Predicting the Final League Tables

of Domestic Football Leagues

Jan Van Haaren and Jesse Davis
KU Leuven, Department of Computer Science
Celestijnenlaan 200A, 3001 Leuven, Belgium
{jan.vanhaaren,jesse.davis}@cs.kuleuven.be

Abstract
In this paper, we investigate how accurately the final league tables of domestic football leagues can
be predicted, both before the start of the season and during the course of the season. To this end, we
perform an extensive empirical evaluation that compares two flavors of the well-established Elo-ratings
and the recently introduced pi-ratings. We validate the different approaches using a large volume of
historical match results from several European football leagues. We assess how well each ranking system
performs on this task, investigate what is the most natural metric to measure the quality of a predicted
final league table, and the minimum number of matches that needs to be played in order to yield useful
predictions. We find that the proportion of correctly predicted relative positions is a natural metric
to assess the quality of the predicted final league tables and that the traditional Elo-rating system
performs well in most settings we considered.

1 Introduction
The prediction of football matches has received significant attention over the past few decades.
Predicting the outcomes of individual football matches is a very challenging task due to the low
number of goals scored [1]. Despite this fact, the ever-growing interest in football betting has this
an active area of research. Initially, the focus was on purely statistical models (e.g., [3, 6, 7, 8, 9])
but in recent years it has somewhat shifted to predictive ranking systems (e.g., [2, 5]). In
contrast to the large body of work in the area of predicting individual match outcomes, the
related task of predicting final league tables has remained almost unexplored to date. However,
the latter task is of much more interest to managers and directors who want to develop a
successful long-term vision for their clubs.
In this paper, we investigate how accurately the final league tables of domestic football
leagues can be predicted, both before the start of the season and during the course of the season.
We focus our study on different flavors of two popular predictive ranking systems, namely the
recently introduced pi-ratings [2] and the well-established Elo-ratings [5]. We validate the
different systems on a large volume of historical match results covering over twenty seasons
of the most important European football leagues. Besides assessing how well each predictive
system performs on this task, we also investigate what is the most natural metric to measure
the quality of a predicted final league table, and the minimum number of matches that needs
to be played in order to yield useful predictions.
An experimental evaluation on four seasons in seven different leagues shows that the propor-
tion of correctly predicted relative positions and the mean squared error in terms of positions
are natural metrics to assess the quality of the predicted final league tables. Furthermore, the
evaluation shows that the traditional Elo-rating system performs well in most settings. The
Pi-rating model performs reasonably well in many settings but excels when only match results
from previous seasons are available.

1
2 Predictive Models
We consider multiple variants of both the widely used Elo-ratings [4, 5] and the more recent
pi-ratings [2]. While these ranking systems are very similar in spirit, they differ in several
important aspects. We now discuss the specifics of both ranking systems.

2.1 Elo Ratings

The Elo system uses a single number or rating to represent the strength of a team at a particular
point in time. This rating increases or decreases based on the outcomes of matches. After each
match, rating points are transferred from the losing team to the winning team. The number
of transferred rating points depends on the difference between the ratings of the teams prior
to the match. More rating points are transferred when a low-ranked team wins against a high-
ranked team than when a high-ranked team beats a low-ranked team. Hence, the Elo system
is self-correcting in the sense that underrated teams will gain rating points until their rating
reflects their true strength, while overrated teams will loose rating points.
More formally, the rating of a team is computed as follows after each match:

Rnew = Rcur + I G (Ract Rexp ) (1)

In this equation, Rcur and Rnew denote the rating of a team before and after a match, re-
spectively. The parameter I is a positive number that denotes the importance of the match, and
higher numbers correspond to more important matches. The parameter G is a positive number
that denotes the importance of the goal difference and a bigger goal difference corresponds to
an higher number. This parameter is typically defined as a function of the goal difference:

1
if GD 1
G = 1.5 if GD = 2 (2)
(GD+11)

8 if GD 3
The parameters Ract and Rexp represent the actual and expected outcome of the match,
respectively. The parameter Ract takes one of three possible values: 1 for a win, 0.5 for a draw,
and 0 for a loss. The parameter Rexp takes a value between 0 and 1 and is computed as follows:
1
Rexp = Raway Rhome

(3)
1 + 10 400

In this equation, Rhome and Raway denote the ratings of the home and away team.

2.2 Probabilistic Intelligence Ratings

The Probabilistic Intelligence system uses two numbers or pi-ratings to represent the strength
of a team at a particular point in time. These numbers represent the expected goal difference
against an average opponent in a league both at home and on the road. Similar to the Elo
system, the ratings increase or decrease based on the outcomes of matches. After each match,
the ratings of the teams involved are adjusted based on the goal difference. The key idea
is that the system updates the ratings to reduce the discrepancy between the predicted and
observed goal differences. The convergence speed depends on two learning rates, which denote
how important recent results are for assessing the current ability of a team.
More formally, the ratings of a team are computed as follows after each match:
 R,H = C,H + H (e) (4)

R,A = C,A + (R,H C,H ) (5)

R,A = C,A + A (e) (6)

R,H = C,H + (R,A C,A ) (7)

In these equations, C,H and C,A are the current home and away ratings for team , C,H
and C,A are the current home and away ratings for team . R,H , R,A , R,H , and R,A are
the respective revised ratings. Furthermore, and are learning rates, and (e) is a function of
the difference between the observed and predicted goal difference, which is computed as follows:

(e) = 3 log10 (1 + e) (8)

Due to space limitations, we refer to [2] for the technical details and a concrete example.

3 Experimental Evaluation
This section presents the experimental evaluation. We first present the dataset, the methodol-
ogy and the predictive models before discussing the experimental results.

3.1 Dataset
In our experimental evaluation, we use a large dataset of historical football match results.1
The dataset contains match results for eleven European football leagues including the English,
German, Spanish, Italian and French top divisions. The dataset dates back to the 1993/1994
season for most leagues and contains nearly 150,000 match results. Due to space limitations,
we restrict the evaluation to the four seasons from 2010 through 2014 and the following seven
leagues: Belgium, England, France, Germany, Italy, The Netherlands, and Spain.

3.2 Methodology
To evaluate each of the models, we split the match results for each league in two sets: a training
set and a test set. We use the training set to learn the parameters of the model and the test
set to assess the performance of the model. We predict the outcomes of the matches in the test
set in an iterative way. We use the model learned on the training set to predict the outcomes
of the matches on the first match day. We then update the parameters of the model using
the expected outcomes and predict the outcomes of the matches on the second match day. We
repeat this procedure until all matches have been predicted.
To account for the fact that performances tend to vary throughout a season, we employ
a probabilistic approach to predict match outcomes. For the Elo-rating models, we predict a
match outcome by first computing the probability distribution over the possible outcomes and
then sampling an outcome from this distribution. For the pi-rating model, we sample a match

1 http://www.football-data.co.uk/data.php
outcome from a normal distribution whose mean is the expected outcome. We repeat each
experiment 10,000 times and report the average results across all runs.
In each league, we predict the outcomes for the 2013/2014 season and use the match results
from the earlier seasons to learn the parameters of the models.

3.3 Models
We compare the following four models in our experimental evaluation:
Random model (RM): This model predicts a random outcome for each match.
Elo-rating model (EM): This is a traditional implementation of the Elo-rating system
(see Section 2.1), which represents each team by a single rating. We account for the home-
field advantage by increasing the rating of the home team with an empirically determined
number of rating points.
Split Elo-rating model (SEM): This is a modified implementation of the Elo-rating
system, which represents each team by two ratings. The first rating represents the strength
at home, while the second rating represents the strength on the road.
Pi-rating model (PM): This is a traditional implementation of the pi-rating system
(see Section 2.2), which represents each team by two ratings.
We have not rigorously tuned the parameters. For each model, we used reasonable parameter
values that work well on our large volume of training matches.

3.4 Results
In our evaluation, we investigate what is the most natural metric to assess a predicted league
table (Q1), how well each model performs on predicting the final league table (Q2), and what
proportion of the matches needs to be played in order to yield useful predictions (Q3).

Q1: What is the most natural metric to assess a predicted league table?
We investigated the following four metrics to assess the models:
Proportion of correct absolute positions: This is the ratio between the number of
correctly predicted absolute positions and the total number of positions in the league table.
Proportion of correct relative positions: This is the ratio between the number of
correctly predicted relative positions and the total number of relative positions in the
league table. This metric compares the relative positions between any two teams.
Mean squared error in terms of positions: This is the mean squared error between
the actual positions and the predicted positions for each team.
Mean squared error in terms of points: This is the mean squared error between the
actual number of points and the predicted number of points for each team.
We found that the proportion of correctly predicted relative positions and the mean squared
error in terms of positions are the two most natural metrics to assess a predicted league table.
Predicting the absolute position for a team is hard because it depends on the positions of all
other teams in the league. Furthermore, both the Elo-rating models and the Pi-rating model
have difficulties to predict draws, which often leads to an higher variance on the predicted
number of points than on the actual number of points for each team.
 80
Relative positions %

40

MSE positions
70
30
60
20
50 10
S I E B S I E B
N F G N F G
40 0
1 2 3 1 2 3
Number of seasons Number of seasons

Figure 1: Learning curves for the proportion of correctly predicted relative positions and the
mean squared error in terms of positions, where we vary the size of the training set.

100 25
S I E B
Relative positions %

20 N F G
MSE positions

90
15
80
10
70 5
S I E B
N F G
60 0
25 50 75 25 50 75
Matches played % Matches played %

Figure 2: Learning curves for the proportion of correctly predicted relative positions and the
mean squared error in terms of positions, where we vary the amount of played matches.

Q2: How well does each model perform on predicting the final league table?
Figure 1 shows learning curves for the proportion of correctly predicted relative positions and
the mean squared error in terms of positions, where we vary the size of the training set from one
season to three seasons. For the former metric, the PM performs best in 18 of the 21 settings
and the SEM in the remaining three settings. For the latter metric, the EM performs best in
17 of the 21 settings, the PM in three settings, and the SEM in one setting.

Q3: What proportion of the matches needs to be played to yield useful predictions?
Figure 2 shows learning curves for the proportion of correctly predicted relative positions and
the mean squared error in terms of positions, where we vary the number of matches played in the
final season. For example, 25% means that the model is trained on the results for the 2012/2013
season as well as the results for the first quarter of the 2013/2014 season. Unsurprisingly, the
predicted final league tables become more accurate as more matches have been played. For the
former metric, the EM performs best in 15 of the 21 settings and both the SEM and PM in
three settings. For the latter metric, the EM performs best in 18 of the 21 settings and the
SEM in the remaining three settings.

4 Conclusions
This paper investigates whether popular predictive rankings are capable of accurately predict-
ing the final league tables in football leagues. The experimental evaluation shows that the
proportion of correctly predicted relative positions and the mean squared error in terms of
positions are the most natural metrics to assess predicted league tables. Furthermore, the eval-
uation shows that the traditional Elo-rating system performs particularly well in most settings
we considered. While the Pi-rating model performs reasonably well in many settings as well, it
excels when only match results from previous seasons are available.
The suggested directions for future research include devising more sophisticated metrics to
assess the predicted final league tables (e.g., assigning weights to positions such that positions
qualifying for European football become more important) and comparing to more advanced
predictive ranking systems (e.g., systems that also consider performance data).

Acknowledgments
Jan Van Haaren is supported by the Agency for Innovation by Science and Technology in Flan-
ders (IWT). Jesse Davis is partially supported by the Research Fund KU Leuven (OT/11/051),
EU FP7 Marie Curie Career Integration Grant (#294068) and FWO-Vlaanderen (G.0356.12).

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