Redefined Bradley-Terry Models

In this final part of my series on Bradley-Terry models I will talk about how the simple concepts behind Bradley-Terry models link with and underpin some more well-known and advanced concepts.

1. Logistic Regression

Let’s start by making a substitution in the formula of \lambda_i = e^{b_i}.

P(i \text{ beats } j) = \cfrac{\lambda_i}{\lambda_i + \lambda_j}

P(i \text{ beats } j) = \cfrac{e^{b_i}}{e^{b_i} + e^{b_j}}

With a bit of mathematical manipulation we can recast this into a more familiar form.

P(i \text{ beats } j) = \cfrac{e^{b_i-b_j}}{e^{b_i-b_j} + 1} = \cfrac{1}{1 + e^{-(b_i-b_j)}}

These look very similar to the form of a logistic regression. This is a regression where the dependent variable can only take two values – it is binary. In our case we only have the outcomes of team i winning or team i losing.

\text{ invlogit}(x) = \cfrac{e^{x}}{e^{x} + 1} = \cfrac{1}{1 + e^{-x}}

Then if we substitute our initial expression into the logistic transformation we obtain the following terms which can be furthered simplified using the fact that \lambda_i = e^{b_i}.

\text{ invlogit}(P(i \text{ beats } j)) = \log{\cfrac{P(i \text{ beats } j)}{1 - P(i \text{ beats } j)}} = \log{\cfrac{\lambda_i}{\lambda_j}} = b_i - b_j

From here it is simple to invert the transformation to get the final result, which is that the probability of team i beating team j is just a logistic regression on b_i-b_j.

P(i \text{ beats } j) = \text{ logit}(b_i - b_j)

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Extended Bradley-Terry Models

I previously talked about the use of the classic Bradley-Terry model and its applicability to a wide variety of situations from ranking in machine learning algorithms through to modelling sports teams. In this post I will briefly outline some of the main modifications to the model over the last 60 years, extending its use into a wider range of situations.

1. Home Advantage

In many sports that are played in front of a partisan crowd there is often a benefit to the team being supported. This is the concept of “Home Advantage“, where the local team tends to perform better than the visiting team. This is not just because of crowd support, it might also include the fact that the home team are more experienced at playing in those conditions – think of the England’s cricketing struggles in the Indian subcontinient!

A mathematical form for this was suggested by Agresti (1990) which is given below,

P(i \text{ beats } j | i \text{ at home }) = \cfrac{\theta\lambda_i}{\theta\lambda_i + \lambda_j}

P(i \text{ beats } j | j \text{ at home }) = \cfrac{\lambda_i}{\lambda_i + \theta\lambda_j}

Here the parameter \theta > 1 represents the size of the home-field advantage, the larger the value the more likely the home team wins.

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Home Advantage

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A Bradley-Terry Model

Suppose that there is a competition between two people with only two outcomes. Either one of the two players, Alice or Bob, can win and the other must lose. This is a zero-sum game: if Alice wins Bob must lose and if Bob wins Alice must lose. Now let’s further assume that the game that they are playing is skill based and the participants success is determined by their relative skill levels. This means that games that are primarily based on luck or chance such as Snakes and Ladders are excluded, but skill based games like chess are acceptable .

This idea is formally known as a Bradley-Terry model (1952), where the chance of Alice or Bob winning are in proportion to their skill levels. If Alice has a skill level of \lambda_A and Bob has a skill level of \lambda_B, then the probability that either one wins is the ratio of their skill level to their combined total skill level.

P(Alice \text{ beats } Bob) = \cfrac{\lambda_A}{\lambda_A + \lambda_B}

P(Bob \text{ beats } Alice) = \cfrac{\lambda_B}{\lambda_A + \lambda_B}

It is clear to see that the law of total probability holds, if we sum up the probabilities of both outcomes we get unity and the basic idea of winning in proportion to the “skill” or “quality” is also obeyed.

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A game of skill

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