src.pm_rank.model.bradley_terry

Generalized Bradley-Terry Model for Ranking Forecasters in Prediction Markets.

This module implements the generalized Bradley-Terry model to rank forecasters based on their probabilistic predictions. The Bradley-Terry model is a statistical method for modeling pairwise comparisons, which we extend to handle prediction market scenarios where each event outcome is viewed as a contest between winning and losing “teams” composed of forecaster contributions.

Reference: https://www.jmlr.org/papers/v7/huang06a.html

Key Concepts:

  • Bradley-Terry Model: A statistical model for analyzing pairwise comparisons that estimates the relative strengths of competitors.

  • Generalized Extension: Adapts the traditional pairwise model to prediction markets by treating each event outcome as a contest between winning and losing teams.

  • Skill Parameters: Each forecaster has a skill parameter (theta) that represents their relative predictive ability.

  • Majorization-Minimization (MM): An iterative algorithm for fitting the model parameters that guarantees convergence.

Classes

GeneralizedBT

Generalized Bradley-Terry model for ranking forecasters in prediction markets.

Module Contents

class src.pm_rank.model.bradley_terry.GeneralizedBT(method: Literal['MM', 'Elo'] = 'MM', num_iter: int = 100, threshold: float = 0.001, verbose: bool = False)

Bases: object

Generalized Bradley-Terry model for ranking forecasters in prediction markets.

This class implements a generalization of the traditional Bradley-Terry model to handle prediction market scenarios. Each event outcome is treated as a contest between two “pseudo-teams”: a winning team (the realized outcome) and a losing team (all other outcomes). Each forecaster contributes fractions of their capability proportional to their predicted probabilities.

The model estimates skill parameters for each forecaster using an iterative Majorization-Minimization (MM) algorithm, which provides convergence guarantees and intuitive comparative scores similar to Elo ratings.

Parameters:

  • method – Optimization method to use (“MM” for Majorization-Minimization).

  • num_iter – Maximum number of iterations for the MM algorithm (default: 100).

  • threshold – Convergence threshold for parameter updates (default: 1e-3).

  • verbose – Whether to enable verbose logging (default: False).

Initialize the generalized Bradley-Terry model.

Parameters:

  • method – Optimization method to use (“MM” for Majorization-Minimization).

  • num_iter – Maximum number of iterations for the MM algorithm (default: 100).

  • threshold – Convergence threshold for parameter updates (default: 1e-3).

  • verbose – Whether to enable verbose logging (default: False).

method = 'MM'
num_iter = 100
threshold = 0.001
verbose = False
logger
fit(problems: List[pm_rank.data.base.ForecastProblem], include_scores: bool = True) → Tuple[Dict[str, Any], Dict[str, int]] | Dict[str, int]

Fit the generalized Bradley-Terry model to the given problems.

This method estimates skill parameters for each forecaster using the MM algorithm and returns rankings based on these parameters. The skill parameters represent the relative predictive ability of each forecaster.

Parameters:

  • problems – List of ForecastProblem instances to evaluate.

  • include_scores – Whether to include scores in the results (default: True).

Returns:

Ranking results, either as a tuple of (scores, rankings) or just rankings.