Bradley-Terry Model

Overview

The Bradley-Terry model is a probabilistic model for paired comparisons, introduced by Ralph Bradley and Milton Terry in 1952. Each competitor is assigned a single latent strength, and the outcome of any head-to-head matchup is modeled as a function of the two competitors’ strengths. It is one of the most widely used models for analyzing paired comparison data across sports analytics, ranking systems, and preference learning.

Unlike Elo, which nudges ratings incrementally after every game, Bradley-Terry estimates the strengths by maximum likelihood over the full set of observed comparisons. In Elote this is done the same way as the Colley Matrix method: results are recorded as they happen, and the strengths of the entire connected group of competitors are re-fit after each result.

How It Works

Each competitor i has a positive strength p_i. The probability that i beats j is

\[\begin{split}P(i \\text{ beats } j) = \\frac{p_i}{p_i + p_j}\end{split}\]

Writing the strengths in the log domain as p_i = e^{\\beta_i} turns this into the logistic function

\[\begin{split}P(i \\text{ beats } j) = \\frac{1}{1 + e^{-(\\beta_i - \\beta_j)}}\end{split}\]

which is exactly the functional form of the Elo expected score. Elote reports Bradley-Terry ratings on an Elo-like scale, rating = 1500 + s \\cdot \\beta with s = 400 / \\ln(10), so that the expected score is numerically identical to Elo’s and the ratings are directly comparable to Elo ratings.

The strengths are fit with the minorization-maximization (MM) update of Hunter (2004). Writing W_i for the total number of wins by i and n_{ij} for the number of games between i and j, each iteration applies

\[\begin{split}p_i' = \\frac{W_i}{\\sum_j n_{ij} / (p_i + p_j)}\end{split}\]

followed by normalization by the geometric mean of the strengths. This update is guaranteed to increase the likelihood at every step and converges to the unique regularized maximum. Because the strengths are only identified up to an overall scale, Elote re-centers them so that the mean log-strength of a connected group is zero.

A small amount of regularization (a virtual win and loss against an average phantom opponent) is added so that a unique, finite maximum-likelihood estimate exists even when a competitor is undefeated or winless within its group – a case where the unregularized estimate would diverge to infinity.

Ties are counted as half a win for each side.

Advantages

  • Order independent: the fit depends only on the set of results, not the order they were played.

  • Statistically principled: produces the maximum-likelihood strengths given the data.

  • Elo-compatible scale: expected scores match Elo, so ratings are easy to interpret.

Limitations

  • Recomputes globally: like Colley, every result re-fits the whole connected group, which is more expensive than Elo’s constant-time update for very large populations.

  • No margin of victory: only win/loss/tie outcomes are used.

  • Serialization: the match graph (opponent references) cannot be serialized, so a competitor restored from state keeps its rating and aggregate win/loss/tie counts but not the live links to its opponents.

Implementation in Elote

from elote import BradleyTerryCompetitor

player_a = BradleyTerryCompetitor(initial_rating=1500)
player_b = BradleyTerryCompetitor(initial_rating=1500)

# Record results; strengths are re-fit after each one.
player_a.beat(player_b)
player_a.beat(player_b)
player_b.beat(player_a)

print(player_a.rating, player_b.rating)
print(player_a.expected_score(player_b))

It also plugs directly into an arena:

from elote import LambdaArena, BradleyTerryCompetitor

arena = LambdaArena(comparison_func, base_competitor=BradleyTerryCompetitor)

Customization

Class-level parameters can be tuned with configure_class:

  • anchor_rating – the rating assigned to the mean log-strength (default 1500).

  • scale – points per unit of log-strength (default 400 / ln(10)).

  • reg – regularization strength (default 0.1).

  • max_iter / tol – iteration budget and convergence tolerance for the fit.

Real-World Applications

  • Ranking sports teams or players from head-to-head results.

  • Aggregating pairwise human preferences (e.g. A/B choices) into a single ranking.

  • Any setting where you want a maximum-likelihood ranking rather than an online estimate.

References

  • Bradley, R. A., & Terry, M. E. (1952). Rank Analysis of Incomplete Block Designs: I. The Method of Paired Comparisons. Biometrika, 39(3/4), 324-345.

  • Hunter, D. R. (2004). MM Algorithms for Generalized Bradley-Terry Models. The Annals of Statistics, 32(1), 384-406.