The Mathematics of Bradley-Terry

Abstract

The Bradley-Terry model is a static probabilistic model that finds the single set of ratings that best explains the entire history of game outcomes simultaneously. Unlike Elo, which is path-dependent, Bradley-Terry converges to the Maximum Likelihood Estimate (MLE) of team strength.

1. The Likelihood Function

Let team $i$ have strength parameter $\beta_i$. The probability that team $i$ beats team $j$ is:

$$ P(i > j) = \frac{e^{\beta_i}}{e^{\beta_i} + e^{\beta_j}} $$

This is mathematically uniform to the Elo logistic function if we set $R_i = \beta_i \cdot \frac{400}{\ln(10)}$.

For a set of games $G$, where $w_{ij}$ is the number of times $i$ beat $j$, the total likelihood is:

$$ L(\beta) = \prod_{i