BradleyTerry

The Bradley–Terry model is a probabilistic framework for analyzing paired comparison data and ranking items based on underlying strengths. It was introduced by Ralph Allan Bradley and Milton E. Terry in 1952. In the model, each item i is assigned a positive strength parameter w_i. The probability that item i defeats item j in a direct comparison is w_i divided by the sum w_i + w_j. Equivalently, using log-strengths λ_i = log w_i, the probability can be written as exp(λ_i) / (exp(λ_i) + exp(λ_j)).

Estimation and interpretation follow from observed outcomes. Given a set of observed pairwise results, the likelihood of the data is maximized with respect to the w_i parameters, typically under a normalization constraint to resolve scale (for example, sum of w_i equal to 1 or fixing one w_i). Estimation is commonly done with iterative methods such as maximum likelihood iterations; the model can also be framed within logistic regression by treating each pair as a binary outcome.

The Bradley–Terry model is a specific case of a broader class of logit models for paired comparisons and is related to Thurstone-type models that use normal distributions for latent abilities. When extended to choice sets, it leads to the Bradley–Terry–Luce formulation, used in contexts where a single choice among multiple alternatives is observed.

Applications of the model include ranking players or teams from head-to-head results, analyzing preferences in psychology, and other settings involving comparative data. Extensions address ties, incomplete comparisons, and hierarchical or random-effects structures to accommodate more complex data.