winning

A package for dealing with races, correlated or not.

Documentation, live demos, and the papers: winning.microprediction.org — watch the lattice race beat GHK and Mendell–Elston on wall time in your browser, then read how it works.

One race, five covariance grammars, two calls. race_probabilities prices every contestant of a correlated Gaussian (or Gumbel/softmax) race in one shared-field pass; abilities_from_race inverts observed probabilities back to abilities. Both accept the same covariance descriptions: factor sugar (V=, D=), any grammar structure= (independent, factor, blocks, nested, tree), or a dense cov= that is fitted to the grammar on the way in.

• winning.factor — the engine: all-share forward pass, inversion, exact Jacobians and tie densities, covariance fitting (fit_covariance), constrained polish; top-k membership at any depth (top_k_probabilities, rank_probabilities) with calibration of those curves (abilities_from_topk) and joint per-runner (location, scale) from the top-1 and top-2 curves (loc_scale_from_win_and_second); ordered-prefix (permutation) probabilities (ordered_probabilities) and the Plackett--Luce likelihood (winning.factor.permutations).
• 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 (racing vocabulary: dividends, state prices, dead heats); see History below.
• winning.methods / winning.bench — every rival method behind one interface, and a seeded accuracy-time benchmark grid: python -m winning.bench.runner.
• winning.research — the density-agnostic research engine for arbitrary bases.

Install

pip install winning        # core depends only on numpy and scipy

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

For the probit literature, winning.probit speaks max-wins utilities and shares directly — the paper's own conventions — and is the one audited reflection onto the internal min-wins race. Both of the paper's calibrations live here: 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

One race, everything a parameter: distribution and correlation chosen per call, with factor probit just 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, not approximate: 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. It is not identifiable from a single race, so inversion holds it fixed.

Arbitrary formulas (skewed, multimodal, anything with a standardized survival/density callable) run through custom base= functions. Arbitrary data — empirical histograms, integer scores, atoms with real dead-heat mass — belong to winning.classic, whose primitive is the lattice atom vector and whose multiplicity calculus prices genuine ties exactly. The rule is provenance: formulas to the front door, atoms to classic; the only error in either workflow is format conversion. See the module docstrings for the measured costs of crossing over.

The papers

Six manuscript projects live here; papers/README.md indexes them all with venue status.

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 (index in its README); 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 at machine-precision parity with the Python, for browser demos; r/winning is a pure-R package; rust/fastrace holds the optional compiled kernels — build with pip install maturin && maturin develop --release, and winning.methods uses them automatically. julia/winning is a dependency-free Julia package covering the factor races and the full top-k module (julia parity/check.jl pins it to the reference); the covariance grammar and classic lattice are its remaining roadmap. Two Julia-facing applications ride it: julia/MultinomialProbit — the first multinomial probit for Julia, exact likelihood with analytic score plus a common-random-numbers GHK for head-to-heads — and julia/MvNormalCDFFast — deterministic MVN rectangle probabilities for factor covariance, falling back to MvNormalCDF.jl on refused cases.

Rating systems (research line)

The renovation-era ratings layer — whole-density beliefs, exact full-finish-order updates, benchmarked against TrueSkill, OpenSkill, Glicko-2 and Elo on twelve datasets — lives in src/ pending integration, with results in BENCHMARKS.md. Headlines: decisive win on Formula 1 (1,158 grands prix), best calibration on chess (ECE 0.0047), statistical ties atop WTA/ATP/EPL, and markets remaining the ceiling wherever they exist. The ThurstoneRating API documented there ships with a future release; it is not importable from the current package.

History

Versions 1.x were the SIAM paper's reference implementation. That original lattice API now lives in winning.classic — maintained, rust-accelerated, and parity-locked against the R and JavaScript ports, just no longer sprawled across the top level. The old import paths (winning.lattice_calibration and friends) keep working as aliases that raise a DeprecationWarning pointing at the new home.

A 2.0 renovation explored splitting the numerical core into the separate thurstone package with winning as an applications layer; the decision went the other way. winning owns the core — heritage and name — the thurstone implementation is vendored here as winning.research (named for what it is; winning.thurstone is a deprecation shim), and the thurstone package is a compatibility shim whose imports resolve to this one. The renovation's migration notes and unported ideas 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}
}