Bradley–Terry (Pairwise MLE)
Finds the set of player strengths that makes your entire match history most probable, re-weighing every match every time rather than only the latest.
P(A beats B) = s_A / (s_A + s_B); MLE optimization
Best at
Dense histories between the same people, and global boards, since it tolerates players who have never met.
Where it misleads
Pairwise data only, needs a lot of matches to stabilise, and it is opaque — hard to explain why a number moved.
How it works
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.