View metadata, citation and similar papers at to you byCOREprovided by Leeds Beckett RepositoryHidden dynamics of soccer leagues: the predictive ‘power’ of partial standingsClive B. Beggs1, Alexander J. Bond1, Stacey Emmonds1, Ben Jones1,2,3,4,5,6Institute for Sport, Physical Activity and Leisure, School of Sport, Leeds Beckett University,Leeds, West Yorkshire, United Kingdom.2Yorkshire Carnegie Rugby Union club, Leeds, United Kingdom.3England Performance Unit, The Rugby Football League, Leeds, United Kingdom.4Leeds Rhinos Rugby League club, Leeds, United Kingdom;5School of Science and Technology, University of New England, Armidale, New SouthWales, Australia;6Division of Exercise Science and Sports Medicine, Department of Human Biology, Facultyof Health Sciences, the University of Cape Town and the Sports Science Institute of SouthAfrica, Cape Town, South Africa1Running Title: Hidden dynamics of soccer leaguesCorresponding AuthorProf. Clive BeggsInstitute for Sport, Physical Activity and LeisureSchool of SportHeadingley CampusLeeds Beckett UniversityLS63QSe.mail: [email protected]

AbstractObjectivesSoccer leagues reflect the partial standings of the teams involved after each round ofcompetition. However, the ability of partial league standings to predict end-of-season positionhas largely been ignored. Here we analyze historical partial standings from English soccer tounderstand the mathematics underpinning league performance and evaluate the predictive‘power’ of partial standings.MethodsMatch data (1995-2017) from the four senior English leagues was analyzed, together withrandom match scores generated for hypothetical leagues of equivalent size. For eachseason the partial standings were computed and Kendall’s normalized tau-distance andSpearman r-values determined. Best-fit power-law and logarithmic functions were applied tothe respective tau-distance and Spearman curves, with the ‘goodness-of-fit’ assessed usingthe R2 value. The predictive ability of the partial standings was evaluated by computing thetransition probabilities between the standings at rounds 10, 20 and 30 and the final end-ofseason standings for the 22 seasons. The impact of reordering match fixtures was alsoevaluated.ResultsAll four English leagues behaved similarly, irrespective of the teams involved, with the taudistance conforming closely to a power law (R2 0.80) and the Spearman r-value obeying alogarithmic function (R2 0.87). The randomized leagues also conformed to a power-law, buthad a different shape. In the English leagues, team position relative to end-of-seasonstanding became ‘fixed’ much earlier in the season than was the case with the randomizedleagues. In the Premier League, 76.9% of the variance in the final standings was explainedby round-10, 87.0% by round-20, and 93.9% by round-30. Reordering of match fixturesappeared to alter the shape of the tau-distance curves.ConclusionsAll soccer leagues appear to conform to mathematical laws, which constrain the leaguestandings as the season progresses. This means that partial standings can be used topredict end-of-season league position with reasonable accuracy.(Abstract word count: 298)KeywordsSoccer; Premier League; partial standings; predicting team performance; transitionprobabilities.

Hidden dynamics of soccer leagues: the predictive ‘power’ of partial standingsIntroductionFor all the complexities of the game itself, how the outcome of soccer matches are recordedand evaluated is a relatively simple process, with most teams belonging to a league thatreflects the standings of the respective clubs after each round of competition. In a typicalsoccer league teams are awarded three points for a win, one point for a draw and no pointsfor loosing, with the respective partial standings (i.e. the standings part way through theseason) at any point in time based on the number of points accumulated. The only variationto this simple rule comes when teams tie on the number of points accumulated. InBundesliga (Germany), Ligue 1 (France) and the English leagues when ties occur, thedifference between the total number of goals scored and conceded is used to ‘fine tune’ theranking order of the teams, whereas in other leagues, such as La Liga (Spain), ties may besettled using the head-to-head goal difference or points depending on the number of teamsinvolved in the tie. Although much money, time and effort is expended by club owners,coaches, players, pundits and fans considering strategy and tactics in an attempt to predictand influence the outcome of matches, it is often forgotten that in a league the only outcomethat ultimately matters is the final position at the end of the season. Although only one teamcan win the league, the top two or three teams (it varies from league to league) may bepromoted to a higher division, or selected to compete in an international competition, such asthe Union of European Football Associations (UEFA) Champions League, while those in thebottom three or four places are generally relegated to a lower division. Other variationsinclude the introduction of ‘play-off’ positions for the tranche of teams just outside theautomatic promotion positions, such as used in the English Championship and Englishleagues one and two. As such, it is the final league standing, rather than the pointsaccumulated per se, that is the measure by which teams and their managers are judged.Indeed, given the financial implications, it is no surprise that the final standings matter. In theEnglish Premier League, for example, each additional placing in the final standings is worth 1.9 million (2017 data) [1], no matter the magnitude of the points differences betweenadjacent teams. Furthermore, when it comes to issues of promotion and relegation thefinancial implications may be immense, with, in some cases, relegated clubs forced intoliquidation [2].Given the huge implications (both financial and reputational) of final league standing, itperhaps comes as no surprise that many stakeholders (e.g. clubs, coaches, fans,bookmakers, gamblers, amongst others) have a keen interest in predicting final leagueposition. After all, if club owners and executive decision-makers can predict with some

confidence, early in a season, the likely final position of their team, then they may be able tomake adjustments to accommodate or avert a negative outcome, as well as setting realistictargets and performance indicators. Indeed many people, particularly those associated withthe gambling industry, have developed methodologies for ranking sporting teams andpredicting team performance. While many of these methodologies remain unpublishedbecause of confidentiality issues, others are in the public domain. For example, wellestablished mathematical techniques such as the Colley [3-5] and Keener [4-6] algorithmsaim to rank teams based on past performance in fractured competition, while techniquessuch as the Pythagorean expectation system [7, 8] attempt to predict future leagueperformance based on past performance. Still other techniques, such as the Bradley-Terry[9, 10] and Elo [4, 11] methods, seek to predict the outcome of single matches using aprobabilistic methodology. The Bradley-Terry and Elo methods have also been adapted foruse as ranking systems [11, 12]. In this capacity, the Bradley-Terry and Elo systems consideronly the outcome of the matches played when rating teams, whereas the Keener and Colleymethods utilize respectively, the total number of scores for and against, and the number ofwins and losses achieved, when determining rankings. The Elo system has also been usedto predict the final league position of soccer teams [13]. By comparison relatively little workhas been done on the dynamics of league tables themselves and the factors that influencetheir behaviour. Indeed, much of the work that has been done in this field originates in theUSA to assess team performance in high scoring sports such as American football, baseballand basketball [14-18] – sports that exploit a conference system, rather than the traditionalleagues preferred in soccer. So while much work has focused on US sports, less work hasbeen published concerning soccer, and that which has been done has tended to focus onpredicting match outcomes [19-25] and computing expected points totals [26-28] rather thanfocusing on the behaviour of the partial standings. Having said this, a few researchers haveinvestigated the structure of soccer leagues. Notably, Lasek and Gagolewski [29] found thatthe traditional ‘round-robin’ league format, in which each team plays the other teams twice,once at home and once away, was not as efficacious at ranking teams as a two-stagecompetition. Shin et al. [30] comparing the dynamics of the English Premier (soccer) Leaguewith those of the National (American) Football League (NFL) in the USA, observedsimilarities between the two leagues despite the marked differences between the sports andthe structures of the two competitions. As such, this suggests that the performance of soccerteams might be constrained by the structure of the league competitions in which theycompete.In mathematical terms, soccer leagues can be viewed as a discrete state-space in which thecompeting teams change ranking state (i.e. standing position) as each round of competitionis completed. So for example, the English Premier League, which consists of 20 teams who

each play 38 games per season, can be viewed as a [20 38] matrix in which each team canoccupy one of 20 unique states at the end of each round of competition. When viewed in thisway it can be shown that leagues possess an inherent dynamic that constrains theirbehaviour, making it more difficult for teams to change league position as the seasonprogresses [30]. This phenomenon is well illustrated in Figure 1, which shows how thestandings of the teams in the English Premier League altered over the 38 rounds ofcompetition during the 2016-17 season. From this, it can be seen that at the start of theseason there was considerable crossing-over in partial standing position between the teamsas they progressed from round to round. However as the competition progressed, thenumber of ‘cross-over’ events greatly diminishes, with only relatively few occurring afterround 30. This is reflected in the normalized Kendall’s tau distance plot in Figure 2 for thesame season, which shows the number of changes in ranking order between the standingsfor successive rounds of competition as the season progressed. This reveals that a rapiddecrease in tau distance occurred at the start of the season, which subsequently sloweddown as the season progressed, with little change occurring in the latter part of the season. Ifone also considers the Spearman rank correlations (r-values) between the team standingsafter each round of competition with the final standings at the end of the season (see Figure2), it can be seen that the resulting correlation curve for season 2016-17 mirrors the taudistance curve, revealing that r-values 0.8 were achieved by round 10, with only relativelymodest changes occurring after that. Given this, it may be that the partial standingsthemselves have potential as predictors of league outcome. However, there is a paucity ofpublished work on the dynamics of partial standings in soccer leagues, with Shin et al.’s [30]study of the English Premier League being a notable exception. This study, although avaluable contribution, was nonetheless small, evaluating just the Premier League forseasons 2011-12 and 2012-13. As such, many unanswered questions remain concerning thegeneral applicability of their findings. Consequently, there is a need for a morecomprehensive study to evaluate whether or not the dynamics observed by Shin et al. arespecific to the English Premier League or generally applicable to all soccer leagues. Wetherefore undertook the extensive study reported here, with the aim of better understandingthe dynamics associated with partial standings in soccer leagues and assessing theirpotential for predicting league outcomes.MethodsThe analysis utilized an ‘in-sample’ study design and was performed using publicly availablehistorical match data (acquired from for all four senior leagues in

English soccer (Premier League, Championship, League 1, and League 2 – details listed inTable 1) for the 22 seasons from 1995-96 to 2016-17. 1995-96 was chosen as a cut-off pointbecause this was the first season in which the Premier League contained just 20 teams –prior to this it comprised 22 teams. Analysis was performed using bespoke ‘in-house’programs written in ‘R’ (version 3.0.2; open source statistical software), which incorporatedan algorithm for computing league standings after each round of competition directly frommatch results [31]. In keeping with the system used in the English Premier League, teamswere ranked firstly by total points, then by goal difference, and finally by goals scored, if theother two metrics were tied.For the respective leagues in each season the team standings after every round ofcompetition (i.e. the partial standings) were computed. These were then used to compute: (i)the changes in the normalized Kendall’s tau distance (equation 1) between the teamstandings in successive rounds of competition; and (ii) the changes in the Spearman r-value(equation 2) between the team standings after each round of competition with the finalstandings at the end of the season. Kendall’s tau distance, which is a metric that counts thenumber of pair order disagreements between two ranking lists [32] was used because itdirectly reflects the number of changes in ranking order that occurred between the teamstandings in successive rounds of competition. In order to allow comparisons to be madebetween the various leagues, the tau distance was normalized to yield the fraction ofdiscordant pairs as follows:τ ndn(n 1) / 2(1)where, τ is the normalized Kendall’s tau distance; n is the number of pairs of observations;and nd is the number of discordant pairs.In order to quantify the variance explained by the respective partial standings after eachround of competition, the Spearman’s rank correlation coefficient was computed as follows:6 d irs 1 n(n 2 1)2(2)where, rs is the Spearman’s rank correlation coefficient; and di is the difference between thetwo ranks for each pair of observations.In order to describe the normalized tau distance and Spearman correlation curves for the

respective leagues, several metrics were computed which captured various characteristics ofthe curves, such as the area under the curve (AUC), and the standardized AUC (i.e. AUCdivided by the number of rounds of competition). Power law and logarithmic functions werealso applied to the tau distance and Spearman correlation curves to establish the leastsquares ‘best-fit’ curves for each league in the respective seasons. For the tau distancecurves, the ‘best-fit’ conformed to the power law function:τ b wa(3)where; τ is the normalized tau distance; w is the round of competition; a is the powercoefficient; and b is the multiplier coefficient. While the ‘best-fit’ for the Spearman correlationcurves of the leagues approximated to a logarithmic function of the form:rs d ln (w) c(4)where; r is the Spearman r-value; w is the round of competition; c is the intercept coefficient;and d is the logarithmic coefficient.For each season, coefficients a and b in the power law ‘best-fit’ function were determined bylog transforming τ and w, and applying the ‘polyfit’ function in the ‘pracma’ package [33] in‘R’. Coefficients c and d in the logarithmic function were computed using a linear model withw log transformed. Goodness-of-fit for the respective ‘best-fit’ curves was assessed using theR2 value.In order to compare the dynamics of the four English leagues with a ‘baseline’, weconstructed two hypothetical random leagues, one containing 20 teams and the othercontaining 24 teams. Computation of the respective ‘round-by-round’ partial standings forthese random leagues was done by applying the same methodology as that outlined for thereal leagues above [31], but using random home and away match scores instead of historicalones. For the 20-team random league, this involved generating 380 random home scoresand 380 random away scores, both drawn from a Poisson distribution with a mean value of1.325 (i.e. half of the average number of goals scored per match (mean 2.650) for thePremier League over the 22-seasons), whereas for the 24-team league, 552 random homeand away scores were generated from a Poisson distribution with a mean value of 1.291,which was half the 22-season average value of 2.582 goals per match for all the otherleagues. We then attributed points (three points for a win, one for a draw, and zero points fora loss) to the random match results and computed the standings after each round of

competition for both leagues.Importantly, in order to ensure that the behaviour of therandom leagues was completely random and unbiased, no weighting was given for homeadvantage when generating the random results. In this way we could observe how thestandings in a 20 or 24-team league would behave if the results were completely random. Inorder to simulate the behaviour of the random leagues over an equivalent number ofseasons to the real leagues, this process was repeated 22 times and the resultingnormalized tau distance and Spearman correlation curves acquired for each season. For therandom leagues, the ‘best-fits’ for both the normalized tau distance and Spearman correlationcurves corresponded to a power law function of the form shown in equation 3.To assess whether or not the order in which matches were played had any impact on thedynamic behaviour of the partial standings, post-hoc analysis was performed using PremierLeague data for season 2016-17. This involved computing the partial standings using thehistorical match results for this season, but with the order in which the matches were playedchanged. Specifically, we restructured the fixture list for Premier League for season 2016-17,so that the top 10 pre-season ranked teams played the teams ranked 11-20 during the firsthalf of the season, with second half of the season reserved solely for fixtures in which the top10 pre-season ranked teams played each other, and the lower ranked teams also playedeach other. In addition, we randomized the fixtures for season 2016-17, to create 22separate ‘shuffled’ leagues. For all the real, restructured and shuffled leagues the normalizedtau distances of the partial standings were computed, together with the respective ‘best-fit’curves.In addition to computing the partial standings, the end-of-season points totals for all theteams in the real and random leagues for the respective seasons were computed using thepoints allocation system outlined above. From this we were able to calculate the mean andstandard deviation of the end-of-season points totals for each place in the final standings ofthe respective real and random leagues. The mean points difference between the adjacentplaces in the end-of-season standings was also computed for the real leagues.Statistical analysis of the real and random leagues was performed using a two-tailed paired ttest applied to key metrics (coefficients a, b, c and d from the respective ‘best-fit’ models,AUC, and standardized AUC) derived from the individual tau distance and Spearmancorrelation curves for the respective seasons. The difference in the points totals for the teamsin the real and randomized leagues was also assessed using a two-tailed paired t-test. For alltests, p values 0.05 were deemed to be significant.In order to assess the predictive ‘power’ of the partial standings throughout the season we

used the ‘round-by-round’ team standings computed for the respective real leagues for all 22seasons from 1995-96 to 2016-17. Specifically, we used the partial standings at rounds 10,20 and 30 and the final end-of-season standings to compute the respective transitionprobabilities. This was done by creating a separate adjacency matrix, A, with dimensions [20 20] for the Premier League and [24 24] for the other three leagues, for each of the threerounds (i.e. rounds 10, 20 and 30). For each respective round, matrix A contained the totalnumber of transitions that occurred between the various partial and final standings over the22 seasons, such that Ai,j represented the number of times a team transitioned from jth placeafter either round 10, 20 or 30, to ith place at the end of the season. The rows and columns ofthe adjacency matrix each summed to 22.Having constructed matrix A, the transition probability matrix, T, was computed as follows: 1 T A 22 (5)The rows and columns of the transition probability matrix both summed to one, with eachelement, Ti,j, representing the historical probability of a team finishing the season in ith placegiven a jth place partial standing after round 10, 20 or 30.ResultsFigure 3 shows the combined normalized tau distance results for the Premier League andChampionship for seasons 1995-2017, together with the corresponding curves for therandom 20-team and 24-team leagues. From this it can be seen that both the PremierLeague and Championship, together with the random leagues all have ‘best-fit’ curves ofsimilar shape which conform to a power law. For the Premier League the correlation betweenthe normalized tau distance and the power law curves for the respective seasons was R2 0.803 (SD 0.079), whereas that for the: Championship was R2 0.838 (SD 0.066);League 1 was R2 0.833 (SD 0.089); and League 2 was R2 0.816 (SD 0.080).Interestingly, while the normalized tau distance curves for the real and random leagues bothconform to a power law, those for the real league declined more rapidly than those for therandom leagues, indicating that in the real leagues the standings tend to become ‘fixed inposition’ relatively early in the season, with less ‘crossing-over’ in the team position occurringthan would otherwise be the case if the results were completely random. With the exceptionof coefficient b, which did not reach significance for League 2, the differences between the

real and corresponding random leagues were significant (p 0.05) for all the recorded taudistance metrics (i.e. AUC, standardized AUC, coefficient a, and coefficient b).The comprehensive normalized tau distance results for all four real leagues and the tworandom leagues are presented in Figure 4 and Tables 2 and 3. These reveal that all the real24-team leagues produced tau distance curves that were almost identical, with no significantdifferences occurring between any of the key metrics. Likewise, the tau distance metrics forthe 20-team Premier League were similar to those of the other English leagues, with the onlymajor difference being that the AUC metric was significantly lower for the Premier League(p 0.001), reflecting the fact that this league has only 38 rounds of competition, comparedwith 46 rounds for the other leagues. However, when the AUC was standardized, thisdifference disappeared. The only other significant difference between Premier League andthe other leagues related to coefficient a, the absolute magnitude of which was slightlygreater for the Premier League compared with that for the Championship (p 0.046) andLeague 2 (p 0.004). Collectively, these findings suggest that with respect to the normalizedtau distance dynamics all the English 24-team leagues behaved in a very similar manner,with the Premier League also similar but with a tau distance curve that has a slightly differentshape (Figure 4).The Spearman correlation results for the real and random leagues shown in Figure 5 andTables 4 and 5 reveal a similar picture to the tau distance results, with notable exception thatthe correlation curves for the real leagues approximated to a logarithmic function, whereasthose of the random leagues conformed to a power law. For the Premier League thecorrelation between the Spearman r-value curves and the ‘best-fit’ logarithmic curves was R2 0.874 (SD 0.058), whereas that for the: Championship was R2 0.899 (SD 0.059);League 1 was R2 0.884 (SD 0.058); and League 2 was R2 0.885 (SD 0.082). Unlikethe real leagues, the Spearman correlation curves of the random leagues conformed veryclosely to a power law, with post-hoc fitting of a power law function to the mean ensemblecurve yielding R2 0.957 for the 20-team random league and R2 0.969 for the 24-teamleague. For all the real leagues the AUC was significantly greater (all p 0.001) than that forthe corresponding random leagues. As such, this mirrors the tau distance results andindicates that for the real leagues the positions of the teams relative to their final standings,tends to become fixed earlier in the season than would otherwise be the case if the resultswere completely random.While little difference was observed in the Spearman correlation behaviour of the three 24team leagues, noticeable differences were observed between the Premier League and theother leagues, with for example the intercept (coefficient c) and logarithmic (coefficient d)

coefficients being larger (p 0.004 and p 0.026 respectively) in the Premier Leaguecompared to the Championship. This indicates that with the Premier League the position ofthe teams relative to their final standings tends to become fixed earlier in the season than inthe other 24-team leagues, which all broadly behave in a similar manner.The results of the post-hoc analysis in which the fixtures for season 2016-17 were firstlyrestructured and then randomly shuffled are presented in Figure 6. These reveal thatchanging the order in which the matches were played, while keeping the match resultsunchanged, did indeed alter the behaviour of the partial standings of the Premier League.While all the normalized tau distance curves closely conformed to a power law (real league,R2 0.734; restructured league, R2 0.761, and shuffled leagues, R2 0.789 (SD 0.116)),it was noticeable that reordering the games did change the shape of the ‘best-fit’ curves. Forthe real Premier League in season 2016-17 the best-fit curve had a power coefficient(coefficient a) -0.455 and a multiplier coefficient (coefficient b) 0.176, whereas for therestructured league the corresponding values were –0.460 and 0.120 respectively. Thepower and multiplier coefficients for the shuffled leagues, where -0.635 (SD 0.082) and0.246 (SD 0.044) respectively.The average points totals per place per season for the real and random leagues arepresented in Table 6. From this it can be seen that the ‘total points’ distributions for therespective real leagues are markedly different from those for the corresponding randomleagues. Noticeably, for each real league the top and bottom teams achieved much higherand lower points totals respectively than the teams with the same standings in thecorresponding random leagues. As such, this indicates that in real life, points acquisition isnot a random process, with the teams at the top of the leagues primarily prospering at theexpense of the teams at the lower end of the leagues. This is supported by the resultsdisplayed in Figure 7, which show the mean points differences between each of therespective end of season standings. These reveal that for all four leagues, the distributionexhibits a slightly skewed ‘U’ shape, with the greatest points difference between therespective standings occurring at either end of the tables.The final standing transition probabilities for respective league positions at rounds 10, 20 and30 of competition for the Premier League, Championship, League 1, and League 2 arepresented in Tables 7-10. The probabilities presented in these tables are based on historicaldata (1995-2017) and show the likely final standing positions for all the league positions afterrounds 10, 20 and 30. So for example, it can be seen that for the Premier League the teamthat is first at round 10 has a 40.9% chance of finishing the season in first place, an 18.2%chance of finishing in second place, and an 18.2% chance of finishing in third place.

However, by round 30, the team in first place in the Premier League has a 72.7% chance offinishing first and an equal 13.6% chance of coming in second or third in the league.Correspondingly, at the other end of the league, the team in bottom position at round 10 hasonly a 27.3% chance of finishing last, whereas by round 30 this increases to 72.7%. As such,this confirms the findings of the tau distance and Spearman correlation analysis presented inFigures 3 and 5. Inspection of Tables 8-10 for the 24-team leagues reveals a similar pictureto that for the Premier League, with the teams in the top positions increasingly likely to finishnear the top of the league as the season progresses, and those near the bottom more likelyto be relegated as the season progresses.DiscussionFrom the analysis presented above it can be seen that a clear and consistent pictureemerges, namely that all the English soccer leagues exhibit very similar behaviour withregard to the dynamics of the partial standings. For all leagues, the number of ‘cross-over’events occurring, indicated by the change in the normalized tau distance betweensuccessive rounds of competition, rapidly decreases as the season progresses, irrespectiveof the actual teams involved in the competition. In all the leagues, this reduction in taudistance conforms closely to a power law (R2 0.8). This indicates that as the seasoncommences the best teams quickly rise to the top of their respective leagues, while thepoorest teams equally quickly sink to the bottom, with the middle-ranking teams occupyingthe space in between. From this it can be surmised that being more consistent than theircompetitors, the best teams in ea

Clive B. Beggs1, Alexander J. Bond1, Stacey Emmonds1, Ben Jones1,2,3,4,5,6. . soccer league teams are awarded three points for a win, one point for a draw and no points . and basketball [14-18] – sports that