The document presents a generalized Bradley-Terry model for modeling football match results that incorporates the probability of a draw. It extends previous models to include parameters for the overall probability of a draw, the dependence of the draw probability on the relative abilities of the teams, and the effect of home advantage. The model is applied to data from the English Premier League and is shown to fit the data better than previous models that did not account for these factors.

1. Generalized Bradley-Terry Modelling of Football
Results
Heather Turner1,2
, David Firth1
and Greg Robertson1
1 University of Warwick, UK
2 Independent statistical/R consultant
11 July 2013
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 1 / 18

2. Bradley-Terry Model
The Bradley-Terry model provides an intuitive framework for
modelling the outcome of a football match between home team i and
away team j:
odds(i beats j) =
pr(i beats j)
pr(j beats i)
=
αi/(αi + αj)
αj/(αi + αj)
=
αi
αj
where αi, αj > 0 are the team abilities.
But what about draws?
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 2 / 18

3. Approximate Method
Treat draws as half a win and half a loss.
Examples in BradleyTerry2
data(ice.hockey) (useR! 2010)
data(CEMS) (JSS 48(9))
Fine if objective is simply to rank teams—not a solution if want to
predict outcomes or characterise pr(draw).
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 3 / 18

4. Davidson Model
The Davidson model extends the Bradley-Terry model to
incorporate the probability of a draw:
pr(draw) =
ν
√
αiαj
αi + αj + ν
√
αiαj
pr(i beats j | not a draw) =
αi
αi + αj
where ν > 0.
Like the Bradley-Terry model, this can be expressed as a log-linear
model and estimated using glm.
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 4 / 18

5. Closer Look at ν
ν → ∞: pr(draw) → 1
ν → 0: pr(draw) ∝ ν
√
αiαj/(αi + αj) (approx.)
The single extra parameter ν conflates
overall (max) probability of a draw
strength of dependence of pr(draw) on αi, αj.
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 5 / 18

6. Dependence of pr(draw) on Relative Ability
Under the Davidson model we have
logit (pr(draw)) = log
ν
√
αiαj
αi + αj
= log ν
αi
αi + αj
1
2 αj
αi + αj
1
2
= log νp
1
2 (1 − p)
1
2
where
p = pr(i beats j | not a draw)
This enables us to visualise the dependence of pr(draw) on the
relative ability of the teams.
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 6 / 18

7. 0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Davidson (1970) model for ties, for 1/4 < nu < 128
pr(i beats j in a non−tied contest)
pr(iandjtie)
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 7 / 18

8. Home Advantage
So far based model purely on team ability.
In football (as other sports) expect home advantage to boost ability
of home team, i:
pr(draw) =
ν
√
µαiαj
µαi + αj + ν
√
µαiαj
pr(i beats j | not a draw) =
µαi
µαi + αj
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 8 / 18

9. Generalised Davidson Model
We propose the following generalisation of the Davidson model:
logit (pr(draw)) = log
δ
c
pσπ
(1 − p)σ(1−π)
where
c is function of σ, π such that expit(δ) is the maximum
probability of a draw
σ scales the dependence on relative ability
0 < π < 1 is the value of p with maximum pr (draw)—if
π = 0.5, implies home advantage effect.
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 9 / 18

10. Changing σ
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
σ = 0.1
pr(i beats j | not a draw)
pr(draw)
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
σ = 1
pr(i beats j | not a draw)
pr(draw)
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
σ = 10
pr(i beats j | not a draw)
pr(draw)
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 10 / 18

11. Changing π
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
π = 0.25
pr(i beats j | not a draw)
pr(draw)
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
π = 0.5
pr(i beats j | not a draw)
pr(draw)
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
π = 0.75
pr(i beats j | not a draw)
pr(draw)
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 11 / 18

12. English Premier League Results
Consider data from the past 5 seasons of the English Premier League
20 teams, 380 games/season
≈ 1
4
games drawn
First convert the the win/lose/draw results to trinomial counts
football.tri <- expandCategorical(football, "result",
idvar = "match")
head(football.tri)
home away season result match count
1 Ars Ast 2008-9 -1 1 1
2 Ars Ast 2008-9 0 1 0
3 Ars Ast 2008-9 1 1 0
4 Ars Ast 2010-11 -1 2 1
5 Ars Ast 2010-11 0 2 0
6 Ars Ast 2010-11 1 2 0
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 12 / 18

13. Generalised Davidson as Poisson Model
Model log expected counts for each match k (log probabilities)
log(pr(i beats j)k) = θijk + log(µαi)
log(pr(draw)k) = θijk + log δ − log c
+ σ(π log(µαi) − (1 − π) log(αj))
+ (1 − σ) log(µαi + αj)
log(pr(j beats i)k) = θijk + log(αj)
where θijk fixes the total count for each match to 1. With gnm:
gnm(count ~ GenDavidson(result == 1, result == 0, result == -1,
home:season, away:season, home.adv = ~1,
tie.max = ~1, tie.scale = ~1, tie.mode = ~1,
at.home1 = home1, at.home2 = home2) - 1,
eliminate = match, family = poisson, data = football.tri)
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 13 / 18

14. Alternative Models
We consider the following models
1 Davidson: σ = 1, π = 0.5
2 Scaled Davidson: σ = ˆσ, π = 0.5
3 Shifted, scaled Davidson: σ = ˆσ, π = ˆπ
Resid. Df Resid. Dev Df Deviance Pr(>Chi)
1 3665 3533.7
2 3664 3523.2 1 10.4657 0.001216 **
3 3663 3517.6 1 5.5575 0.018402 *
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 14 / 18

15. q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqq
q
q
qq
q
q
q
q
q
qq
qqq
qq
Davidson
Proportion
q
q
Non−drawn matches won
Matches drawn
q
qq
qq
qq
qq
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
pr(i beats j | not a draw)
0.0 0.5 1.0
0.000.321.00
σ = 1, q = 0.5
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 15 / 18

16. q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
qqq
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
qqq
q
qq
q
qq
q
Scaled Davidson
Proportion
q
q
Non−drawn matches won
Matches drawn
q
q
q
qq
qq
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
qq
q
qq
q
q
q
q
q
q
qqqq
q
qq
q
q
q
q
q
qq
q
q
q
qq
q
q
pr(i beats j | not a draw)
0.0 0.5 1.0
0.000.351.00
σ = 2.22, q = 0.5
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 16 / 18

17. q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
qq
q
qq
q
Shifted & Scaled Davidson
Proportion
q
q
Non−drawn matches won
Matches drawn
q
q
q
q
qqq
q
q
qq
q
q
q
q
q
q
qq
q
q
qq
qq
q
q
qq
q
qqq
qq
q
q
q
q
q
qq
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
qq
qq
q
qq
q
q
pr(i beats j | not a draw)
0.00 0.58 1.00
0.000.361.00
σ = 3.12, q = 0.58
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 17 / 18

18. Conclusions
Proposed model
explicitly models draws
distinguishes overall pr(draw) from dependence of pr(draw) on
relative ability
Expressing pr(draw) in terms of pr(i beats j | not a draw) helps to
assess model fit.
plotProportions and GenDavidson to be provided as example in
BradleyTerry2.
Turner, Firth & Robertson (Warwick, UK) Bradley-Terry Models with Ties useR! 2013 18 / 18