Chapter 18 · Worked solutions
Chapter 18 workings: Rating the Teams: Elo, Pi and the Art of Forgetting
Chapter eighteen worked: two Elo updates by hand, a favourite held to a draw and a three-goal away win, and the case for a faster K-factor after promotion.
The conventions are the World Football Elo ones the chapter uses: 100 points of home advantage, K = 40, and a goal multiplier G of 1 for a draw or one-goal margin, 1.5 for two goals and 1.75 for three. Expected scores are kept to four decimal places so the updates match the book to the penny.
The exercises
Anchor Town 1-1 Redbrick Rovers
Anchor Town (1600, at home) draw 1-1 with Redbrick Rovers (1550). Update both ratings.
Rating gap, with home advantage
1600 + 100 − 1550 = 150 points.
Expected scores
EH = 1 ÷ (1 + 10−(150)/400) = 0.7034, and EA = 1 − EH = 0.2966.
Score the result
A draw: WH = 0.5, WA = 0.5. The goal multiplier G is 1.
Update both sides
Anchor Town: 1600 + 40 × 1 × (0.5 − 0.7034) = 1591.86. Redbrick Rovers: 1550 + 40 × 1 × (0.5 − 0.2966) = 1558.14. What one side gains the other loses.
The answer
Anchor Town 1591.86, Redbrick Rovers 1558.14: the favourite drops about eight points for failing to win.
Colliery United 0-3 Ferry Athletic
Colliery United (1500, at home) lose 0-3 to Ferry Athletic (1650). Update both ratings.
Rating gap, with home advantage
1500 + 100 − 1650 = −50 points.
Expected scores
EH = 1 ÷ (1 + 10−(−50)/400) = 0.4285, and EA = 1 − EH = 0.5715.
Score the result
An away win: WH = 0, WA = 1. The goal multiplier G is 1.75.
Update both sides
Colliery United: 1500 + 40 × 1.75 × (0 − 0.4285) = 1470.00. Ferry Athletic: 1650 + 40 × 1.75 × (1 − 0.5715) = 1680.00. What one side gains the other loses.
The answer
Colliery United 1470.00, Ferry Athletic 1680.00: thirty points, mostly because of the three-goal margin.
K for a newly promoted side
A promoted club has six new signings and a first-time manager. Should its K sit above or below 40?
What K does
K is how far one result can move a rating. A high K believes new evidence quickly; a low K trusts the history.
Why the history is weak here
Last season's rating was earned in a different division by a different squad under a different manager. It is a poor guide to this side.
See the difference
Rerun exercise 1 as if Anchor Town were the promoted club. At K = 60 the same draw moves them half as far again as at K = 40, as the table shows.
Then taper
Raise K for the opening weeks, then bring it back towards 40 over the first ten or so matches, so one freak result in August does not define the season.
| Setting | Anchor Town after | Redbrick after | Anchor Town change |
|---|---|---|---|
| K = 40 | 1591.86 | 1558.14 | −8.14 |
| K = 60 | 1587.80 | 1562.20 | −12.20 |
The answer
Above 40 for the first several matches, then tapered back, because the old rating describes a side that no longer exists.
These are the workings. The argument for them, and the exercises themselves, are in chapter 18 of the book, Rating the Teams: Elo, Pi and the Art of Forgetting.
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