenakit

Python 3.11+. Runtime dependencies are numpy, pydantic and matplotlib. The two case studies additionally need pyarrow to read the published parquet files.

Quickstart

arenakit audit battles.csv

where battles.csv has model_a, model_b, winner, and optionally item_id and voter_id. model_a is the model shown first.

from arenakit import Arena, load_battles, run_audit

audit = run_audit(Arena(load_battles("battles.csv")))

for row in audit.leaderboard[:5]:
    print(row["rank"], row["model"], row["score"], row["rank_ci"])

for probe in audit.triggered:
    print(probe.name, "-", probe.reason)

Fitting on its own, without the audit:

from arenakit import fit_bt, to_elo

fit = fit_bt(arena.counts(), ties="half", order=True)
print(fit.order_log)      # advantage of being shown first, in log-odds
print(to_elo(fit.scores)) # the same scores in the usual leaderboard units

What it checks

Thresholds are named constants in src/arenakit/probes.py; the table above is a description of that file, not a second source of truth. A probe that cannot answer its question says so and skips — for instance, the position probes stand down when the data records which model sorts first rather than which was shown first, which is exactly what MT-Bench's GPT-4 split does.

How the numbers are defended

• Separation is not interval overlap. Two intervals can overlap while the difference between them is significant, so the overlap rule quietly calls real gaps unresolved. arenakit takes the bootstrap distribution of the difference, studentises it, and compares it against a critical value. There is a test that constructs a board where the two rules disagree.
• A leaderboard makes every comparison at once. Ranking 55 models means 54 adjacent tests chosen by the same data that will judge them, so separation reports both a marginal verdict and a simultaneous one using a max-t critical value across the whole board. On the arena data the two answers are 5 and 1.
• The fits are checked against a reference implementation. Bradley-Terry is logistic regression on signed model indicators, and the first-position term is its intercept; both are asserted equal to statsmodels to 1e-5. Two-model boards are checked against the closed form, and Rao-Kupper is checked to collapse onto plain Bradley-Terry when there are no ties.
• Every probe has a board built to make it fire. make results audits nine planted boards and asserts the exact set of flags each one raises, so a probe that becomes over-eager shows up as a clean board raising a flag, and one that goes blind shows up as a planted defect nobody noticed.
• Reruns are identical. One seed, fixed convergence tolerance and iteration cap, integer matrix products where a count has to be exact, and no BLAS in any path that produces a published number.

Design notes

Sufficient statistics, not battle lists. Every model here is fitted from a matrix of ordered pair counts, which is all the likelihood depends on. A bootstrap resample therefore costs three weighted bincount calls plus an iteration on a 55x55 matrix, and two thousand resamples over a hundred thousand battles take seconds rather than an afternoon.

The default resampling is the most conservative the data allows — voters if they are recorded, then prompts, then battles — because treating correlated votes as independent is the commoner mistake and the quieter one. When the chosen column has too few distinct values to resample, the audit drops to a coarser scheme and says so, rather than reporting the zero-width intervals a one-annotator resample would produce.

Disconnected boards are refused, not fudged. Two models joined by no chain of opponents have no estimable difference; the likelihood is flat in exactly that direction, and any number printed for the gap comes from the prior. run_audit keeps the largest connected group, reports what it dropped, and fit_bt raises rather than returning a number nobody should use.

Ties arrive as one outcome on purpose. Real boards ship richer tie labels — "both bad", "both good", "inconsistent" — and collapsing them changes the ranking, so the loader refuses to guess and asks you to map them yourself. Both case studies show the mapping in their source.

A weak prior, so the fit always exists. Each model plays half a virtual battle against an average opponent, which keeps an undefeated model's score finite. With prior=0.0 you get the unpenalised estimate, and a clear error when it does not exist.

Limitations

• Which pairs are adjacent is decided by the point estimate. The simultaneous test controls the family-wise error rate across those 54 comparisons, but the comparisons themselves were chosen after seeing the data. A fully simultaneous treatment of the ranking would be more conservative still, so the separation counts here are, if anything, generous.
• Bradley-Terry assumes one number per model. intransitivity checks that assumption but does not repair it; a board with genuine rock-paper-scissors structure needs a different model, not a wider interval.
• Half-credit ties are a convention, not a likelihood. With ties="half" the fit maximises a weighted objective rather than a probability model. fit_rao_kupper gives ties a parameter instead, and tie_convention reports whether the choice mattered.
• The position term is one number for the whole board. A judge that favours the first slot only for long answers, or only in one language, will not show up as a single advantage.
• The case studies are public samples, not the live leaderboards. The LMArena file is a 106k-battle release covering 55 models, not the full vote history behind the published board.

Related work

• Bradley and Terry (1952), Rank Analysis of Incomplete Block Designs (doi:10.2307/2334029) — the model.
• Rao and Kupper (1967), Ties in Paired-Comparison Experiments (doi:10.1080/01621459.1967.10482901) — the tie model used here.
• Hunter (2004), MM algorithms for generalized Bradley-Terry models (doi:10.1214/aos/1079120141) — the fitting algorithm, including the position and tie extensions.
• Ford (1957), Solution of a Ranking Problem from Binary Comparisons (doi:10.2307/2308513) — when a finite estimate exists at all.
• Chiang et al. (2024), Chatbot Arena: An Open Platform for Evaluating LLMs by Human Preference (arXiv:2403.04132) — the leaderboard this package audits.
• Zheng et al. (2023), Judging LLM-as-a-Judge with MT-Bench and Chatbot Arena (arXiv:2306.05685) — the source of the human and GPT-4 judgments, and of the position-bias measurement.
• Westfall and Young (1993), Resampling-Based Multiple Testing — the max-t idea behind the simultaneous critical value.

Companion projects

• judgekit — bias probes for LLM judges, each with a bootstrap confidence interval.
• judgepanel — estimate judge accuracy without gold labels.
• raterkit — reliability, rogue-rater, drift and leakage diagnostics for labelled data.
• calikit — reliability diagrams, ECE and Brier decomposition for a model's probabilities.
• abkit — sample-ratio, peeking, multiple-testing and winner's-curse checks for experiment readouts.

Citation

@software{mohammadi_arenakit,
  author  = {Mohammadi, Hadi},
  title   = {arenakit: audit a pairwise model leaderboard before you trust its order},
  year    = {2026},
  doi     = {10.5281/zenodo.21813309},
  url     = {https://github.com/mohammadi-hadi/arenakit}
}

License

MIT