Bradley–Terry (Pairwise MLE)
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Maximum-likelihood models — the most rigorous skill estimates. The formula and explanation below describe Bradley–Terry (Pairwise MLE).
How it works About 1 min read
The Bradley-Terry model is a probabilistic approach to ranking from pairwise comparisons. It estimates the probability that any player beats any other player using maximum likelihood estimation.
How it works: Each player has a latent "strength" parameter. The probability that Player A beats Player B is strength_A / (strength_A + strength_B). The model finds the strength values that best explain the observed win/loss data using iterative optimization (MLE).
When to use it: When you want mathematically principled skill estimates from 1v1 data. The model produces predicted win probabilities for any matchup, which is useful for seeding and prediction.
Watch out for: Only works with pairwise (1v1) data. Computationally intensive -- requires iterative optimization. Can be unstable with sparse data (e.g., if two players have never met). Use regularization to handle small samples.
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Also in Statistical Models
Maximum-likelihood models — the most rigorous skill estimates.