An Intuitive Explanation of the Bradley-Terry Model

The Bradley-Terry model is a classical statistics idea related to paired comparisons. A typical example is a set of 6 tennis players who play against each other many times. Each match is a paired comparison between two of the players.

The Bradley-Terry model says that each of the 6 players has an associated value called p(i) where i is the index of the player. Suppose that

p1 = 0.15
p2 = 0.17
p3 = 0.20
p4 = 0.16
p5 = 0.14
p6 = 0.18

These pi values are determined by the results of all the previous matches (explained below). If you have the pi values, then the Bradley-Terry model says that in a new match between players i and j, the probability that player i will win is pi / (pi + pj). For example, the probability that player 3 will beat player 1 is

Prob(3 beats 1) = p3 / (p3 + p1) 
                = 0.20 / (0.20 + 0.15)
                = 0.20 / 0.35
                = 0.5714

Notice that the higher the pi score, the stronger the player is, and therefore, the pi values also can be used to rank players from best (player 3) to worst (player 5).

OK, pretty simple if you have the pi values, but where do the pi values come from? Well, you have previous results that might look something like:

     T1  T2  T3  T4  T5  T6  Tot Wins
T1    0   7   6   7   8   7     35
T2    8   0   7   8   8   7     38
T3    9   8   0   8   9   8     42
T4    8   7   7   0   8   7     37
T5    7   7   6   7   0   7     34
T6    8   8   7   8   8   0     39

The first row means, T1 (tennis player 1) beat T1 0 times, beat T2 7 times, beat T3 6 times, beat T4 7 times, beat T5 8 times, beat T6 7 times, and so T1 had a total of 35 wins.

You use these previous results to find the pi values. In the abstract, you use an algorithm to find the pi values that would best generate the observed results. This is called maximum likelihood estimation. The Wikipedia article on Bradley-Terry shows one possible way to get the pi values from a set of pairwise comparisons:

Maybe I’ll code the algorithm up someday when I have free time. In the end, Bradley-Terry is a very interesting idea but not very practical for actually predicting things such as sports results.

Wimbledon. Left: JB Ward and “Mrs. Armstrong” in the first mixed doubles competition (1913). Center: The first tennis superstar Anthony Wilding (1914). He died in battle in WW I just months after this photo was taken. Right: Alice Marble and Kay Stammers just months before the start of WW II (1938).

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1 Response to An Intuitive Explanation of the Bradley-Terry Model

  1. Thorsten Kleppe says:

    That was a intensiv week on your blog.

    Thank you for this wonderful explanation. The Bradley-Terry Model is a neat inspiration.

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