# JOST A MON

Mar 24, 2010

## Randomly Premier

Gresham College, that ancient institution of public education, never ceases to satisfy one's intellectual cravings. The lectures there dwell on every possible topic of interest. Everyone - from jock to geek - may find themselves satisfied. And while geeks may not be particularly good at sport, they can certainly analyse it to discern patterns and see what's driven by skill, and what is random.

Consider, for example, the Barclays Premiership, object of veneration for football junkies all over the world, but in particular in England. There are twenty teams that play each other twice, once on their home ground and once away. A winner of a match wins 3 points, while tied matches obtain 1 point for each side. What is the distribution of points among the various teams? What is the spread between the topper of the table and the bottom team? What is the average point?

It may or may not be surprising to you, but a very simple probabilistic analysis can reveal much about the Premiership. Take the 2008/2009 season, for example. There were 380 matches played in all, of which 97 were tied. In other words, a quarter of the matches are drawn. In the previous season, 100 matches were tied; in the year before, 98. We may, therefore, assume that the probability of a match ending in a draw is 25%.

This means that 75% of matches have a result. If we assume that the likelihood of winning at home is the same as that away, we have probabilities of 37.5% of a home win, and 37.5% of an away win.

Given these numbers, we can easily simulate a season of the Premiership. If you make an octahedron, two of whose faces have 'Draw' written on them, three have 'Home' and three have 'Away', and toss it 380 times, and create the points table for the entire season, you will obtain a list that will look something like this (in decreasing rank order):

67 65 64 59 59 58 52 52 52 51 49 48 48 47 47 45 40 39 38 34

(I was too lazy to make the octahedron and toss it, so I wrote a little program in R to do the job for me.)

It is clear, for example, that the average points are roughly 50. The minimum score is 34 and the maximum is 68.

How representative is this of reality? Well, here's a table of the nine completed seasons of the Premiership in the decade of 2000.

Rank2008/20092007/20082006/20072005/20062004/20052003/20042002/20032001/20022000/2001
1908789919590838780
 2
86
 85
 83
 83
 83
 79
 78
 80
 70
 3
83
 83
 68
 82
 77
 75
 69
 77
 69
 4
72
 76
 68
 67
 61
 60
 67
 71
 68
 5
63
 65
 60
 65
 58
 56
 64
 66
 66
 6
62
 60
 58
 63
 58
 56
 60
 64
 61
 7
53
 58
 56
 58
 55
 53
 59
 53
 57
 8
51
 57
 55
 56
 52
 53
 52
 50
 54
 9
51
 55
 54
 55
 52
 52
 51
 50
 52
 10
50
 49
 52
 51
 47
 50
 50
 46
 52
 11
45
 46
 50
 50
 46
 48
 49
 45
 51
 12
45
 43
 46
 48
 45
 47
 49
 45
 49
 13
41
 42
 43
 47
 44
 45
 48
 44
 48
 14
41
 40
 42
 45
 44
 45
 48
 44
 42
 15
41
 39
 41
 43
 42
 44
 47
 43
 42
 16
36
 37
 39
 42
 39
 41
 45
 40
 42
 17
35
 36
 38
 38
 34
 39
 44
 40
 42
 18
34
 36
 38
 34
 33
 33
 42
 36
 34
 19
32
 35
 34
 30
 33
 33
 26
 30
 34
 20
32
 11
 28
 15
 32
 33
 19
 28
 26

In every case, the top three teams have scored more points than we expect from our simulation. All the others, though, have points very similar to those we obtained. See, for example, the scores of the team ranked eighth in every year: 51, 57, 55, 56, 52, 53, 52, 50, 54. And what did our simulation tell us? 52. Or take the thirteenth team: 41, 42, 43, 47, 44, 45, 48, 44, 48. Our simulation: 47. Except for a couple of years when the bottom teams were absolutely disastrous, our predicted points (34) for the 20th is not far from those observed (32, 33, ...)

What do we infer from this? Other than the top 3 teams, the results of the rest are entirely random. They draw a quarter of their games, they win as many as they lose. The top teams outperform completely. No wonder that the toppers have been pretty much the same teams year after year. Their overall skill is, by far, overwhelming.

Has this differentiation always existed? I don't have the history back to the eighties or seventies, but in the first year that 20 teams competed in the Premiership, 1995/1996, the top teams scored 82, 78, 71. The remainder fell exactly in the same pattern as we limned above. The bottom-scorer was at 29.

So here's the question that we have attempted to answer: is the Premiership a purely random process? Is there any skill to any team over and above that expected by chance?

The answer appears to be: most of the Premiership results are as random as expected. Only the top three teams demonstrate skill exceeding this randomness.

It might be worthwhile checking the other great leagues to see if the conclusion holds for them. I'll leave that as an exercise.

Reference
John D Barrow, A Physicist Looks at Sport, Gresham Lecture, Oct 25, 2005.