winning

A package for dealing with races, correlated or not.

Documentation, live demos, and the papers: winning.microprediction.org. The convergence demo runs the lattice race against GHK and Mendell–Elston in your browser.

A race has contestants with abilities and a joint distribution of performances. race_probabilities prices every contestant's chance of winning in one pass over a shared field, for Gaussian, Gumbel, or any other base, and abilities_from_race inverts observed probabilities back to abilities. Both accept the same covariance descriptions: factor loadings (V=, D=), a named grammar (structure= independent, factor, blocks, nested, tree), or a dense cov= that is fitted to the grammar on the way in.

Install

pip install winning        # core depends only on numpy and scipy

winning.certified (optional) returns race win probabilities as Arb balls that provably contain the answer, for checking the fast engines; it needs pip install winning[certified] (python-flint) and is never imported by import winning.

Quick start

Independent race: shares from abilities, and back.

import numpy as np
from winning import race_probabilities, calibrate_abilities

mu = np.array([-0.5, 0.0, 0.2, 0.3])          # lower is better (min-wins)
p = race_probabilities(mu)                     # array([0.443, 0.232, 0.175, 0.150])
mu_back = calibrate_abilities(p)               # recovers mu (mean zero)

Correlated race: a hundred runners moved by two common factors, all shares in one pass over a shared survival field, then inverted.

rng = np.random.default_rng(0)
N, k = 100, 2
mu = rng.normal(0, 1, N); mu -= mu.mean()
V = rng.normal(0, 0.4, (N, k))                # factor loadings
D = rng.uniform(0.5, 1.5, N)                  # idiosyncratic variances

p = race_probabilities(mu, V=V, D=D)          # all N shares, O(QNL)
mu_hat = calibrate_abilities(p, V=V, D=D)     # inversion

Counterfactuals and structure from the same shared field:

from winning import removal_shares, tie_densities

q = removal_shares(mu, V=V, D=D)   # q[i][j] = P(j wins | i removed)
w = tie_densities(mu, V=V, D=D)    # photo-finish weights: the Jacobian's
                                   # graph-Laplacian (circuit) conductances

The probit conventions. winning.probit takes utilities, with higher better, and returns shares. It carries both of the paper's calibrations: utilities from observed shares, and the factor structure itself from a supplied covariance.

from winning.probit import shares, utilities_from_shares, fit_factor_model

utilities = -mu                            # higher is better on this side
p = shares(utilities, V=V, D=D)            # all N choice probabilities
u = utilities_from_shares(p, V=V, D=D)     # the paper's calibration
Sigma = V @ V.T + np.diag(D)
V_hat, D_hat = fit_factor_model(Sigma, k=2)  # certified rank-k contrast fit
p2 = shares(utilities, Sigma=Sigma, k=2)   # same fit applied en route

Distribution and correlation are chosen per call, and factor probit is one named point in the family.

from winning.factor import race_probabilities

race_probabilities(mu)                       # the classic independent race
race_probabilities(mu, V=V, D=D)             # factor probit
race_probabilities(mu, base="gumbel")        # Luce / softmax, exactly
race_probabilities(mu, V=V, base="gumbel")   # correlated softmax
race_probabilities(mu, temperature=0.7)      # E[softmin(X/tau)]: soft credit

Temperature is exact. By the Gumbel-argmin identity the softmin expectation equals the hard race with each base convolved with the tau-scaled Gumbel kernel, so the same engine serves it. Inversion holds the temperature fixed; a single race pins down the abilities alone.

Custom densities go in as formulas: any standardized survival and density pair, skewed or multimodal, is a base= callable; give it cdf (and log_sf) attributes and rare places such as last place and bottom-k keep full tail precision, as the named bases do. Empirical data go to winning.classic: histograms, integer scores, and dead heats with real mass are lattice atoms there, and its multiplicity calculus prices ties exactly. Formulas to base=, atoms to classic; the module docstrings give the measured cost of crossing over.

What is in the package

• winning.factor: the engine. The all-share forward pass, inversion, exact Jacobians and tie densities, covariance fitting (fit_covariance(C, k, blocks=...), where blocks is a count to cluster or a known label per entrant; fit_tree(C, parent) onto a known genealogy, priced as race_probabilities(mu, structure=tree)), and constrained polish.

• winning.factor also prices finishing positions: top-k membership at any depth and factor rank, under the same V, D, F, W the race takes (top_k_probabilities(mu, k, V, D, F=F, W=W) with the four from fit_covariance prices a shortlist from a dense covariance), and rank_probabilities; calibration of those curves (abilities_from_topk), joint location and scale per runner from the win and place curves (loc_scale_from_win_and_second), ordered-prefix probabilities (ordered_probabilities), and the Plackett-Luce likelihood (winning.factor.permutations).

• winning.factor keeps beliefs in the race grammar: exact factor-form posteriors for common-random-number replicates (crn_posterior, and crn_posterior_replicates when entrants ran on different replicates) and for a shared-feature bandit (feature_bandit_posterior), each returning (m, d, W) ready for race_probabilities(-m, V=-W, D=d). crn_posterior_replicates also takes, still exactly, a factor or dense prior (prior=(V0, d0, m0)), per-cell noise (D as a matrix, or counts= of runs averaged per cell), shared trial and seed shocks (groups=[(trial_id, U_t), (seed_id, U_s)]) and a nuisance column level (common="free"). fit_factor_model_missing(Y, k) estimates the V, D those take from a trials x entrants score matrix with missing cells (maximum likelihood by EM, with a convergence report). update_contrast conditions a dense or factor belief on one linear observation a'theta ~ N(mean, var), such as a judgment of child minus parent (exact; a factor belief stays exact at one extra column per two-entrant contrast, or is refitted on request).

• winning.probit: the same machine in the probit literature's max-wins, utilities-and-shares conventions.

• winning.classic: the original SIAM-paper lattice ability transform, in racing vocabulary (dividends, state prices, dead heats).

Ratings

winning.ratings turns contest results into ratings and predictions with the race engine underneath, so the model that is fitted is the model that is priced.

AbilityTracker keeps one belief per entity and updates it from a finishing order, a winner, scores, or market prices, at a cost proportional to the entrants of each contest. rate_history and walk_forward carry a full joint covariance instead and score held-out predictions. Finishing orders are passed best first, and order_from_positions, order_from_performance and order_from_times build one from whatever form the data arrive in.

fit_factor_ratings estimates a level plus offsets over conditions observed on the contest, such as time control or task category, with one ridge penalty per covariate: the level pools every contest and the offsets shrink toward it. On Lichess it beats the site's own per-time-control Glicko-2 by 0.008 to 0.012 nats per game across three splits with both sides tuned, and on both Lichess and Chatbot Arena it beats a single rating and a separate rating per category. sweep_offset_ridge tunes the penalty, and covariate_contrast_report tells you whether a covariate was assigned or chosen, which decides how much a null result means.

Block correlation, AbilityTracker(rho=..., groups=...), gives same-group entrants a shared per-contest shock, fitted and priced under one model. tune_block_rho selects the share by the filter's own evidence and can sweep rho and the noise scale together, since the two trade off under an order likelihood. On Formula 1 the team correlation is real and changes from season to season, so tune it on a recent window; the ratings page has the measurements.

python -m winning.ratings.verify runs the identities and calibration checks the engine is held to, from a fast profile in CI to a full nightly one.

A second ratings line, with whole-density beliefs and exact full-finish-order updates, is being integrated from src/ and will ship as ThurstoneRating. Its benchmarks against TrueSkill, OpenSkill, Glicko-2 and Elo on twelve datasets are in BENCHMARKS.md: a decisive win on Formula 1 over 1,158 grands prix, the best calibration on chess (ECE 0.0047), statistical ties at the top on WTA, ATP and the EPL, and markets as the ceiling wherever they exist.

The papers

The manuscripts live under papers/, one folder each.

The correlated calibration is documented in Scalable Share Calibration for Factor Multinomial Probit Models (papers/factor-probit-transform): all shares of a correlated Gaussian race in one O(QNL) pass, matrix-free graph-Laplacian derivatives, and inversion at ten thousand alternatives in under a minute. Every number comes from a committed, seeded script in research/experiments, and research/experiments/run_all_paper.py regenerates the lot.

Demos and other languages

research/demos holds explanatory scripts: the shared survival field, the cavity downdate. js/factor is a dependency-free JavaScript port checked against the Python (forward shares to ~1e-15, calibrated abilities to 2e-6), for browser demos. r/winning is a pure-R package. rust/fastrace holds the optional compiled kernels, built with pip install maturin && maturin develop --release, and winning.methods uses them automatically.

julia/winning covers the factor races and the full top-k module, pinned to the Python reference by Pkg.test and julia parity/check.jl; the covariance grammar and classic lattice are next on its roadmap.

Three Julia packages build on it. Two are in the General registry, ] add MultinomialProbit and ] add GMRFExtremes. julia/MultinomialProbit is the first multinomial probit for Julia, with exact likelihood, analytic score, and a common-random-numbers GHK for head-to-heads. julia/GMRFExtremes answers order-statistic queries on Gauss-Markov chains, argmax marginals, max CDF, first passage and expected maximum, and takes a chain from GaussianMarkovRandomFields.jl through its precision matrix. julia/FactorMvNormalCDF computes deterministic MVN rectangle probabilities for factor covariance and hands every other covariance to MvNormalCDF.jl, which it depends on; its registration is in progress.

History

Versions 1.x were the SIAM paper's reference implementation. That API lives on as winning.classic, maintained, rust-accelerated, and parity-locked against the R and JavaScript ports, and the old import paths such as winning.lattice_calibration still work, pointing at the new home with a DeprecationWarning.

The 2.0 renovation built a density-agnostic engine as a separate thurstone package. It lives here as winning.research, the retired research engine: kept for the pipelines built on it, but for calibration use calibrate_abilities, which is materially faster and more accurate and uses the compiled kernels. import winning.thurstone now raises an ImportError with migration guidance, so the external thurstone shim (which imports through it) fails with the same message; its consumers should import winning.research directly. The renovation's migration notes are preserved in planning/ and attic/.

Cite

For the correlated engine (the shared field, the covariance grammars, the substitution Jacobian, removal counterfactuals, and inversion at scale):

@article{cotton2026inversion,
author = {Cotton, Peter},
title = {Scalable Inversion of Contests with Correlated Performances,
         Including Softmax and Multinomial Probit},
year = {2026},
eprint = {2609.01133},
archivePrefix = {arXiv},
primaryClass = {stat.ME},
doi = {10.2139/ssrn.7307363},
note = {arXiv:2609.01133; also SSRN working paper 7307363},
URL = {https://arxiv.org/abs/2609.01133}
}

For the original independent lattice transform (winning.classic):

@article{cotton2021inferring,
author = {Cotton, Peter},
title = {Inferring Relative Ability from Winning Probability in Multientrant Contests},
journal = {SIAM Journal on Financial Mathematics},
volume = {12},
number = {1},
pages = {295-317},
year = {2021},
doi = {10.1137/19M1276261},
URL = {https://doi.org/10.1137/19M1276261}
}