In this study we investigate the recently introduced competition format for the top association football division in Poland (similar to one used in, e.g., Belgium and Kazakhstan). We compare it to the double round-robin tournament which is the most prevalent league format among European leagues. In a simulation study we show that the new league format has better ability to determine the strongest competitor as the winner of a league as well as it yields higher correlation with theoretical latent teams' strength parameters in the model.
The issue of tournament design has several various dimensions. On one hand, one would like to design a competition so as to maximise the probability that the best team overall wins. On the other hand, the uncertainty of the outcome contributes to the excitement accompanying sport contests. Nothing is determined in advance and this very fact makes the competition interesting. Additionally, we are often facing time and space limitations that impose a set of constraints on both the schedule of the contest and the number of games that can be played. Moreover, there are economic factors that are of interest for a tournament organiser.
The design of sport contests have been of interest of authors in various
research areas. For example, Appleton [
In this paper we focus on the predictive e cacy of the league format that was introduced in Polish Ekstraklasa { the top division of the football competition in Poland { as of the 2013/2014 season. We compare it with a standard roundrobin league system that operated previously. The two tournament formats are compared with respect to their ability to produce the strongest contestant as the winner in a simulation study. Additionally, we investigate the level of agreement between the ranking of teams produced at the end of the competition and the one based on the teams' latent strength parameters in a simulation study.
This contribution is structured as follows. In Section 2, we present in detail the new tournament form. In Section 3, we describe a simulation experiment comparing the predictive e cacy of the two systems and in Section 4 we present the obtained results. Finally, the last section concludes the work. 2
Over the years Polish Ekstraklasa operated on most occasions as a double roundrobin tournament. In such a tournament, each team plays against all the other ones twice, home and away. It requires 2 n2 matches to be played, where n denotes the number of teams in a league. Since season 2005/06, 16 teams compete in the top division. However, as of season 2013/14 the competition format has changed. We shall now recall the rules of awarding points adopted with the introduction of the new system.
First of all, the season is divided into two phases. In the rst phase, the teams compete in a standard double round-robin tournament (30 rounds). Next, the league table is divided into two groups: the championship (\top eight") and the relegation (\bottom eight") groups. At this stage, the number of points accumulated by the teams is divided by two (with possible rounding halves up) and the competition is extended to a single round-robin tournament within each group (with additional 7 rounds), which we call the nal round. In this round, the points are awarded in a standard manner: 3 points for a win, 1 points for a draw and 0 points for a loss. The nal ranking of teams is obtained by summing the points from the two rounds (within the championship and relegation group separately). This competition format requires 2 n2 +2 n2=2 games to be played (with n even). Since the number of rounds in the nal round is odd, some teams are playing one more game at home ground. These are the rst four teams in each group after the rst phase of a season. Moreover, in the nal round, the top team in each group plays the second one at home.
As far as the top division leagues in the countries belonging to UEFA { the governing body for association football competition in Europe { are concerned, currently, the round-robin contest is most prevalent tournament format for domestic football competitions. In season 2014/15 the top division leagues in several countries operated in a general two-phase format resembling the one employed in Poland (with some special minor rules' modi cations not discussed here), e.g., in Andorra, Belarus, Belgium, Bulgaria, Cyprus, Israel, Kazakhstan, Macedonia, Scotland and in Wales. In case of the leagues in Belgium and Kazakhstan, the points gained by the teams after the rst phase are divided by two. Competition formats in these two countries are closely related to the league format which is currently in force in Poland. In fact, the Kazakh league is based on exactly the same rules but fewer teams are involved in the competition: 12 as compared to 16 in Poland. 3
Let us discuss the set up of an simulation experiment carried out for comparison of the new league system to the standard round-robin contest. We discuss the choice of model parameters. In the described computations we used data for four Polish league seasons (from 2011/12 to 2014/15) available at http://www. 90minut.pl/. The data on European leagues' summary statistics were obtained from http://www.football-data.co.uk/.
Game outcome model. In the experiment, we employ an ordered logistic
regression as the match results model [
Rij = 8Hij if dij + > <
Dij if dij + >:Aij if dij +
c; 2 ( c; c]; < c; (1) where c > 0 is an intercept and is an i.i.d. (for all the games) random component. Under the assumption that the random component follows the logistic distribution with mean equal to 0 and scale parameter equal to 1, we have: 1 P(Hij ) = 1 1 + e c+dij ;
1 P(Dij ) = 1 + e c+dij
1 P(Aij ) = 1 + ec+dij :
1 1 + ec+dij ; with P( ) denoting the probability of a particular outcome.
The ordinal logistic regression is a simple model for match outcome. Each
team is characterised by a single parameter indicating its overall strength. An
alternative model could be the Poisson regression [
Team ratings' distribution. In our simulations we assume that the team
ratings are samples from a certain probability distribution. In an analogous study,
Ryvkin [
We also propose taking samples from the estimated team ratings from the
last four game seasons (2011/12{2014/15). To this end, we estimate team ratings
for these seasons. Next, we construct a kernel density estimator (KDE) based
on these ratings with the Gaussian kernel. Random variate generation according
to the obtained density estimate is done via sampling with replacement teams'
from the teams' ratings and adding a Gaussian noise term with the standard
deviation equal to the kernel's bandwidth b. To estimate a team's strength
parameters we use the ordered logistic regression model discussed above with
elastic net regularisation [
1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 The prediction error is minimised for parameter setup ( ; ) = (1:5; 1) for this particular season. The value of parameter = 1 means that the Lasso regularisation yields the best performing model.
Model calibration. To set parameters for ratings distribution of the game outcome model discussed above, we look at the overall proportion of (H; D; A) results from European countries' leagues in season 2014/15 (Belgium, England, France, Germany, Greece, Italy, The Netherlands, Poland, Portugal, Scotland, Spain and Turkey). The proportion of the home team wins varies from 40% in Italy up to 53% in Greece. The fraction of draws ranges from 19% in Scottish Premier League to 31% in Italian Serie A. The highest fraction of away teams' wins is observed again in Scotland (36%) and the lowest in Greece (22%). The dispersion parameters of the assumed distribution functions are chosen so as to the simulated proportion of the results is approximately equal to the observed frequencies in the discussed football leagues. The intercept and the home team advantage parameters are set to (c; h) = (0:6; 0:4). In this way, the probabilities of results for equally rated teams rj = rj are equal (P(H); P(D); P(A)) = (0:45; 0:28; 0:27),
which approximately corresponds to the empirical averages observed for 2014/15 Ekstraklasa season equal to (0:46; 0:27; 0:27).
Evaluation metrics. We employ two chosen metrics for comparison of the two discussed competition formats. The main aim of our study is to investigate which of the two league forms (the classic one or the new one) indicates the best team as the winner. For that, we calculate the percentage of simulations in which the best team won. Additionally, we compute Kendall's rank correlation coe cient for the two lists of teams: the rst one based on pre-tournament ordering of teams according to their strength and the second one based on the nal league ranking. We employ it as a measure of concordance between the two rankings. Other metrics studied in the literature can be well used. In this particular study we restrict our attention to the two given metrics and leave other ones for further research.
Based on the results of our simulation experiment, we conclude that the newly introduced format has better predictive e cacy with respect to the both evaluation criteria: the selection of the strongest competition as well as the Kendall's correlation coe cient for the two list of teams { based on their true ratings and the nal league ratings. The di erences in the predictive e cacy of the two systems are signi cant though by not a large margin. We also note that the new system requires more games to be played in a given season. Under the presented simple model, more games played provides better results in the new league format. These results are in line with the principle that the more samples are obtained, the better are the estimates. On the other hand, the results are contradictory to common fans' believes experienced by the authors in the informal conversations: most of them claim that it is harder to win the league now than in the past. However, our simulation results show that the more the teams play, the higher the fraction of wins of the best contestant. The same applies to the Kendall's measure. We also observe that this fraction as well as Kendall's is increasing with the variance of ratings distribution. This means that the lower the competitive balance in the league, the higher proportion of better teams' wins is observed.
Limitations of the study. In this study, we analysed two league formats and
their ability to rank stronger teams higher. We should stress the fact that the
introduction of the new league format changed the rules of the competition. This
may in uence an individual team's behaviour and performance.4 For example,
4 Perhaps one of the most radical e ects of rule changes on the competition is the
example of Barbados{Grenada match in 1994 Caribbean Cup quali ers. Due to
introduction of a special rule that goals scored in extra-time counted double, the
Barbados team deliberately scored an own goal by the end of the match for the
extra-time to take place. This in turn allowed them to win by a margin of two goals
by scoring e ectively one goal, which allowed them to progress to the next round
of the competition (see https://en.wikipedia.org/wiki/Barbados_4-2_Grenada_
(Caribbean_Cup_qualification), last access date July 23, 2015).
in mid-1990's, FIFA { the governing body for association football competition
over the world { introduced 3-points for a win format, replacing an old 2-points
for a win system. Several authors considered the in uence of this change on the
game. The conclusion is that such a change does have an impact on the strategy
and tactics employed by the teams [
Acknowledgements. Jan Lasek would like to acknowledge the support by the European Union from resources of the European Social Fund, Project PO KL \Information technologies: Research and their interdisciplinary applications", agreement UDA-POKL.04.01.01-00-051/10-00 via the Interdisciplinary PhD Studies Program.