Calculations

The GAA Elo ratings system borrows heavily from the work of Bob Runyan’s similar system for the World Football Elo Ratings system – while both are indebted to the work of Arpad Elo’s original system devised to rank chess players.

The Elo system – and anything derived from it – is generally considered to produce a reliable ranking for each participant after around 30 games. This would mean that, over the course of four or five full League and Championship seasons (and assuming all the other ‘ingredients’ in the mix are fair and reflective), a ranking should become a reliable indication of exactly how strong each team is when compared to its peers.

There are two mathematical formulae involved in this system. The first is the effect that a match result will have on the rating of the winning team.

The winning team’s new rating – RN – is calculated as follows:

RN = RO + W * M * (OE)

where:

• RO denotes the team’s old rating
• W denotes the relative weight (i.e. importance) of the fixture
• M denotes the margin of victory,
• O denotes the actual outcome of the match, and
• E denotes the expectation that the side would win.

In all instances, the new rating is rounded to the nearest whole number.

W – the weight attached to a fixture – depends on the tournament within which the teams are participating. In my model I’ve chosen to differentiate between matches in the National League, Provincial championships, All-Ireland qualifier series (the ‘back door’) and the All-Ireland series proper (i.e. from quarter-finals onwards). In hurling, I’ve further differentiated the Ulster championship as it carries less influence among the selection of teams who participate in it, and also the secondary competitions (the Christy Ring, Nicky Rackard and Lory Meagher Cups) from the ‘main’ Liam MacCarthy Championships.

Fixtures have been assigned the following weights:

• All-Ireland series (both codes): 100
• Football provincial championships: 70
• Hurling provincial championships: 60
(This is to reflect the extra workload of the round-robin system in Leinster on lower-rated teams, and the generally smaller population of the MacCarthy tier)
• All-Ireland Qualifier series: 60
• National Football Leagues: 45
• National Hurling Leagues: 40
• Secondary hurling competitions (Ring/Rackard/Meagher): 50
• Ulster Hurling championship: 35

Matches in ‘pre-season’ (Waterford Crystal / O’Byrne / Kehoe / McGrath, etc) competitions are not counted – predominantly because they rarely offer matches between two ‘full’ county sides, with no college sides involved, and also because the county sides who line out in such competitions are usually experimental and bear little relationship to those who feature in later competitions.

M denotes the margin of victory in the match. This is calculated by taking the winning team’s total score (in points) and dividing it by that of the losing team. That is to say, a team who wins on a margin of 1-11 to 0-10 is considered to have won by a margin of 14/10 = 1.4. Similarly, a team which wins a low-scoring match by 0-8 to 0-6 is considered to have won by a margin of 8/6 = 1.333.

The M value is capped at 2 (two) – the reason being that, beyond double scores, scorelines can often be freakishly large and it is senseless to give a team a disproportionate reward for an extraordinary scoreline, or likewise to punish a team’s rating for allowing the scoreline to slip when a game has already been lost.

O is the outcome of the match. This is a numerical expression of how the match has finished – with a 1 for a victory, 0 for a defeat, and 0.5 for a draw.

Finally, the calculation of E – the expectation that the side would win. In this formula, dr denotes the rating of the team in question, minus that of its opponent:

1 / (1 + 10dr/500)

In my system – as with the World Elo Football Ratings model – home advantage is considered to be worth 100 extra ratings points. That is to say, a side playing a team with a rating 100 higher than its own, but which plays the fixture at home, is considered to have an equal rating for the purposes of calculating points.

However, a significant departure for the World Elo Football Ratings model is that the denominator used here is 500, rather than 400. This is to introduce some stability in the system, to take account of the high-scoring nature of gaelic games (as opposed to soccer) which leads to a greater degree of unpredictability. With a denominator of 400, an advantage of 200 ratings points means a 75% expectation of success – which falls to 71% when the denominator rises to 500. This modest adjustment has a significant stabilising effect – particularly when a larger number of results are taken into account.

The relationship between O and E is key to the system, as it ensures a proportional change in ratings when a team beats a rival with a rating far higher or lower than its own. A gap of 400 ratings points between two teams will mean the higher-rated team has an E value of 0.863 – i.e. the ‘stronger’ team should win with 86% certainty. If the stronger team wins (O = 1), OE is therefore 0.137 – meaning even a victory by a high margin will have a relatively small impact on its rating. Conversely, in the case of the weaker team, E is 0.137 and if it wins, OE is 0.863 – so its win has a higher impact on the exchange of rating points between the two sides.

An example

A traditionally ‘weaker’ county – with a rating of 1350 – scores a famous win in its provincial championship against a more fancied, ‘stronger’ rival with a rating of 1600 and which enjoyed home advantage for the match. The weaker side wins on a scoreline of 2-13 to 0-15.

As it enjoyed home advantage, the strong side’s outgoing rating is inflated by 100 points for the purposes of calculating its new rating – i.e. it is considered to have a rating of 1700 for the purposes of the match in question. dr is therefore (1600+100) – 1350 = 350, while M is 19/15 = 1.267. A provincial championship match carries a weight of 70.

The stronger team’s E expectation of victory is 0.834, while that of the weaker team is 0.166. Therefore OE in the case of the stronger team is -0.834, and 0.834 in the case of the weaker.

The stronger team’s new rating is as follows:

RN = 1600 + 70 * 1.267 * (0-0.834) = 1526

The ‘stronger’ team’s rating falls by 74 points, which are accordingly gained by the weaker team, whose rating rises to 1424:

RN = 1350 + 70 * 1.267 * (1-0.166) = 1424

Had the margin of victory been more convincing – perhaps the ‘stronger’ team only scored 0-10 – the sides would have exchanged 111 points. Had it been narrower (say the losing team scored 1-15 and lost only by a point, 19 to 18) they would have exchanged only 62 points.

Had the two sides drawn – 2-13 to 1-16, for example (though the combination of goals and points is irrelevant) – the stronger side’s O value rises to 0.5 and the new rating would be as follows:

RN = 1600 + 70 * 1 * (0.5-0.834) = 1577

The teams therefore exchange a more modest 23 points – the losing team rising to 1373. Assuming the replay is a home game for the weaker side, with a 100 ratings boost as a result, the dr value will be just 104 – giving the ‘stronger’ away side an E of 0.617 and the ‘weaker’ side an E of 0.383.

Clearly, the replay ought to be a much closer contest than the original match had been.