The Bradley–Terry model is a probability model for the outcome of pairwise comparisons between items, teams, or objects. Given a pair of items i and j drawn from some population, it estimates the probability that the pairwise comparisoni > j turns out true, as
Pr ( i > j ) = p i p i + p j {\displaystyle \Pr(i>j)={\frac {p_{i}}{p_{i}+p_{j}}}} (1)
where pi is a positive real-valued score assigned to individual i. The comparison i > j can be read as “i is preferred to j”, “i ranks higher than j”, or “i beats j”, depending on the application.
For example, pi might represent the skill of a team in a sports tournament and Pr ( i > j ) {\displaystyle \Pr(i>j)} the probability that i wins a game against j.[1][2] Or pi might represent the quality or desirability of a commercial product and Pr ( i > j ) {\displaystyle \Pr(i>j)} the probability that a consumer will prefer product i over product j.
The Bradley–Terry model can be used in the forward direction to predict outcomes, as described, but is more commonly used in reverse to infer the scores pi given an observed set of outcomes.[2] In this type of application pi represents some measure of the strength or quality of i {\displaystyle i} and the model lets us estimate the strengths from a series of pairwise comparisons. In a survey of wine preferences, for instance, it might be difficult for respondents to give a complete ranking of a large set of wines, but relatively easy for them to compare sample pairs of wines and say which they feel is better. Based on a set of such pairwise comparisons, the Bradley–Terry model can then be used to derive a full ranking of the wines.
Once the values of the scores pi have been calculated, the model can then also be used in the forward direction, for instance to predict the likely outcome of comparisons that have not yet actually occurred. In the wine survey example, for instance, one could calculate the probability that someone will prefer wine i {\displaystyle i} over wine j {\displaystyle j} , even if no one in the survey directly compared that particular pair. (View Highlight)
The Bradley–Terry model can be parametrized in various ways. (View Highlight)
This formulation highlights the similarity between the Bradley–Terry model and logistic regression. Both employ essentially the same model but in different ways. In logistic regression one typically knows the parameters β i {\displaystyle \beta _{i}} and attempts to infer the functional form of Pr ( i > j ) {\displaystyle \Pr(i>j)} ; in ranking under the Bradley–Terry model one knows the functional form and attempts to infer the parameters. (View Highlight)
(1) where pi is a positive real-valued score assigned to individual i. The comparison i > j can be read as “i is preferred to j”, “i ranks higher than j”, or “i beats j”, depending on the application. For example, pi might represent the skill of a team in a sports tournament and Pr ( i > j ) {\displaystyle \Pr(i>j)}
the probability that i wins a game against j.[1][2] Or pi might represent the quality or desirability of a commercial product and Pr ( i > j ) {\displaystyle \Pr(i>j)}
and the model lets us estimate the strengths from a series of pairwise comparisons. In a survey of wine preferences, for instance, it might be difficult for respondents to give a complete ranking of a large set of wines, but relatively easy for them to compare sample pairs of wines and say which they feel is better. Based on a set of such pairwise comparisons, the Bradley–Terry model can then be used to derive a full ranking of the wines. Once the values of the scores pi have been calculated, the model can then also be used in the forward direction, for instance to predict the likely outcome of comparisons that have not yet actually occurred. In the wine survey example, for instance, one could calculate the probability that someone will prefer wine i {\displaystyle i}
, even if no one in the survey directly compared that particular pair. (View Highlight)
and attempts to infer the functional form of Pr ( i > j ) {\displaystyle \Pr(i>j)}