Note (August 2026): the research line has moved home. New development of the factor-correlated race machinery — forward shares, scalable probit calibration, Jacobian-vector products, the methods arena, and the benchmark database — now lives in the winning package, where this line of work began (SIAM J. Financial Mathematics, 2021). thurstone remains maintained as a compatibility layer, and its documentation below stays current for the existing API; new work should import from winning.

thurstone

Convert winning probabilities to relative abilities using the fast ability transform.

What it does

Given market odds or winning probabilities, infer the relative abilities of competitors.

Input: Market odds [3.2, 4.8, 12.0, 7.5, 20.0]
Output: Relative abilities [1.15, 0.73, -0.88, 0.21, -1.21]

The model assumes each competitor's performance = true ability + random noise, and the best performance wins.

Usage

pip install thurstone

from thurstone import UniformLattice, Density, AbilityCalibrator, STD_L, STD_UNIT

# Setup
lattice = UniformLattice(L=STD_L, unit=STD_UNIT)
base = Density.skew_normal(lattice, loc=0.0, scale=1.0, a=0.0)
calibrator = AbilityCalibrator(base)

# Convert odds to abilities
odds = [3.2, 4.8, 12.0, 7.5, 20.0]
abilities = calibrator.solve_from_dividends(odds)
probabilities = calibrator.state_prices_from_ability(abilities)

Exact joint inversion (Laplacian Newton–CG)

The Jacobian of the map from abilities to win probabilities is minus a weighted graph Laplacian, and it can be applied to a vector in O(nM) without ever being formed. invert_outright_probabilities exploits this for exact joint Newton–CG inversion — all runners move together, no per-runner curve approximation:

from thurstone import Density, UniformLattice, invert_outright_probabilities

lattice = UniformLattice(L=400, unit=0.05)
bases = [Density.skew_normal(lattice, loc=0.0, scale=s, a=0.0) for s in (0.8, 1.0, 1.2)]

result = invert_outright_probabilities(bases, [0.5, 0.3, 0.2])  # normalized prices
result.abilities  # mean-zero abilities reproducing the price ratios
result.converged  # honest flag; result.message explains any failure

Targets are matched in ratio (a single multiplicative renormalization absorbs the lattice tie mass, so longshot probabilities keep their meaning), per-runner distributions may be heterogeneous, and non-convergence is diagnosed — e.g. a target requiring more ability spread than the lattice represents. The building blocks laplacian_weights, laplacian_matvec, and LaplacianOperator are exported for direct use.

Applications

• E-commerce product ranking
• Search result relevance scoring
• Financial instrument comparison
• Sports betting analysis
• Any competitive scenario with market-implied rankings

Examples

python examples/global_calibration_demo.py      # 500 competitors
python examples/dynamic_calibration_demo.py     # Time-varying abilities
python examples/diffeomorphism_demo.py          # Advanced mappings
python examples/laplacian_newton_demo.py        # Laplacian Jacobian + Newton-CG inversion

Documentation

📖 Full Documentation & Interactive Demos

Citation

Cotton, Peter. "Inferring Relative Ability from Winning Probability in Multientrant Contests." SIAM Journal on Financial Mathematics 12.1 (2021): 295-317.

Development

pip install -e ".[test,viz]"
python scripts/format-code.py
pytest