Bradley-Terry Model

A Bradley-Terry Model is a probability model that predicts the outcome of pairwise comparisons between entities by estimating the probability that one item will be preferred over another.

• Context:
  • It can (typically) be applied in scenarios where items, such as teams in a sports tournament or products in a consumer survey, are compared in pairs.
  • It can (often) use the strength or quality score of each item to predict the outcome of unobserved pairwise comparisons.
  • ...
  • It can be a simple application in sports rankings
  • It can be a complex scenario in market research.
  • It can be employed in fields like statistics, machine learning, and decision theory.
  • It can utilize algorithms like maximum likelihood estimation to infer the scores of items from observed comparisons.
  • ...
• Example(s):
  • An application in sports analytics: The Bradley-Terry Model is used to rank teams based on game outcomes. By comparing teams in pairs, the model predicts the probability of one team winning against another, which helps in forecasting future match outcomes.
  • A market research study: Consumer preferences are analyzed using the Bradley-Terry Model by comparing products in pairs. The model helps derive a comprehensive ranking of products, which informs product development and marketing strategies.
  • Extended Bradley-Terry Model in sports: This variation of the Bradley-Terry Model includes parameters for contextual factors such as home-field advantage. It is used in scenarios where these factors significantly impact the outcomes of pairwise comparisons, leading to more accurate predictions.
  • Bradley-Terry-Luce Model in multi-item comparisons: In talent competitions or ranking scenarios involving multiple candidates, the Bradley-Terry-Luce Model allows for simultaneous comparisons among several contestants, providing a robust ranking based on pairwise preferences.
  • Bayesian Bradley-Terry Model in prior-informed ranking: This variation incorporates prior knowledge, such as historical performance data, into the Bradley-Terry Model. It is used in contexts where this prior information is crucial for refining predictions, such as in ranking historical sports teams or long-term consumer product preferences.
  • ...
• Counter-Example(s):
  • a Rank Aggregation Method that does not rely on pairwise comparisons but instead aggregates full rankings provided by participants.
  • a Bayesian Ranking Model that incorporates prior information into the ranking process.
• See: Thurstone's Law of Comparative Judgment, Probability Theory, Pairwise Comparison.

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References

2024

• (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/Bradley–Terry_model Retrieved:2024-8-14.
  • The Bradley–Terry model is a probability model for the outcome of pairwise comparisons between items, teams, or objects. Given a pair of items and drawn from some population, it estimates the probability that the pairwise comparison i > j turns out true, as

    where is a positive real-valued score assigned to individual . The comparison i > j can be read as "is preferred to ", "ranks higher than ", or "beats ", depending on the application.

    For example, might represent the skill of a team in a sports tournament and [math]\displaystyle{ \Pr(i\gt j) }[/math] the probability that wins a game against . Or might represent the quality or desirability of a commercial product and [math]\displaystyle{ \Pr(i\gt j) }[/math] the probability that a consumer will prefer product over product .

    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 given an observed set of outcomes. In this type of application represents some measure of the strength or quality of [math]\displaystyle{ i }[/math] 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 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 [math]\displaystyle{ i }[/math] over wine [math]\displaystyle{ j }[/math] , even if no one in the survey directly compared that particular pair.

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