Identification of skill in an online game: The case of Fantasy Premier League
IIdentification of skill in an online game: The case of Fantasy Premier League
Joseph D. O’Brien, James P. Gleeson, and David J. P. O’Sullivan
MACSI, Department of Mathematics and Statistics,University of Limerick, Limerick V94 T9PX, Ireland (Dated: September 3, 2020)In all competitions where results are based upon an individual’s performance the question ofwhether the outcome is a consequence of skill or luck arises. We explore this question through ananalysis of a large dataset of approximately one million contestants playing
Fantasy Premier League ,an online fantasy sport where managers choose players from the English football (soccer) league.We show that managers’ ranks over multiple seasons are correlated and we analyse the actions takenby managers to increase their likelihood of success. The prime factors in determining a manager’ssuccess are found to be long-term planning and consistently good decision-making in the face ofthe noisy contests upon which this game is based. Similarities between managers’ decisions overtime that result in the emergence of ‘template’ teams, suggesting a form of herding dynamics takingplace within the game, are also observed. Taken together, these findings indicate common strategicconsiderations and consensus among successful managers on crucial decision points over an extendedtemporal period.
I. INTRODUCTION
Hundreds of millions of people consume sporting con-tent each week, motivated by several factors. These mo-tivations include the fact that the spectator enjoys boththe quality of sport on display and the feeling of eustressarising from the possibility of an upset [1, 2]. This sug-gests that there are two important elements present insporting competition: a high level of skill among playersthat provides aesthetic satisfaction for the spectator andalso an inherent randomness within the contests due tofactors such as weather, injuries, and in particular luck.The desire for consumers to get further value from theirspectating of sporting content has resulted in the emer-gence of fantasy sports [3–6], in which the consumers, or managers as we shall refer to them throughout this ar-ticle, begin the season with a virtual budget from whichto build a team of players who, as a result of partak-ing in the real physical games, receive points based upontheir statistical performances. The relationship betweenthe fantasy game and its physical counterpart raises thequestion of whether those who take part in the formersuffer (or gain) from the same combination of skill andluck that makes their physical counterpart enjoyable.The emergence of large scale quantities of detailed datadescribing the dynamics of sporting games has opened upnew opportunities for quantitative analysis, both froma team perspective [7–14] and also at an individuallevel [15–21]. This has resulted in analyses aiming to de-termine two elements within the individual sports; firstlyquantifying the level of skill in comparison to luck inthese games [9, 22–25] while, secondly, identifying char-acteristics that suggest a difference in skill levels amongthe competing athletes [17, 26]. Such detailed quanti-tative analysis is not, however, present in the realm offantasy sports, despite their burgeoning popularity withan estimated 45.9 million players in the United Statesalone in 2019 [27]. One notable exception is a recentstudy [28], which derived an analytical quantity to de- termine the role chance plays in many contests includingfantasy sports based on American sports, and suggestedthat skill was a more important factor than luck in thegames.Motivated by this body of work, we consider a datasetdescribing the
Fantasy Premier League (FPL) [29], whichis the online fantasy game based upon the top divisionof England’s football league. This game consists of overseven million managers , each of whom builds a virtualteam based upon real-life players. Before proceeding, wehere introduce a brief summary of the rules underlyingthe game, to the level required to comprehend the follow-ing analysis [30]. The (physical) Premier League consistsof 20 teams, each of whom play each other twice, resultingin a season of 380 fixtures split into 38 unique gameweeks ,with each gameweek generally containing ten fixtures. Amanager in FPL has a virtual budget of £100m at theinitiation of the season from which they must build asquad of 15 players from the approximately 600 avail-able. Each player’s price is set initially by the game’sdevelopers based upon their perceived value to the man-ager within the game, rather than their real-life trans-fer value. The squad of 15 players is composed under ahighly constrained set of restrictions which are detailedin Supplementary Note I.In each gameweek the manager must choose 11 play-ers from their squad as their team for that week andis awarded a points total from the sum of the perfor-mances of these players (see Supplementary Table 1).The manager also designates a single player of the 11to be the captain, with the manager receiving doublethis players’ points total in that week. Between consec-utive gameweeks the manager may also make one un-penalised change to their team, with additional changescoming as a deduction in their points total. The price ofa given player then fluctuates as a result of the supply-and-demand dynamic arising from the transfers acrossall managers’ rosters. The intricate rules present multipledecisions to the manager and also encourages longer-term a r X i v : . [ phy s i c s . s o c - ph ] S e p FIG. 1:
Relationship between the performance of managers over seasons of FPL . (a) The relationshipbetween managers’ ranks in the 2018/19 and 2017/18 seasons. Each bin is of width 5,000 with the colourhighlighting the number of managers in each bin; note the logarithmic scale in colour. (b) The pairwise Pearsoncorrelation between a manager’s points totals over multiple seasons of the game, calculated over all managers whoappeared in both seasons.strategising that factors in team value, player potential,and many other elements.In Section II A we analyse the historical performanceof managers in terms of where they have ranked withinthe competition alongside their points totals in multi-ple seasons, in some cases over a time interval of up tothirteen years. We find a consistent level of correlationbetween managers’ performances over seasons, suggest-ing a persistent level of skill over an extended temporalscale. Taking this as our starting point, in Section II B weaim to understand the decisions taken by managers whichare indicative of this skill level over the shorter temporalperiod of the 38 gameweeks making up the 2018/19 sea-son by analysing the entire dataset of actions taken bythe majority of the top one million managers over thecourse of the season. Even at this shorter scale we findconsistent tiers of managers who, on a persistent basis,outperform those at a lower tier.With the aim of identifying why these differences oc-cur, we present (Section II C) evidence of consistentlygood decision making with regard to team selection andstrategy. This would be consistent with some commonform of information providing these skilled managerswith an ‘edge’ of sorts, for example in the US it hasbeen suggested that 30% of fantasy sports participantstake advantage of further websites when building theirteams [31]. Arguably most interesting of all, in Sec- Due to data availability issues at the time of collection such asmanagers not taking part in the entire season, the final numberof managers identified was actually 901,912. We will however,for the sake of brevity, refer to these as the top 1 million man-agers over the course of this article. It is also important to notethat data from previous seasons is unattainable, which is why werestrict this detailed study to the 2018/19 season. tion II D we demonstrate how at points throughout theseason there occurs temporary herding behaviour in thesense that managers appear to converge to consensus ona template team . However, the consensus does not per-sist in time, with managers subsequently differentiatingthemselves from the others. We consider possible rea-sons and mechanisms for the emergence of these templateteams.
II. RESULTSA. Historical Performance of Players
We consider two measures of a manager’s performancein a given season of FPL: the total number of points theirteam has obtained over the season and also their result-ing rank based on this points total in comparison to allother managers. A strong relationship between the man-agers’ performances over multiple seasons of the game isobserved. For example, in panel (a) of Fig. 1 we comparethe ranks of managers who competed in both the 2018-19and 2017-18 seasons. The density near the diagonal ofthis plot suggests a correlation between performances inconsecutive seasons. Furthermore, we highlight specifi-cally the bottom left corner which indicates that thosemanagers who are among the most highly ranked appearto perform well in both seasons. Importantly, if we con-sider the top left corner of this plot it can be readilyseen that the highest performing managers in the 2017-18 season, in a considerable number of cases, did not fin-ish within the lowest positions in the following season asdemonstrated by the speckled bins with no observations.This is further corroborated in panel (b), in which weshow the pairwise Pearson correlation between the total
Season average points − 57.05
Gameweek P o i n t s All Managers
Average Points per Gameweek (a) −100102030 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
Gameweek P o i n t s − A v e r age top top top top By Tier (b)
FIG. 2:
Summary of points obtained by managers over the course of the 2018/19 season. (a) The meannumber of points over all managers for each GW. The shaded regions denote the 95% percentiles of the points’distribution. (b) The difference between the average number of points for four disjoint tiers of manager, the top , , , and , and the overall average points as per panel (a). Note that managers are considered to be inonly one tier so, for example, the top- tier contains managers ranked from 1001 to .points obtained by managers from seasons over a periodof 12 years. While the number of managers who partookin two seasons tends to decrease with time, a considerablenumber are present in each comparison. Between the twoseasons shown in Fig. 1(a) for example, we observe resultsfor approximately three million managers and find a cor-relation of 0.42 among their points totals. Full resultsfrom 13 consecutive seasons, including the number man-agers present in each pair and the corresponding Pearsoncorrelation coefficients, are given in Supplementary Table4. Using a linear regression fit to the total points scoredin the 2018/19 season as a function of the number ofprevious seasons in which the manager has played (Sup-plementary Figure 3) we find that each additional year ofexperience is worth on average 22.1 (R = 0 . ) addi-tional points (the overall winner in this season obtained2659 points). This analysis suggests that while there arefluctuations present in a manager’s performance duringeach season of the game, there is also some consistencyin terms of performance levels, suggesting a combinationof luck and skill being present in fantasy sports just aswas observed in their physical analogue in [28]. B. Focus on Season 2018-19
In Sec. II A we considered, over multiple seasons, theperformance of managers at a season level in terms oftheir cumulative performance over the 38 gameweeks ofeach season. We now focus at a finer time resolution, toconsider the actions of managers at the gameweek levelfor the single season 2018/19, in order to identify ele-ments of their decision making which determined theiroverall performance in the game. The average points earned by all managers through-out the season is shown in Fig. 2(a) along with the 95inter-percentile range, i.e., the values between which themanagers ranked in quantiles 0.025 to 0.975 appear. Thisquantity exhibits more frequent fluctuations about itslong-term average (57.05 points per gameweek) in thelater stages of the season, suggesting that some elementsof this stage of the season cause different behaviour inthese gameweeks. There may of course be many rea-sons for this e.g., difficult fixtures or injuries for generallyhigh-scoring players or even simply a low/high scoringgameweek, which are themselves factors of luck withinthe sport itself (see Supplementary Table 2 for a detailedbreak down of points per gameweek). However, in Sec-tion II C 3 we analyse an important driver of the fluctu-ations related to strategic decisions of managers in thesegameweeks.In each season some fixtures must be rescheduled dueto a number of reasons, e.g., clashing fixtures in Europeancompetitions, which results in certain gameweeks thatlack some of the complete set of ten fixtures. Such scenar-ios are known as blank-gameweeks (BGW) and their fix-tures are rescheduled to another gameweek in which someteams play twice; these are known as double-gameweeks (DGWs). In the case of the 2018/19 season these BGWstook place in GWs 27 (where there were eight fixtures),31 (five fixtures), and 33 (six fixtures), making it difficultfor some managers to have 11 starting players in theirteam. The DGWs feature some clubs with two gamesand therefore players in a manager’s team who feature inthese weeks will have twice the opportunity for points; inthe 2018/19 season these took place in GWs 25 (where11 games were played), 32 (15), 34 (11), and 35 (14).We see that the main swings in the average number ofpoints are actually occurring in these gameweeks (asidefrom the last peak in GW 36 which we will comment onlater in the article). In Section II C 3 we show that themanagers’ attitude and preparation towards these game-weeks are in fact indicators of their skill and ability as afantasy manager.To analyse the impact of decision-making upon finalranks, we define tiers of managers by rank-ordering themby their final scores and then splitting into the top ,top , top , and top positions. These dis-joint tiers of managers, i.e., the top is the managerswith ranks between 1 and 1000, the top those withranks between 1001 and 10,000 and so on, range fromthe most successful (top ) to the relatively unsuccess-ful (top ) and so provide a basis for comparison (seeSupplementary Table 2 and Supplementary Table 3 forsummaries of points obtained by each tier). The aver-age performance of the managers in each tier (relative tothe baseline average over the entire dataset) are shown inpanel (b) of Fig. 1. Note that the points for the top tier are generally close to zero as the calculation of thebaseline value is heavily dependent upon this large bulkof managers. A detailed summary of each tier’s points to-tal, along with visualisation of the distribution of pointstotal may be found in Supplementary Table 1 and Sup-plementary Figure 1. It appears that the top tier man-agers outperform those in other tiers, not only in specificweeks but consistently throughout the season which re-sults in the competition for places in this top tier moredifficult to obtain as the season progresses (Supplemen-tary Figure 2). This is particularly noticeable in the firstgameweek, where the top managers tended to per-form very strongly, suggesting a high level of preparation(in terms of squad-building) prior to the physical leaguestarting. We also comment that the largest gaps betweenthe best tier and the worst tier occur not only in two ofthe special gameweeks (DGW 35 and BGW 33) but alsoin GW 1, which suggests that prior to the start of theseason these managers have built a better-prepared teamto take advantage of the underlying fixtures. We notehowever that all tiers show remarkably similar temporalvariations in their points totals, in the sense that they allexperience simultaneous peaks and troughs during theseason. See Supplementary Table 2 for a full breakdownof these values alongside their variation for each game-week.Having identified both differences and similarities un-derlying the performance in terms of total points for dif-ferent tiers of managers we now turn to analysis of theactions that have resulted in these dynamics. C. Decision-Making
1. Transfers
The performance of a manager over the season may beviewed as the consequence of a sequence of decisions thatthe manager made at multiple points in time. These deci-sions include which players in their squad should feature lllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll −2000200400600 t op t op t op t op S u m o f P o i n t s f r o m T r an s f e r s Distribution of Total Points from Transfers (a)
Better decision making L i k e li hood o f w o r s e de c i s i on −2 −1 Fraction of Better Transfers
CCD F top top top top Quality of Transfers (b) llllll lllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll 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t op t op t op t op S u m o f P o i n t s f r o m C ap t a i n cy Distribution of Total Points from Captaincy (c)
FIG. 3:
Decisions of managers by tier. (a) Distributions of the total net points earned bymanagers in the gameweek following a transfer, i.e., thepoints scored by the player brought in minus that of theplayer transferred out. The average net points for eachtier is also shown below; note the difference between thetop three tiers and the bottom tier. (b) Distribution ofthe fraction of better transfers a manager could havemade based upon points scored in the followinggameweek. Faster-decreasing distributions reflectmanagers in that tier being more successful with theirtransfers. (c) The distribution of points from captaincyalong with the average total for each tier.in the starting team, the formation in which they shouldset up their team, and many more. In the following sec-tions we consider multiple scenarios faced by managersand show that those who finished within a higher tiertended to consistently outperform those in lower tiers.One decision the manager must make each gameweekis whether to change a player in their team by using atransfer. If the manager wants to make more than onetransfer they may also do so but at the cost of a pointsdeduction for each extra transfer. The distribution oftotal points made from transfers, which we determineby the difference between points attained by the playerthe manager brought in for the following gameweek com-pared to the player whom they transferred out, over theentire season for each tier is shown in Fig. 3(a). Theaverage number for each tier is also shown. To furtheranalyse this scenario we calculate, for each gameweek, thenumber of better transfers the managers could have madewith the benefit of perfect foresight, given the player theytransferred out. This involves taking all players witha price less than or equal to that of the player trans-ferred out and calculating the fraction of options whichwere better than the one selected, i.e., those who receivedmore points the following gameweek (see Methods). Fig-ure 3(b) shows the complementary cumulative distribu-tion function (CCDF) of this quantity for each tier, notethe steeper decrease of the CCDFs for the higher tiersimplies that these managers were more likely to choose astrong candidate when replacing a player.A second decision faced by managers in each gameweekis the choice of player to nominate as captain, which re-sults in the manager receiving double points for this play-ers’ actions during the GW. This is, of course, a difficultquestion to answer as the points received by a playercan be a function of both their own actions i.e., scor-ing or assisting a goal, and also their team’s collectiveperformance (such as a defender’s team not conceding agoal). This topic is an identification question which maybe well suited to further research making use of the datadescribing the players and teams but with additional dataabout active managers who are making the same decision.For example, an analysis of the captaincy choice of man-agers based upon their social media activity was recentlypresented in [32] and showed that the wisdom of crowds concept performs comparably to that of the game’s topmanagers. Panel (c) of Fig. 3 shows the distribution ofpoints obtained by managers in each tier from their cap-taincy picks. Again we observe that the distribution ofpoints obtained over the season is generally larger forthose managers in higher tiers.
2. Financial Cognizance
The financial ecosystem underlying online games hasbeen a focus of recent research [33, 34]. With this inmind, we consider the importance of managers’ finan-cial awareness in impacting their performance. As men- l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l −20246 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
Gameweek C hange i n T ea m V a l ue ( M ) All Managers
Average Change in Team Value (a)
Team Value (M) G a m e w ee k Top MillionTop 10k
Top 10k vs. Top 1M
Team Value Distribution (b)
Change in Team Value F i na l P o i n t s Gameweek 19
Final Points and Team Value (c)
FIG. 4:
Analysis of the team value of managers. (a) The change in average team value from the initial£100M of all managers, along with 95 percentiles; notethe general upward trend of team value over the courseof the season. (b) Distributions of team values for eachgameweek for those who finished in the top tenthousand positions (i.e., the combination of those in thetop and tiers) versus lower-ranked managers.The distribution for those with higher rank is generallyto the right of that describing the other managers froman early stage of the season, indicating higher teamvalue being a priority for successful managers. (c) Therelationship between a manager’s team value at GW 19versus their final points total, where the heat mapindicates the number of managers within a given bin.The black line indicates the fitted linear regression line,showing that an increase in team value by £1M at thispoint in the season results in an average final pointsincrease of 21.8 points. G W Top MillionTop 10k
Bench Boost Chip (a) P r obab ili t y Top MillionTop 10k
Distribution of Points from Bench Boost (b)
FIG. 5:
Summary of use and point returns of the bench boost chip.
The managers are grouped into twogroups: those who finished in the top ten-thousand positions (Top 10k) and the remainder (Top Million). (a)Fraction of managers who had used the bench boost chip by each gameweek. We see a clear strategy for use indouble gameweek 35, particularly for the top managers, 79.4% of whom used it at this stage. (b) Distribution ofpoints earned from using this chip along with the average points—23.2 for the Top 10k and 13.8 for the TopMillion—shown by the dashed lines.tioned previously, each manager is initially given a bud-get of £100 million to build their team, constrained bythe prices of the players which, themselves fluctuate overtime. While the dynamics of player price changes occurvia an undisclosed mechanism, attempts to understandthis process within the community of Fantasy PremierLeague managers have resulted in numerous tools to helpmanagers predict player price changes during the season,for example see [35]. The resulting algorithms are in gen-eral agreement that the driving force behind the changesis the supply and demand levels for players.These price fluctuations offer an opportunity for theastute manager to ‘play the market’ and achieve a pos-sible edge over their rivals and allow their budget to bemore efficiently spent (see Supplementary Figure 4 for adescription of player value and their corresponding pointstotals and Supplementary Figure 5 for an indication ofhow the managers distribute their budget by player posi-tion). At a macro level this phenomenon of price changesis governed by the aforementioned supply and demand,but these forces are themselves governed by a number offactors affecting the player including, but not limited to,injuries, form, and future fixture difficulty. As such, man-agers who are well-informed on such aspects may profitfrom trading via what is in essence a fundamental analy-sis of players’ values by having them in their team prior tothe price rises [36]. Interestingly, we note that the gen-eral trend of team value is increasing over time amongour managers as shown in panel (a) of Fig 4 along withcorresponding 95 percentiles of the distribution, althoughthere is an indicative decrease between weeks towards theseason’s end (GWs 31-35) suggesting the team value be-comes less important to the managers towards the gamesconclusion. Equivalent plots for each tier are shown inSupplementary Figure 6. Probing further into the relationship between financeand the managers’ rank, we show in Fig. 4(b) the dis-tribution of team values for the top two tiers (top and top ), compared with that for the bottom twotiers (top and top ) There is a clear divergencebetween the two groups from an early point in the sea-son, indicating an immediate importance being placedupon the value of their team. A manager who has a ris-ing team value is at an advantage relative to one whodoes not due to their increased purchasing power in thefuture transfer market. This can be seen in panel (c) ofFig. 4 which shows the change in team value for managersat gameweek 19, the halfway point of the season, versustheir final points total. A positive relationship appearsto exist and this is validated by fitting an OLS LinearRegression with a slope of 21.8 ( R = 0 . ), i.e., anincrease of team value by £1M at the halfway point isworth, on average, an additional 21.8 points by the endof the game (for the same analysis in other gameweekssee Supplementary Table 5). The rather small R valuesuggests, however, that the variation in a managers’ finalperformance is not entirely explained by their team valueand as such we proceed to analyse further factors whichcan play a part in their final ranking.
3. Chip Usage
A further nuance to the rules of FPL is the presence offour game-chips , which are single use ‘tricks’ that may beused by a manager in any GW to increase their team’sperformance, by providing additional opportunities toobtain points. The time at which these chips are playedand the corresponding points obtained are one observableelement of a managers’ strategy. A detailed descriptionfor each of the chips and analysis of the approach takenby the managers in using them is given in SupplementaryNote V.For the sake of brevity we focus here only on one spe-cific chip, the bench boost . When this chip is played, themanager receives points from all fifteen players in theirsquad in that GW, rather than only the starting elevenas is customary. This clearly offers the potential for alarge upswing in points if this chip is played in an effi-cient manner, and as such it should ideally be used inGWs where the manager may otherwise struggle to earnpoints with their current team or weeks in which many oftheir players have a good opportunity of returning largepoint scores. The double and blank GWs might naivelyappear to be optimal times to deploy this chip howeverwhen the managers’ actions are analysed we see differingapproaches (and corresponding returns).Figure 5 shows the proportion of managers who hadused the bench boost chip by each GW alongside the cor-responding distribution of points the manager receivedfrom this choice, where we have grouped the two highertiers into one group and the remaining managers in an-other for visualization purposes (see Supplementary Fig-ure 10 & Supplementary Figure 11 and SupplementaryTable 7-Supplementary Table 10 for a breakdown of useand point returns by each tier). It is clear that the ma-jority of better performing managers generally focused onusing these chips during the double and blank GWs with79.4% choosing to play their BB chip during DGW35in comparison to only 28.9% of those in the rest of thedataset. We also observe the difference in point returns asa result of playing the chip, with the distribution for thetop managers being centred around considerable highervalues, demonstrating that their squads were better pre-pared to take advantage of this chip. The fact that themanagers were willing to wait until one of the final game-weeks is also indicative of the long-term planning thatseparates them from those lower ranked. Similar resultscan be observed for the other game-chips (SupplementaryTable 8-Supplementary Table 10). We also highlight thata large proportion of managers made use of other chipsin GW36, which was the later gameweek in which therewas a large fluctuation from the average shown in Fig. 2.Finally, we comment on the fact that some managersdid not employ their chips by the game’s conclusionwhich suggests that either they were not aware of themor, more likely, the mangers in question had simply lostinterest in the game at this point. As such, the quan-tity of managers who had not used their chip gives us anaive estimation of the retention rate for active managersin Fantasy Premier League ( . of managers in ourdataset). We note that this is a biased estimate in thesense that our dataset is only considering the top tiersof managers, or at least those who finished in the toptiers, and one would expect the drop out rate to be infact much higher in lower bands. D. Template Team
While the preceding analysis proposes reasons for thedifferences between points obtained by tiers shown inFig. 2, the question remains as to why the managers’gameweek points totals show similar temporal dynam-ics. In order to understand this we consider here theunderlying structure of the managers’ teams. We showthat a majority of teams feature a core group of play-ers that results in a large proportion of teams havinga similar make-up. We call this phenomenon the tem-plate team which appears to emerge at different pointsin the season; this type of collective behaviour has beenobserved in such social settings previously, see, for exam-ple [37, 38]. We identify the template team by using thenetwork structure describing the teams of all managers,which is described by the adjacency matrix A Gij , wherebyan edge between two players i and j appearing in n teamsfor a given gameweek G describes a value in the matrixgiven by A Gij = n . This matrix is similar in nature to theco-citation matrix used within the field of bibliometrics[39], see Fig. 6 for a representation of the process.With these structures in place we proceed to performhierarchical clustering on the matrices in order to iden-tify groups of players constituting the common buildingblocks of the managers’ teams. By performing the algo-rithm with k = 4 clusters we find that three clusters con-tain only a small number of the 624 players, suggestingthat most teams include this small group of core players(see Supplementary Table 6 for the identities of those inthe first cluster each gameweek). Figure 7(a) shows thesize of these first three clusters over all managers for eachgameweek of the season (Supplementary Figure 8 showsthe equivalent values for each tier). To understand thisresult further, consider that at their largest these threeclusters only consist of 5.13% (32/624) of the availableplayers in the game, highlighting that the teams are con-gregated around a small group of players. For an exam-ple representation of this matrix alongside its constituentclusters we show the structure in panel (b) of Fig. 7 forgameweek 38, which was the point in time at which thethree clusters were largest.To further examine the closeness between managers’decisions we consider the Jaccard similarity between setsof teams, which is a distance measure that considers boththe overlap and also total size of the sets for comparison(see Methods for details). Figure 7(c) shows the averageof this measure over pairwise combinations of managersfrom all tiers and also between pairs of managers who arein the same tier. Fluctuations in the level of similarityover the course of the season can be seen among all tiersindicating times at which teams become closer to a tem-plate followed by periods in which managers appear todifferentiate themselves more from the peers. Also notethat the level of similarity between tiers increases withrank suggesting that as we start to consider higher per-forming managers, their teams are more like one anothernot only at certain parts of the season but, on average,FIG. 6: Schematic representation of the approaches taken to identify similarity between thecomposition of managers’ teams in each GW.
We view the connections between managers and players as abipartite network such that an edge exists if the player is in the managers’ team. To determine the relationshipbetween players’ levels of popularity we use the co-occurrence matrix which has entries corresponding to the numberof teams in which two players co-appear. Using this matrix we perform hierarchical clustering techniques to identifygroups of players who are similarly popular within the game, where the number of clusters is determined byanalysing the within-cluster sum of squared errors. The similarity between the teams of two managers is determinedby calculating the Jaccard similarity, which is determined by the number of players that appear in both teams.over its entirety (see Supplementary Figure 9 for corre-sponding plots for each tier individually). The high levelof similarity between the better managers’ teams in thefirst gameweek (and the corresponding large points totalsseen in Supplementary Figure 1) is particularly interest-ing given that this is before they have observed a physicalgame being played in the actual season. This suggests asimilar approach in identifying players based purely upontheir historical performance and corresponding value bythe more skilled managers.
III. DISCUSSION
The increasing popularity of fantasy sports in recentyears [27] enables the quantitative analysis of managers’decision-making through the study of their digital traces.The analysis we present in this article considers the gameof Fantasy Premier League, which is played by approxi-mately seven million managers. We observe a consistentlevel of skill among managers in the sense that there ex-ists a considerable correlation between their performanceover multiple seasons of the game, in some cases over thir-teen years. This result is particularly striking given thestochastic nature of the underlying game upon which itis based. Encouraged by these findings, we proceeded to conducta deeper analysis of the actions taken by a large propor-tion of the top one million managers from the 2018-19season of the game. This allowed each decision madeby these managers to be analysed using a variety of sta-tistical and graphical tools. We divided the managersinto tiers based upon their final position in the game andobserved that the managers in the upper echelons con-sistently outperformed those in lower ones, suggestingthat their skill levels are present throughout the seasonand that their corresponding rank is not dependent onjust a small number of events. The skill-based decisionswere apparent in all facets of the game, including mak-ing good use of transfers, strong financial awareness, andtaking advantage of short- and long-term strategic op-portunities, such as their choice of captaincy and use ofthe chips mechanic, see Section II C 3.Arguably the most remarkable observation presentedin this article is, however, the emergence of what we coina template team that suggests a form of common collec-tive behaviour occurring between managers. We showthat most teams feature a common core group of con-stituent players at multiple time points in the season.This occurs despite the wide range of possible optionsfor each decision, suggesting that the managers are actingsimilarly, and particularly so for the top-tier managers as
All Managers − First Three Clusters
Cluster Size by Gameweek (a)
Edge WeightGW38 − All Managers
Structure of Clusters (b) lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll lllll GW A v e r age J a cc a r d S i m il a r i t y N u m be r o f P l a y e r s l l l l l All tiers top top top top All and Within Tiers
Jaccard Similarity of Teams (c)
FIG. 7:
Analysis of team similarity of managers. (a) Size of each of the first three identified clusters overall managers for each gameweek. Note that the firstcluster is generally of size one, simply containing themost-owned player in the game. (b) An example of thenetwork structure of these three clusters for gameweek38, where we can see the ownership level decreasing inthe larger clusters. The diagonal elements of thisstructure are the fraction of teams in which the playeris present. (c) The Jaccard similarity between the tiersof managers and also over all managers; note that thehigher-performing managers tend to be more like oneanother than those in lower tiers, also note thefluctuations in similarity over the course of the seasonindicating that a template team emerges at differenttime points. evident by their higher similarity metrics. Such coordi-nated behaviour by managers suggests an occurrence ofthe so-called ‘superstar effect’ within fantasy sports justas per their physical equivalent [40], whereby managersindependently arrive at a common conclusion on a coregroup of players who are viewed as crucial to optimalplay. A further dimension is added by the fact that thesimilarity between the teams of better managers is evi-dent even prior to the first event of the season, i.e., theyhad apparently all made similar (good) decisions even‘before a ball was kicked’.In this article we have focussed on the behaviour ofthe managers and their decision-making that constitutestheir skill levels. The availability of such detailed dataoffers the potential for further research from a wide rangeof areas within the field of computational social science.For example, analysis of the complex financial dynamicstaking place within the game as a result of the chang-ing player values and the buying/selling decisions madeby the managers would be interesting. A second com-plementary area of research would be the developmentof algorithms that consider the range of possible optionsavailable to managers and give advice on optimizing pointreturns. Initial analysis has recently been conducted [32]in this area, including the optimal captaincy choice ina given gameweek, and has demonstrated promising re-sults.In summary, we believe the results presented here offeran insight into the behaviour of top fantasy sport man-agers that is indicative of both long-term planning andcollective behaviour within their peer group, demonstrat-ing the intrinsic level of skill required to remain amongthe top positions over several seasons, as observed in thisstudy. We are however aware that the correlations be-tween decisions and corresponding points demonstratedare not perfect, which is in some sense to be expecteddue to the non-deterministic nature which makes thesport upon which the game is based so interesting to themillions of individuals who enjoy it each week. Theseoutcomes suggest a combination of skill and luck beingpresent in fantasy sport just as in their physical equiva-lent.
IV. METHODSA. Data Collection
We obtained the data used in this study by access-ing approximately 50 million unique URLs throughthe Fantasy Premier League API. The rankingsat the end of the 2018/19 season were obtainedthrough https://fantasy.premierleague.com/api/leagues-classic/{league-id}/standings/ from which we could obtain the entry ID of thetop 1 million ranked managers. Using these IDs wethen proceeded to obtain the team selections alongwith other manager quantities for each gameweek0of this season that were used in the study through https://fantasy.premierleague.com/api/entry/{entry-id}/event/{GW}/picks/ , we then filteredthe data to include only managers for whom wehad data for the entirety of the season which con-sisted of , unique managers. The data forindividual footballers and their performances were cap-tured via https://fantasy.premierleague.com/api/bootstrap-static/ . Finally, the historical performancedata was obtained for 6 million active managers through https://fantasy.premierleague.com/api/entry/{entry-id}/history/ . B. Calculation of Transfer Quality
In order to calculate the transfer quality plot shownin Fig. 3(b) we consider the gameweeks in which man-agers made one transfer and, based upon the value of theplayer whom they transferred in, determine what frac-tion of players with the same price or lower the managercould have instead bought for their team. Suppose thatin gameweek G the manager transferred out player x i ,who had value q G ( x i ) , for player x j who scored p G ( x j ) points in the corresponding gameweek. The calculationinvolves firstly finding all players the manager could havetransferred in, i.e., those with price less than or equal to q G ( x i ) and then determining the fraction y G ( x i , x j ) ofthese players who scored more points than the chosenplayer given the player whom was transferred out. Thisis calculated by using y G ( x i , x j ) = (cid:80) k [ q G ( x k ) ≤ q G ( x i )] · [ p G ( x k ) > p G ( x j )] (cid:80) (cid:96) [ q G ( x (cid:96) ) ≤ q G ( x i )] , where represents the indicator function. Using thisquantity we proceed to group over the entire season foreach tier of manager which allows us to obtain the dis-tribution of the measure itself and finally the probabilityof making a better transfer which is shown in panel (b)of Fig. 3. C. Team Similarity
With the aim of identifying levels of similarity betweenthe teams of two managers i and j we make use of theJaccard similarity which is a measure used to describethe overlap between two sets. Denoting by T Gi the set ofplayers that appeared in the squad of manager i duringgameweek G we consider the Jaccard similarity betweenthe teams of managers i and j for gameweek G given by J G ( i, j ) = (cid:12)(cid:12) T Gi ∩ T Gj (cid:12)(cid:12)(cid:12)(cid:12) T Gi ∪ T Gj (cid:12)(cid:12) , where | · | represents the cardinality of the set. We thenproceed to calculate this measure for all n managerswhich results in a n × n symmetric matrix J G , the ( i, j ) element of which is given by the above equation, notethat the diagonal elements of this matrix are unity. Cal-culation of this quantity over all teams is computationallyexpensive in the sense that one must perform pair-wisecomparison of the n teams for each gameweek. As suchwe instead calculated an estimate of this quantity by tak-ing random samples without replacement of 100 teamsfrom each tier and calculating the measure both over allteams and also within tiers for each gameweek. We re-peat this calculation 10,000 times and the average resultsare those used in the main text and Supplementary NoteIV. D. Cluster Identification of Player Ownership
As described in the main text, the calculation of clus-ters within which groups of players co-appear involvestaking advantage of the underlying network structure ofall sets of teams. The adjacency matrix describing thisnetwork is defined by the matrix A Gij that has entry ( i, j ) equal to the number of teams within which player i and j co-appear in gameweek G . Note that the diagonal en-tries of this matrix describe the number of teams in whicha given player appears gameweek G . Using this matrixwe identify the clusters via a hierarchical clustering ap-proach, with k = 4 clusters determined via analysing thewithin-cluster sum of squared errors of k -means for eachcluster using the elbow method as shown in Supplemen-tary Figure 7. ACKNOWLEDGMENTS
Helpful discussions with Kevin Burke, James Fannon,Peter Grinrod, Stephen Kinsella, Renaud Lambiotte,and Sean McBrearty are gratefully acknowledged. Thiswork was supported by Science Foundation Ireland grantnumbers 16/IA/4470, 16/RC/3918, 12/RC/2289 P2 and18/CRT/6049), co-funded by the European RegionalDevelopment Fund. (J.D.O’B and J.P.G). We acknowl-edge the DJEI/DES/SFI/HEA Irish Centre for High-EndComputing (ICHEC) for the provision of computationalfacilities and support. The funders had no role in studydesign, data collection, and analysis, decision to publish,or preparation of the manuscript. [1] S. Mumford,
Watching sport: Aesthetics, ethics and emo-tion (Routledge, 2013). [2] D. L. Wann, Preliminary validation of the sport fan mo-tivation scale, Journal of Sport & Social Issues , 377 (1995).[3] S. Lee, W. J. Seo, and B. C. Green, Understanding whypeople play fantasy sport: development of the FantasySport Motivation Inventory (FanSMI), European SportManagement Quarterly , 166 (2013).[4] B. Dwyer and Y. Kim, For love or money: Develop-ing and validating a motivational scale for fantasy foot-ball participation, Journal of Sport Management , 70(2011).[5] A. J. Karg and H. McDonald, Fantasy sport participationas a complement to traditional sport consumption, SportManagement Review , 327 (2011).[6] L. K. Farquhar and R. Meeds, Types of fantasy sportsusers and their motivations, Journal of Computer-Mediated Communication , 1208 (2007).[7] J. Park and M. E. Newman, A network-based rankingsystem for US college football, Journal of Statistical Me-chanics: Theory and Experiment , P10014 (2005).[8] Y. Yamamoto and K. Yokoyama, Common and uniquenetwork dynamics in football games, PLoS ONE ,e29638 (2011).[9] T. U. Grund, Network structure and team performance:The case of English Premier League soccer teams, SocialNetworks , 682 (2012).[10] A. Gabel and S. Redner, Random walk picture of basket-ball scoring, Journal of Quantitative Analysis in Sports (2012).[11] H. V. Ribeiro, S. Mukherjee, and X. H. T. Zeng, Theadvantage of playing home in NBA: Microscopic, team-specific and evolving features, PLoS ONE , e0152440(2016).[12] J. Gudmundsson and M. Horton, Spatio-temporal analy-sis of team sports, ACM Computing Surveys , 1 (2017).[13] B. Gonçalves, D. Coutinho, S. Santos, C. Lago-Penas,S. Jiménez, and J. Sampaio, Exploring team passing net-works and player movement dynamics in youth associa-tion football, PLoS ONE , e0171156 (2017).[14] J. M. Buldú, J. Busquets, I. Echegoyen, and F. Seirul.lo,Defining a historic football team: Using Network Scienceto analyze Guardiola’s F.C. Barcelona, Scientific Reports , 1 (2019).[15] R. N. Onody and P. A. de Castro, Complex networkstudy of Brazilian soccer players, Physical Review E ,037103 (2004).[16] S. Saavedra, S. Powers, T. McCotter, M. A. Porter, andP. J. Mucha, Mutually-antagonistic interactions in base-ball networks, Physica A: Statistical Mechanics and itsApplications , 1131 (2010).[17] J. Duch, J. S. Waitzman, and L. A. Nunes Amaral, Quan-tifying the performance of individual players in a teamactivity, PLoS ONE , e10937 (2010).[18] F. Radicchi, Who is the best player ever? a complex net-work analysis of the history of professional tennis, PLoSONE , e17249 (2011).[19] S. Mukherjee, Identifying the greatest team and captain -A complex network approach to cricket matches, PhysicaA: Statistical Mechanics and its Applications , 6066(2012).[20] P. Cintia, L. Pappalardo, and D. Pedreschi, ’Engine mat-ters’: A first large scale data driven study on cyclists’performance, in Proceedings of the IEEE 13th Interna-tional Conference on Data Mining Workshops, ICDMW 2013 (2013) pp. 147–153.[21] J. Brooks, M. Kerr, and J. Guttag, Developing a data-driven player ranking in soccer using predictive modelweights, in
Proceedings of the ACM SIGKDD Interna-tional Conference on Knowledge Discovery and DataMining (2016) pp. 49–55.[22] B. Yucesoy and A. L. Barabási, Untangling performancefrom success, EPJ Data Science , 1 (2016).[23] E. Ben-Naim, N. W. Hengartner, S. Redner, andF. Vazquez, Randomness in Competitions, Journal ofStatistical Physics , 458 (2013).[24] R. Y. Aoki, R. M. Assunção, and P. O. Vaz De Melo,Luck is hard to beat: The difficulty of sports prediction,in Proceedings of the ACM SIGKDD International Con-ference on Knowledge Discovery and Data Mining (2017)pp. 1367–1376.[25] L. Pappalardo and P. Cintia, Quantifying the relationbetween performance and success in soccer, Advances inComplex Systems , 1750014 (2018).[26] F. Radicchi, Universality, limits and predictability ofgold-medal performances at the olympic games, PLoSONE , e40335 (2012).[27] Fantasy Sports Trade Association. Industry demograph-ics (2020).[28] D. Getty, H. Li, M. Yano, C. Gao, and A. E. Hosoi, Luckand the law: Quantifying chance in fantasy sports andother contests, SIAM Review , 869 (2018).[29] Fantasy Premier League (2020).[30] The rules of Fantasy Premier League (2020).[31] D. D. Burke, B. P. Meek, and J. M. Norwood, Explor-ing the legality of the lucrative world of fantasy sports,Journal of Legal, Ethical and Regulatory Issues , 38(2016).[32] S. Bhatt, K. Chen, V. L. Shalin, A. P. Sheth, and B. Min-nery, Who should be the captain this week?leveraginginferred diversity-enhanced crowd wisdom for a fantasypremier league captain prediction (2019) pp. 103–113.[33] S. Papagiannidis, M. Bourlakis, and F. Li, Making realmoney in virtual worlds: MMORPGs and emerging busi-ness opportunities, challenges and ethical implications inmetaverses, Technological Forecasting and Social Change , 610 (2008).[34] H. Yamaguchi et al. , An analysis of virtual currencies inonline games, The Journal of Social Science , 57 (2004).[35] FPL statistics for fantasy managers (2020).[36] P. M. Dechow, A. P. Hutton, L. Meulbroek, and R. G.Sloan, Short-sellers, fundamental analysis, and stock re-turns, Journal of Financial Economics , 77 (2001).[37] T. L. Ross and L. B. Collister, A social scientific frame-work for social systems in online video games: Building abetter looking for raid loot system in World of Warcraft,Computers in Human Behavior , 1 (2014).[38] A. Aleta and Y. Moreno, The dynamics of collective so-cial behavior in a crowd controlled game, EPJ Data Sci-ence , 1 (2019).[39] M. Newman, Networks: An Introduction (Oxford Uni-versity Press, Oxford).[40] C. Lucifora and R. Simmons, Superstar Effects in Sport:Evidence From Italian Soccer, Journal of Sports Eco-nomics , 35 (2003). Supplemental Materials
Supplementary Note I. Summary of Rules of Fantasy Premier League
The decisions made by the managers of Fantasy Premier League are governed by a stringent set of rules [30]. Theinitial restrictions of the game is that the manager must select a squad of 15 players consisting of two goalkeepers, fivedefenders, five midfielders, and three forwards. The total value of these players may not exceed £100M and a furtherrestriction is that no more than three players from one club may appear in a given squad. Each week the managermust then select a starting 11 players which must include one goalkeeper and a minimum of three defenders, threemidfielders, and one forward, this restriction is known as the formation criterion. These players are the ones whoseperformance contributes to the managers’ points total. The remaining players feature on the ‘bench’ and are orderedby the manager such that if one of their starting players does not appear the first placed bench players’ points aregiven to the manager (provided the formation criterion remains satisfied by said first choice bench player).Players selected by a manager are rewarded points based upon their statistical performance during the physicalmatches they compete in. Points for specific actions vary by the players’ position, for example, a defender receivesmore points for scoring a goal than a forward due to the relative rarity of such an event. In Supplementary Table1 we show the points per position for each of the possible actions the players may be rewarded/penalised for. For agoalkeeper or defender to be classified as keeping a clean sheet they must have played at least 60 minutes, excludingstoppage time. For example, if a defender was substituted in the physical game with the score at 0-0 in the 63rdminute, and their team proceeded to concede a goal then the defender in question would receive clean sheet pointsbut the remaining players would not. Also, in the case of a goal scored directly from a set piece i.e., a free-kick orpenalty, the player who was fouled in the awarding of the set piece receives the assist.Managers may proceed to make changes to their team between gameweeks which involves transferring a player fromtheir team for another with the same position. Each week a manager is entitled to one such transfer known as a freetransfer . Additional transfers may be made but at a cost of four points each from their points total. If the managerdoes not make use of their free transfer the following gameweek they may then make two free transfers however it isnot possible to accumulate more than two free transfers. The aforementioned restrictions regarding positions, clubs,and value must be satisfied for each transfer.In terms of the points amassed by the managers observed in the study we show the points totals and the numberof manager who obtained them in panel (a) of Supplementary Figure 1. We firstly comment on the skewness ofthe distribution with the number of managers obtaining a certain number of points decreasing quickly as the pointsbecomes larger. The large gap between the points obtained by the overall winner (2659) and those of second place(2602) is also interesting. The distribution of points within each tier is shown in Supplementary Figure 1(b), wesee the same skewness being present in each of these distributions, and also the presence of outliers among the topranked positions. Summary statistics for each tier and also all managers are given in Supplementary Table 2 andSupplementary Table 3. To view how the ranks of managers change over the course of the season we show the flowof manager position in Supplementary Figure 2. All managers are considered in panel (a) while those who finishedin the top positions are shown in (b). We see the competition for the top positions by the fact that no managerwho is outside the top ranks at gameweek 20 finishes within the top tier.Supplementary Table 1: Points awarded to players based on action by position. Action Goalkeeper Defender Midfielder Forward
For playing up to 60 minutes 1 1 1 1For playing 60 minutes or more 2 2 2 2For each goal scored 6 6 5 4For each assist 3 3 3 3For keeping a clean sheet 4 4 1 -For every 3 saves made 1 - - -For each penalty saved 5 - - -For each penalty missed -2 -2 -2 -2For every two goals conceded -1 -1 - -For each yellow card -1 -1 -1 -1For each red card -3 -3 -3 -3For each own goal -2 -2 -2 -2
Overall winner 2659 pointsRunner up 2602 points
Total Points N u m be r o f M anage r s top top top top All Managers
Total Points (a) lllllllllllllllllllllllllllllllll top top top T o t a l P o i n t s Distribution by Tier (b)
Supplementary Figure 1:
Summary of points obtained by managers in the 2018/19 season. (a) Thenumber of managers that obtained each points total where the bins are by tier. The overall winner (2659 points)and second place manager (2602) are highlighted. (b) Distribution of points totals earned by each tier.
Supplementary Note II. Historical Correlations
To obtain the historical data we considered managers taking part in the 2019/20 season of the game for which wefound ≈ million managers, of whom ≈ . million had taken part in a previous season. We then determined theirhistorical performance in terms of both points earned and overall rank for each season in which they partook in thegame. These quantities are the only two available at the historical level, unlike the gameweek level of resolution studiedin the main text. We then proceeded to identify the number of managers for whom we have data in each pairwisecombination of seasons, shown in the lower elements of Supplementary Table 4, and calculated Pearson correlationsbetween their points totals in each case represented by the upper elements of the same table. These correlations arevisualised in Fig. 1 of the main text. The two single blocks in the figure represent the winner (who had played in sixprevious seasons) and the runner-up (three previous seasons).We also consider the relationship between the number of previous seasons that the managers took part in andtheir points totals in the 2018/19 seasons as shown in Supplementary Figure 3. We comment on the small number ofmanagers present in the bottom right corner of the plot, suggesting that managers who have played for multiple yearsare less likely to have obtained a lower points total, in line with the correlation results obtained in SupplementaryTable 4. This positive relationship is also evident when one fits a linear regression to the data which suggests eachadditional year’s experience is worth, on average, an additional 22 points. GW2 GW11 GW20 GW29 GW38top top top top > top All Managers
Change in Tier Over the Season (a)
GW2 GW11 GW20 GW29 GW38top top top top > top top and (b) Supplementary Figure 2:
Alluvial graph describing the flow of manager rank at multiple time pointsduring the season. (a) Change in tier over all managers in the dataset. (b) The same analysis but on those whofinished in the top and tiers.Supplementary Table 2: Average points and standard deviation of points earned for each tier in each gameweek. GW Mean SD Mean SD Mean SD Mean SD1 88.16 11.73 84.97 12.65 76.84 14.62 63.17 15.062 85.44 11.09 83.58 12.18 78.15 14.57 68.49 16.283 51.12 8.52 50.04 8.66 49.42 9.73 49.81 11.124 51.85 7.74 50.80 7.99 49.70 8.97 47.00 10.235 71.64 14.77 68.42 14.67 64.03 14.99 54.69 15.516 62.97 8.18 61.84 8.27 59.51 8.89 55.40 9.527 62.80 10.36 60.83 10.22 58.13 10.61 53.96 11.338 70.82 13.01 66.73 13.06 62.52 13.41 56.79 13.869 45.70 7.82 44.79 8.39 43.35 9.12 42.78 10.2810 75.62 11.34 73.70 12.02 70.56 13.93 65.21 15.7011 70.94 12.03 68.71 12.62 65.65 13.31 59.63 14.1512 62.71 7.19 60.57 7.87 57.62 8.58 53.44 8.9713 48.36 10.92 48.18 11.27 49.19 11.56 50.79 11.9914 59.28 7.91 58.17 8.35 56.20 9.07 53.60 10.2615 60.74 11.77 58.48 11.72 55.27 11.95 51.20 12.1616 72.55 16.74 66.80 16.36 62.70 16.04 58.82 16.4117 55.06 9.12 53.83 9.73 51.50 10.43 47.06 11.1918 59.70 13.33 58.41 13.37 57.85 14.13 57.44 15.0219 78.08 12.44 75.98 12.60 73.45 12.63 68.13 12.6320 61.98 11.63 58.77 11.75 55.47 12.43 52.15 12.9521 56.22 11.29 56.24 11.29 56.49 11.41 56.17 11.6122 67.43 9.04 65.45 9.47 61.50 9.95 55.73 10.3123 68.75 8.69 67.69 9.50 65.51 10.69 60.86 12.1424 47.98 9.15 46.86 9.42 46.36 10.19 45.84 11.1525 86.11 17.93 81.53 18.19 78.68 18.62 73.11 18.4826 74.72 10.57 72.30 11.05 69.19 11.97 64.48 13.4627 42.87 9.86 41.67 9.63 40.97 9.97 39.32 10.5728 59.38 10.03 58.44 10.65 57.36 11.60 56.91 12.9029 47.24 8.15 45.75 8.51 45.15 9.10 43.86 9.4930 63.76 16.04 60.68 16.08 57.02 15.66 53.62 14.9031 35.11 10.41 35.26 10.96 36.44 11.76 34.57 13.4532 98.99 12.11 96.09 12.93 90.78 14.46 80.69 15.6233 71.58 12.06 68.07 13.36 60.56 16.10 46.85 17.5934 50.87 11.80 51.96 12.95 54.25 14.33 58.19 15.2135 106.96 17.31 100.75 18.72 89.80 19.20 76.07 15.4636 100.31 16.82 96.12 16.73 90.28 17.11 82.24 17.2237 49.19 10.50 48.02 11.16 48.67 12.35 50.77 13.2038 66.81 11.53 66.20 11.96 64.68 12.63 61.25 13.74
Supplementary Note III. Financial Analysis
Here we briefly consider implications of the financial aspects of the game. Firstly, the price of players themselvesdemonstrate some interesting dynamics. We remind the reader that initially the price of each player is set by thedevelopers of the game and this price subsequently fluctuates over the season depending on the supply and demandof managers transferring the player in and out of their teams. A summary of the distribution of the average valueover the season for each of the ≈
600 players coloured by their corresponding position is provided in panel (a) ofSupplementary Figure 4. We comment on the skewed distribution of points obtained in all cases but in particular formidfielders and forwards. The corresponding points earned over the season by each of the players versus their averageprice is shown in panel (b). We see that there is, in general, a positive relationship between the price of a player andtheir corresponding points totals. As per their prices, we see a handful of midfielders and forwards who earn the mostpoints.The make-up of the managers’ squad consists of two goalkeepers, five defenders, five midfielders, and three forwards.Supplementary Figure 5 demonstrates the proportion of budget spend in these three positions (we have groupedgoalkeepers and defenders together) by all managers at GW 1. We observe some remarkable variation in where thebudget is spent, with some managers spending over half of their budget on midfield players despite of them onlyaccounting for a third of their squad. Due to the price of players fluctuating throughout the season, like an investorSupplementary Table 3: Summary statistics of the points obtained by the managers in the dataset. Both over allmanagers and the tiers used in this study.
TierEveryone n 901912 1000 8493 83897 808522Max 2659 2659 2464 2385 2269Min 2082 2464 2385 2269 2082Mean 2167.91 2489.82 2412.68 2310.82 2150.11Median 2150 2482 2409 2306 2140Std. dev 71.72 24.21 20.68 30.44 49.42IQR 96 29 32 47 78 Supplementary Table 4: Correlation between a managers’ historical performance. The lower triangular elements ofthe table represent the number of managers who were present in both of the seasons, such that the diagonal elementsdescribe the number of managers for each season for whom we could obtain data. Upper elements of the tablerepresent the pairwise Pearson correlation coefficient between the points obtained by the manager in the two seasons.
Season holding a varying stock, the managers’ overall team value changes. Supplementary Figure 6 shows the average teamvalue of the managers by tier over each gameweek of the season along with corresponding 95% intervals of thedistribution. As per the main text we fit a linear regression to the total points obtained by all managers as a functionof their team value each gameweek, with results shown in Supplementary Table 5.
Supplementary Note IV. Team Similarity and Cluster Analysis
In this section we show more information regarding the hierarchical clustering analysis described in the main text.Supplementary Figure 7 shows the scaled within sum of squared errors (WSS) for each number of clusters where theerror is calculated using k -means. We decide upon four clusters as the decrease in errors slows down at this point.In order to give equal weighting to each gameweek we firstly calculate the WSS for each gameweek and rescale thesebefore averaging over these rescaled values over all gameweeks.The sizes of these first three clusters for each tier are shown in Supplementary Figure 8 and follow a similar patternto that found for all managers in the main text. The top two tiers are shown in panel (a) and (b), however, doappear to make use of fewer players which may be a function of the smaller number of teams to analyse It may alsobe further evidence of the higher similarity between the teams in these tiers as suggested in Supplementary Figure 9,which shows the Jaccard similarity (calculated as in the main text) for each of the tiers versus all other tiers. Finally,for the interested reader we provide the identity of those players who appear in the first cluster when the analysis isperformed on all managers in Supplementary Table 6. We see frequent appearance of some higher priced players suchas Mohammed Salah and Sergio Agüero throughout the season. However, also interesting is the presence of someextremely inexpensive players, in particular Aaron Wan-Bissaka who was appearing in his debut campaign and waspriced at the cheapest level as a £4M defender but surprisingly made consistent appearances throughout the season,which made him a very attractive option for skilful managers in order to spend more of their budget elsewhere. Number of Previous Seasons P o i n t s / Points and Number of Previous Seasons
Supplementary Figure 3:
Manager points in the 2018/19 season versus the number of previous seasonswhich they had registered for.
The bins, each of which cover a 100 point range, are coloured by the number ofmanagers in each, note the logarithmic scale. We comment on the small number present in the bottom right corner(in comparison to the top right corner), which indicates that players who have played for multiple years did not tendto perform poorly.
Position A v e r age V a l ue Distirbution of Average Player Price
Player Summary (a) ll l llll l ll l lll l ll lll lll lll lll l ll ll ll ll llll l llll ll lllllll l lll l lllll ll ll ll l ll ll ll lll lll l l ll ll ll llll llll lll lll lll lll llll ll l lll l ll l ll ll l ll llllllll ll ll l lll l lll ll lll ll l ll l llll l ll l ll lll ll ll lll ll ll lll lll ll ll l ll lll llllll ll llll l ll lll l ll l ll lll l ll l lllll l ll l ll ll l ll lll lll l lll ll lll lll ll ll lll ll lll llll ll lll lll lll ll ll ll lll l ll l llllll l lllllll l ll l ll l llll lll l ll ll l lll ll lll ll ll l llll ll ll ll lll ll l ll ll l llll lll l l llll lll llll ll lll ll ll lllll l ll lll l lll llll lll ll lll lll l lll lll llll ll lll ll l ll ll ll ll lll lll lll lll ll ll ll lll lll ll ll lll llll l l ll ll l l lll l llll lll ll l llll l llllll l ll l lll l l ll ll l ll lll l ll l lll l lll ll ll lll ll ll lllll l ll l ll lll l ll lll ll l llll l llll l lll llll ll lll ll llll
Alexis SánchezAndrew Robertson Eden HazardGabriel Fernando de Jesus Harry KaneKevin De Bruyne Mohamed SalahPierre−Emerick Aubameyang Raheem SterlingRomelu LukakuSadio Mané Sergio AgüeroVirgil van Dijk
Average Value T o t a l P o i n t s l a l a l a l a GK DEF MID FWDPlayer Points and Average Value (b)
Supplementary Figure 4:
The value of player values and the corresponding points earned. (a) Distributionof average player price over the course of the season factored by the player position, we comment on the skewednature of the distribution particularly for midfielders and forwards. (b) The same average value is shown versus thecorresponding points earned over the season by said players, which shows the largest points totals being provided bygenerally the higher priced players. The identities of some players with higher prices and points totals are alsoshown.
Supplementary Note V. Chip Usage
As described in the main text we described three chips which essentially are tricks a manager can make use of inany given gameweek (note that more than one chip can not be used in any single gameweek). The chip properties aresummarised below
20 25 30 % M i d f i e l de r s % G K s and D e f ende r s % Forwards MIDDEF FWD
DensityAll Managers − Gameweek 1
Percentage of Budget by Position
Supplementary Figure 5:
Ternary diagram demonstrating the make-up of managers’ squads ingameweek one.
The combination of proportions spent in each position (where DEF represents both goalkeepersand defenders) is shown, where the colour in each bin represents the fraction of managers who used a givencombination of proportions.1.
Bench Boost (BB) - The manager receives the points awarded by all 15 players in their squad in comparisonto the usual starting 11 players.2.
Free Hit (FH) - The manager may make unlimited changes to their team for one gameweek, at the end ofwhich their team reverts to the squad from the previous GW, under the standard restriction, i.e., they mustremain under their budget, satisfy the formation criterion, and have no more than three players from any oneclub.3.
Triple Captain (TC) - For the GW this chip is played in, the captain’s points are tripled rather than doubled.If the captain does not play the triple points are awarded to the vice-captain and, as usual, if neither play noone is awarded triple points.The points obtained for each of these chips, the distributions of which are shown in Supplementary Figure 10, arecalculated by1.
Bench Boost (BB) - We identify the four players on the manager bench the week the chip was played andtally their points total.2.
Free Hit (FH) - The amount of points the manger received that week is used in this case as the free hitessentially acts like a free week to choose the eleven players of their desire with the aim of maximizing pointsfor one week i.e., no long-term planning is needed.3.
Triple Captain (TC) - The captain’s points total is shown in the distribution. We assume they would havechosen this player to be captain regardless of the chip so would have received double points regardless and assuch the difference is only the single points score. llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll −20246810 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
Gameweek C hange i n T ea m V a l ue ( M ) l l l l top top top top By Class
Average Change in Team Value
Supplementary Figure 6:
Average team value along with 95 percentiles for each tier over the season.
Wecomment on the general upward trend, but observe the higher-placed managers having larger team valuesthroughout the season. l l l l l l l l l ll l l l l l l l l ll l l l l l l l l ll l l l l l l l l ll l l l l l l l l l l l l l l l l l l ll l l l l l l l l ll l l l l l l l l ll l l l l l l l l ll l l l l l l l l l
Number of Clusters S c a l ed W SS lllll All tierstop top top top All Managers and Within Tiers
Within Sum of Squared Errors by Cluster Size
Supplementary Figure 7:
Scaled within sum of squared errors for the k -means cluster analysis Thehorizontal axis represents the number of clusters used and the vertical the within sum of squared errors. Themeasure is calculated for each tier in each of the 38 gameweeks before being rescaled in order to give equal weightingto each gameweek. We note that each tier follows a similar pattern.We repeated this calculation for each gameweek for every manager in our dataset and determined both the numberof individuals who played the chip that gameweek alongside the average number of points those that did earned fromSupplementary Table 5: Regression coefficients for final points as a function of each additional million pounds inteam value at each gameweek over all managers.
GW Intercept Co-efficient p R < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − < − doing so. The corresponding figures are shown in Supplementary Table 7, Supplementary Table 8, and SupplementaryTable 9. A fourth chip also exists in the game and is known as the wildcard , this chip allows the manager to make asmany transfers as they like in the week it is played thus offering a chance to totally redefine their team. The managersreceive this chip twice in the season, the first may only be used between gameweeks 1 and 21, while the second inone of the remaining gameweeks. It proves however much more difficult to quantify the return from this chip e.g.,one could consider the wildcarded teams return versus their original team in the following m gameweeks, however inpractice the manager would make transfers to their original team in the following gameweeks, another issue is thepossibility that the manager ‘dead-ends’ their team up to the week of their wildcard gameweek in the sense that theystop planning for beyond the wildcard opening up the possibility of an extremely biased comparison as the team beingchanged arguably would not be there without the wildcard. We may still, however, consider the gameweek in whichthe managers played each of their two chips and this is shown in Supplementary Figure 11 alongside the quantitiesthemselves in Supplementary Table 10. Looking at the point of season in which these chips are used we again noticean evident pattern among the actions of the top managers particularly when the second wildcard chip is considered.It appears as though the strategy of choice for those who finished in the top two tiers was to use their free hit duringdouble gameweek 32, wildcard in gameweek 34, and thus having what they believed to be an optimal squad such thatthey could optimise their bench boost chip which most played in double gameweek 35. top Cluster Size by Gameweek (a) top (b) top (c) top (d) Supplementary Figure 8:
Cluster size analysis.
Size of the first three clusters identified by the hierarchicalclustering approach described in the main text for each tier. Note all appear to follow a similar pattern.0 llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll A v e r age J a cc a r d S i m il a r i t y l l l l top top top top top Jaccard Similarity of Teams (a) llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll l l l l top top top top top (b) llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll GW A v e r age J a cc a r d S i m il a r i t y l l l l top top top top top (c) llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll GW l l l l top top top top top (d) Supplementary Figure 9:
Jaccard similarity between managers in each tier with those in other tiers.
Calculation is repeated as described in Sec. IV C of the main text. The Jaccard similarity of teams in each tier iscompared with all other tiers, where we observe a stronger similarity between those in higher tiers indicating thatthe better managers are more likely to have a similar structure.1Supplementary Table 6: Summary of players who appeared in the first cluster over the course of the season amongthe different tiers of managers.
TierGameweek Everyone G W Top MillionTop 10k
Bench Boost Chip (a) P r obab ili t y Top MillionTop 10k
Distribution of Points from Bench Boost (b)
DGW25DGW32 26.6%43.1%2468101214161820222426283032343638 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 G W Top MillionTop 10k
Triple Captain Chip (c) P r obab ili t y Top MillionTop 10k
Distribution of Points from Triple Captain (d)
Proportion Used G W Top MillionTop 10k
Free Hit Chip (e)
Points P r obab ili t y Top MillionTop 10k
Distribution of Points from Free Hit (f)
Supplementary Figure 10:
Summary results for each of the three chips.
We show the time of the chips’ usealong with the points received by the manager who played them, for manager who finished in the top and tiers (top 10k) in comparison to the remaining managers (top million) in the dataset. The left panels show theproportion of managers who had used the corresponding chip by each GW (the complementary cumulativedistribution function), particularly highlighting the large usages in the ‘special’ gameweeks for each chip. The rightpanels show the distribution of points received from the chip’s use by the two groups of managers, while the meannumber of points for each group are also shown by the dashed vertical lines. We comment on the fact that the top10k received more points on average for each of the three chips.3 G W Top MillionTop 10k
Wildcard Chip
Supplementary Figure 11:
Analysis of the wildcard chip’s use.
Fraction of managers in the two groups who hadused their wildcard chip by each gameweek. Note that the count resets in gameweek 22 when the chip is replenished.4Supplementary Table 7: Usage and Average Points from Bench Boost Chip GW Rel. Freq Cum. Freq Mean Points Rel. Freq Cum. Freq Mean Points Rel. Freq Cum. Freq Mean Points Rel. Ferq Cum. Freq Mean Points1 0.003 0.003 20.333 0.004 0.004 21.939 0.006 0.006 16.930 0.010 0.010 11.0232 0.000 0.003 — 0.000 0.004 4.667 0.001 0.008 7.959 0.004 0.014 7.4513 0.000 0.003 — 0.000 0.004 8.000 0.001 0.009 9.907 0.004 0.018 10.0224 0.001 0.004 12.000 0.002 0.006 8.667 0.003 0.012 9.373 0.007 0.025 8.3335 0.002 0.006 10.000 0.003 0.009 14.042 0.004 0.016 10.143 0.007 0.032 8.7836 0.000 0.006 — 0.001 0.010 12.500 0.003 0.019 12.431 0.007 0.039 10.5437 0.003 0.009 13.667 0.005 0.015 12.436 0.008 0.028 11.223 0.012 0.051 10.3228 0.001 0.010 17.000 0.002 0.017 13.700 0.005 0.033 13.732 0.009 0.060 12.4919 0.000 0.010 — 0.001 0.018 6.375 0.004 0.037 8.076 0.008 0.068 7.71710 0.001 0.011 2.000 0.002 0.020 6.214 0.005 0.041 7.409 0.010 0.078 7.37611 0.001 0.012 7.000 0.001 0.021 9.364 0.002 0.043 9.960 0.006 0.084 9.03712 0.000 0.012 — 0.001 0.023 11.083 0.003 0.046 10.810 0.007 0.091 10.31713 0.000 0.012 — 0.001 0.024 12.250 0.002 0.048 9.361 0.005 0.096 8.30014 0.000 0.012 — 0.003 0.027 7.793 0.005 0.053 9.610 0.007 0.103 8.48815 0.001 0.013 13.000 0.001 0.028 11.222 0.003 0.056 8.466 0.007 0.110 7.01816 0.000 0.013 — 0.001 0.029 6.125 0.003 0.059 8.080 0.006 0.116 6.82217 0.002 0.015 5.500 0.001 0.030 13.000 0.004 0.063 11.663 0.007 0.123 10.20918 0.000 0.015 — 0.001 0.031 16.444 0.002 0.065 9.754 0.006 0.129 9.49219 0.005 0.020 16.600 0.008 0.039 15.232 0.012 0.076 11.900 0.017 0.147 10.24920 0.001 0.021 11.000 0.001 0.040 7.857 0.003 0.079 10.795 0.008 0.155 9.66721 0.002 0.023 19.000 0.002 0.042 13.333 0.006 0.085 12.410 0.011 0.166 11.18722 0.000 0.023 — 0.002 0.044 11.538 0.003 0.088 10.874 0.008 0.174 10.08323 0.001 0.024 13.000 0.001 0.045 6.000 0.004 0.092 9.313 0.008 0.182 8.70824 0.004 0.028 9.250 0.005 0.051 8.200 0.010 0.102 9.220 0.012 0.194 8.20825 0.001 0.029 12.000 0.004 0.054 14.467 0.008 0.110 14.635 0.012 0.206 13.29926 0.009 0.038 11.889 0.007 0.061 8.860 0.013 0.123 7.053 0.015 0.221 6.52227 0.001 0.039 31.000 0.000 0.061 8.000 0.000 0.123 8.387 0.001 0.223 4.78928 0.000 0.039 — 0.001 0.062 12.583 0.002 0.125 12.816 0.003 0.226 10.37629 0.006 0.045 7.167 0.005 0.067 10.326 0.012 0.137 11.184 0.016 0.242 9.97930 0.004 0.049 19.250 0.006 0.073 13.922 0.010 0.147 12.906 0.014 0.256 11.59931 0.000 0.049 — 0.000 0.073 8.000 0.000 0.147 5.455 0.001 0.257 3.90532 0.020 0.069 17.750 0.019 0.093 19.261 0.041 0.188 17.578 0.057 0.314 15.57333 0.000 0.069 — 0.000 0.093 19.000 0.000 0.189 10.320 0.001 0.315 4.62834 0.006 0.075 16.333 0.007 0.100 15.082 0.013 0.201 12.463 0.019 0.334 10.64735 0.856 0.931 27.474 0.787 0.888 25.414 0.594 0.795 23.147 0.257 0.590 18.84936 0.033 0.964 15.000 0.045 0.933 15.592 0.052 0.847 15.775 0.052 0.642 14.83237 0.018 0.982 13.778 0.028 0.960 13.233 0.059 0.905 12.486 0.081 0.723 11.44738 0.017 0.999 11.588 0.035 0.995 10.414 0.074 0.980 9.413 0.113 0.835 8.668 GW Rel. Freq Cum. Freq Mean Points Rel. Freq Cum. Freq Mean Points Rel. Freq Cum. Freq Mean Points Rel. Freq Cum. Freq Mean Points1 0.000 0.000 — 0.000 0.000 — 0.000 0.000 — 0.000 0.000 —2 0.001 0.001 67.000 0.001 0.001 84.500 0.003 0.003 78.689 0.011 0.011 71.8963 0.000 0.001 — 0.002 0.003 53.308 0.004 0.007 55.192 0.012 0.023 53.3754 0.003 0.004 56.667 0.001 0.004 62.000 0.004 0.011 53.742 0.010 0.033 51.7995 0.002 0.006 91.000 0.002 0.006 76.706 0.005 0.017 72.022 0.012 0.045 62.2256 0.001 0.007 64.000 0.002 0.008 64.250 0.004 0.021 60.878 0.010 0.056 56.9287 0.006 0.013 79.000 0.007 0.015 81.172 0.011 0.031 76.225 0.015 0.070 68.0908 0.003 0.016 57.667 0.005 0.020 57.667 0.013 0.044 56.857 0.017 0.087 56.1429 0.000 0.016 — 0.002 0.022 49.214 0.005 0.049 48.288 0.011 0.098 45.35110 0.006 0.022 97.000 0.005 0.027 88.000 0.010 0.059 83.850 0.015 0.114 78.13611 0.002 0.024 81.500 0.002 0.030 80.143 0.006 0.066 75.512 0.010 0.124 66.41312 0.004 0.028 59.250 0.002 0.032 59.062 0.004 0.070 56.708 0.009 0.133 54.17913 0.001 0.029 90.000 0.003 0.035 69.731 0.008 0.078 66.031 0.014 0.147 61.90214 0.000 0.029 — 0.000 0.035 61.000 0.002 0.080 58.169 0.006 0.153 57.07515 0.005 0.034 64.400 0.004 0.039 69.581 0.007 0.087 64.119 0.013 0.166 58.87216 0.003 0.037 64.000 0.006 0.045 78.957 0.011 0.098 72.813 0.020 0.186 65.90617 0.000 0.037 — 0.001 0.045 55.714 0.004 0.102 54.592 0.010 0.196 50.40618 0.000 0.037 — 0.002 0.047 53.231 0.003 0.105 53.367 0.006 0.202 55.98519 0.001 0.038 94.000 0.002 0.049 84.737 0.005 0.110 80.305 0.010 0.212 74.25020 0.001 0.039 40.000 0.003 0.052 60.409 0.006 0.117 61.189 0.013 0.225 57.58321 0.003 0.042 66.333 0.002 0.054 65.429 0.004 0.121 65.541 0.006 0.231 62.01622 0.000 0.042 — 0.001 0.055 62.750 0.003 0.124 62.424 0.005 0.236 58.84323 0.002 0.044 94.000 0.002 0.057 75.706 0.005 0.129 76.457 0.009 0.245 69.24824 0.000 0.044 — 0.000 0.057 47.667 0.001 0.130 52.558 0.003 0.248 47.56225 0.009 0.053 94.556 0.016 0.074 91.221 0.032 0.162 88.109 0.042 0.290 82.06226 0.000 0.053 — 0.000 0.074 79.500 0.002 0.163 70.571 0.005 0.295 68.97727 0.016 0.069 45.000 0.027 0.101 46.240 0.058 0.221 44.662 0.079 0.373 42.86028 0.000 0.069 — 0.000 0.101 79.500 0.002 0.223 69.053 0.005 0.378 65.06329 0.000 0.069 — 0.000 0.101 — 0.001 0.223 51.585 0.002 0.380 47.06930 0.000 0.069 — 0.000 0.101 59.000 0.001 0.224 65.154 0.002 0.382 57.93331 0.114 0.183 39.561 0.178 0.279 40.827 0.300 0.524 42.048 0.291 0.674 43.87932 0.788 0.971 99.999 0.681 0.960 98.233 0.399 0.923 96.261 0.103 0.777 92.51033 0.021 0.992 79.762 0.025 0.985 76.359 0.037 0.961 74.060 0.033 0.809 68.15234 0.001 0.993 36.000 0.001 0.986 56.400 0.002 0.962 54.674 0.003 0.812 54.86635 0.003 0.996 84.000 0.007 0.993 77.541 0.013 0.975 76.154 0.019 0.831 74.45636 0.001 0.997 111.000 0.001 0.993 100.000 0.002 0.977 93.881 0.005 0.835 86.75437 0.000 0.997 — 0.001 0.994 63.000 0.003 0.980 56.013 0.006 0.841 55.20138 0.003 1.000 62.000 0.004 0.998 72.833 0.008 0.988 70.057 0.019 0.860 66.632 GW Rel. Freq Cum. Freq Mean Points Rel. Freq Cum. Freq Mean Points Rel. Freq Cum. Freq Mean Points Rel. Freq Cum. Freq Mean Points1 0.000 0.000 — 0.001 0.001 8.571 0.003 0.003 7.100 0.011 0.011 5.8962 0.003 0.003 20.000 0.002 0.002 18.214 0.004 0.007 17.104 0.012 0.023 12.8463 0.003 0.006 8.000 0.003 0.006 7.000 0.006 0.013 7.015 0.014 0.037 5.9714 0.006 0.012 6.000 0.013 0.019 6.083 0.017 0.030 6.090 0.021 0.058 5.9345 0.005 0.017 9.600 0.004 0.023 11.105 0.007 0.037 11.845 0.013 0.071 10.0326 0.001 0.018 8.000 0.001 0.024 5.000 0.003 0.039 6.167 0.010 0.081 5.8687 0.001 0.019 8.000 0.003 0.026 8.000 0.004 0.044 8.491 0.012 0.093 8.4858 0.001 0.020 1.000 0.003 0.030 1.500 0.006 0.049 2.554 0.013 0.106 3.9099 0.000 0.020 — 0.001 0.031 6.222 0.002 0.052 5.387 0.008 0.114 5.09410 0.007 0.027 15.000 0.007 0.037 15.000 0.010 0.062 14.529 0.014 0.128 13.51311 0.001 0.028 13.000 0.000 0.038 15.000 0.003 0.065 12.974 0.009 0.137 11.46812 0.010 0.038 8.000 0.012 0.050 7.670 0.013 0.078 7.538 0.015 0.152 6.61113 0.001 0.039 3.000 0.000 0.050 3.000 0.001 0.079 6.391 0.004 0.156 5.25214 0.000 0.039 — 0.000 0.051 6.500 0.002 0.081 4.089 0.005 0.161 3.56515 0.000 0.039 — 0.001 0.051 12.000 0.003 0.083 9.582 0.007 0.169 7.42416 0.000 0.039 — 0.001 0.052 4.500 0.003 0.086 5.891 0.010 0.178 6.04517 0.001 0.040 5.000 0.001 0.054 4.300 0.003 0.089 6.175 0.008 0.186 5.54918 0.000 0.040 — 0.001 0.055 4.625 0.002 0.091 6.882 0.008 0.194 7.40819 0.001 0.041 6.000 0.001 0.056 11.500 0.004 0.096 10.410 0.014 0.208 10.07320 0.000 0.041 — 0.001 0.057 10.286 0.002 0.098 9.729 0.006 0.214 8.04121 0.000 0.041 — 0.001 0.058 8.250 0.002 0.100 8.131 0.005 0.219 7.46722 0.000 0.041 — 0.000 0.058 8.000 0.001 0.101 8.686 0.006 0.225 8.11623 0.000 0.041 — 0.002 0.060 13.929 0.003 0.104 12.465 0.009 0.234 12.13824 0.001 0.042 2.000 0.001 0.061 7.182 0.003 0.107 6.650 0.007 0.241 6.41825 0.152 0.194 14.066 0.115 0.176 12.720 0.077 0.184 12.068 0.046 0.287 12.98926 0.005 0.199 8.000 0.004 0.180 8.000 0.006 0.190 7.874 0.009 0.296 7.66227 0.001 0.200 2.000 0.001 0.181 1.400 0.000 0.191 2.310 0.001 0.297 2.20828 0.000 0.200 — 0.001 0.182 5.571 0.002 0.193 5.558 0.004 0.301 5.50929 0.001 0.201 3.000 0.001 0.183 2.500 0.002 0.195 2.947 0.004 0.305 2.71930 0.002 0.203 5.000 0.002 0.185 10.056 0.004 0.199 7.042 0.009 0.314 6.59131 0.027 0.230 2.000 0.036 0.221 2.581 0.025 0.224 3.083 0.011 0.324 4.74132 0.121 0.351 9.405 0.182 0.403 9.370 0.291 0.516 9.093 0.264 0.588 8.45133 0.002 0.353 16.000 0.004 0.406 14.387 0.004 0.520 12.335 0.005 0.593 8.98334 0.001 0.354 1.000 0.004 0.411 2.216 0.007 0.526 2.938 0.008 0.601 3.58035 0.056 0.410 6.893 0.056 0.467 6.686 0.077 0.603 6.570 0.088 0.689 6.46936 0.505 0.915 15.986 0.422 0.889 15.376 0.266 0.869 15.136 0.098 0.787 14.58637 0.038 0.953 4.368 0.046 0.935 4.205 0.045 0.914 4.106 0.039 0.826 4.16538 0.045 0.998 6.111 0.062 0.997 6.779 0.074 0.988 6.781 0.075 0.900 6.7966