A PROBABILITY BASED APPROACH FOR THE ALLOCATION OF PLAYER DRAFT SELECTIONS IN AUSTRALIAN RULES

Journal of Sports Scence and Medcne (2006) 5, 509-516 http://www.jssm.org Research artcle The 8th Australasan Conference on Mathematcs and Computers n Sport, 3-5 July 2006, Queensland, Australa A PROBABILITY BASED APPROACH FOR THE ALLOCATION OF PLAYER DRAFT SELECTIONS IN AUSTRALIAN RULES FOOTBALL Anthony Bedford and Adran J. Schembr School of Mathematcal and Geospatal Scences, RMIT Unversty, Bundoora, VIC, Australa Publshed (onlne): 15 December 2006 ABSTRACT Australan Rules Football, governed by the Australan Football League (AFL) s the most popular wnter sport played n Australa. Lke North Amercan team based leagues such as the NFL, NBA and NHL, the AFL uses a draft system for rooke players to jon a team s lst. The exstng method of allocatng draft selectons n the AFL s smply based on the reverse order of each team s fnshng poston for that season, wth teams wnnng less than or equal to 5 regular season matches obtanng an addtonal early round prorty draft pck. Much crtcsm has been levelled at the exstng system snce t rewards losng teams and does not encourage poorly performng teams to wn matches once ther season s effectvely over. We propose a probablty-based system that allocates a score based on teams that wn unmportant matches (akn to Carl Morrs defnton of mportance). We base the calculaton of unmportance on the lkelhood of a team makng the fnal eght followng each round of the season. We then nvestgate a varety of approaches based on the unmportance measure to derve a score for unmportant and unlkely wns. We explore dervatves of ths system, compare past draft pcks wth those obtaned under our system, and dscuss the attractveness of teams knowng the draft reward for wnnng each match n a season. KEY WORDS: AFL, probablty, draft, mportance. INTRODUCTION The AFL draft system has been desgned to favour teams anchored to the bottom of the ladder. Ths enables those teams to mprove ther player lsts and propel themselves up the ladder durng future seasons by havng a frst choce of pckng rooke players. Currently, the order of the AFL draft concdes wth the nverse order of the ladder as t stands at the concluson of the home and away season. Ths system allocates the frst draft choce to the team that fnshed last (or sxteenth), the second draft choce to the team that fnshed ffteenth, and the sxteenth draft choce to the team that fnshed frst. Subsequent rounds of the draft replcate the exact order of the frst round. A hghly contentous ssue surroundng the AFL draft has been the allocaton of prorty pcks. A prorty pck s a draft choce provded to a team pror to the frst round of the natonal draft. From 1997 to 2005, prorty pcks were provded to teams that won fewer than fve games durng the regular season. In effect, f a team fnshed last wth less than fve wns, they receved a prorty pck n

510 Probablstc AFL draft model addton to the frst choce n the natonal draft, thus enablng the club to receve the two best avalable players. Ths system provded teams wth poor wn/loss records wth lttle ncentve to wn games durng the latter part of the season and actually provded clubs wth an ncentve not to wn fve games. The draft systems of other major world sports are somewhat comparable to that of the AFL. Major League Baseball (MLB) and the Amercan Natonal Football League (NFL) both allocate draft choces usng nversed fnal season standngs (Grer and Tollson, 1994; Spurr, 2000). However, nether competton provdes prorty pcks and thus does not provde addtonal reward for wnnng only a handful of games. Several studes have assessed the ncentve effects of draft systems and ther mpact on team performance. Taylor and Trogdon (2002) assessed the performance of teams followng ntatves by the NBA to reduce team ncentves to wn or lose games. After controllng for venue and the qualty of each team, these authors found that when draft choces were decded on nverse rankngs (1983-84 season), non-playoff teams were 2.5 tmes more lkely to lose games than teams lkely to feature n the playoffs. However, when the NBA modfed the draft system and gave all teams an equal probablty of obtanng the frst draft choce (1984-85 season), non-playoff teams were as lkely to wn as play-off bound teams. Fnally, when the lottery system became weghted durng the 1989-1990 season, non-playoff teams were 2.2 tmes more lkely to lose when compared to teams qualfyng for the playoffs (Taylor and Trogdon, 2002). These fndngs demonstrate the profound mpact of provdng ncentves for teams to lose games and head towards the bottom of the ladder. Furthermore, t hghlghts that an ncentve based system such as the process employed by the AFL ncreases the lkelhood that teams wll lose addtonal matches durng the latter part of the season snce they are unlkely to feature n the fnals. In ths paper we use a model that allocates a Draft Pont Reward (DPR) to each team when they wn a match. Ths reward vares n value from 0 to 1 dependng upon the Unmportance of the match. The cumulatve sum of DPR, known as the DScore, s used to determne the fnal draft pcks at the concluson of the regular season. We wll begn our work by defnng the crtera our model should meet. We then outlne the methods employed, and consder the operatonal aspects of the model. METHODS Our system s based on Carl Morrs famous work on the most mportant ponts n tenns (Morrs, 1977). He defned the mportance of a pont as the dfference between two condtonal probabltes: the probablty a server wns the game gven that he wns the next pont, mnus the probablty a server wns the game gven that he loses the next pont. Here we are not consderng ponts n a game, but rather matches n a season, and t s the Unmportant matches that appeal to us. We calculated Unmportance so that t s ndependent of the opposng team. Crtera In devsng the system of selecton for the AFL natonal draft, we desgned our model based on the followng: Teams wth a reduced probablty of makng the fnals are rewarded ncrementally hgher for wnnng matches of hgh Unmportance Teams that have qualfed for the fnals are nelgble for any reward. DPR s restrcted to a 16-week perod commencng from the end of Round 6. DPR s hgher for teams unlkely to wn, and s further enhanced by the Unmportance of a match n terms of makng the fnals. No DPR s gven n defeat, so teams must wn to obtan a reward. A prorty system s n place to protect teams that have contnuous runs of losses, but t s not mplemented at the expense of rewardng vctory. The way n whch the number of matches needed to wn s calculated s based upon the mnmum number of wns needed by a team n the remander of the season based solely upon makng the fnal 8 (F8). Obvously ths s not precsely known untl the end of the season; however a reasonable estmaton can be made. Probablstc model The heart of our model s based upon reworkng Morrs equaton to sut our purpose of determnng how Unmportant a match s to a team s fnals aspratons. There are a number of thngs that we need to evaluate frst, such as what measures are requred n our assessment of what makes a match Important, and, n turn, Unmportant. A regular AFL season consttutes 22 matches and we need to consder the probablty of a team makng the fnals based upon the number of matches won at round r. There are a number of features n our probablstc model that were used to determne how much a

Bedford and Schembr 511 team was rewarded for wnnng a match. The process s as follows: Determne the mnmum number of wns (Par Wns) requred for team to make the fnal 8 after round r. Check f team at round r has already made the fnal 8 or cannot make the fnal 8. If nether of these events are true, we determne the probablty of team makng the fnals at the completon of round r. Calculate the Unmportance of match r +1 for team usng the above results. Allocate the Draft Ponts Reward (DPR) based on the above measures. Determnaton of projected wns to make the fnal8 There are two possble approaches to determnng the number of wns requred to make the fnal 8 at round r for team. We could ether use the fnal season s requred wns and mpose that retrospectvely on the completed season, or use a projected requrement durng the season and keep ths result even at the end of the season. For example, n season 2004, the eghth placed team won 12 of 22 matches to make the fnals. Ultmately, dfferng results make t dffcult to predct ths result durng the season. However, the attracton of our model s that teams must know the rewards of wnnng ther next match pror to the game as an ncentve to wn. They should also be confdent ths reward does not change post game. So we used a projected fnal 8 wns, or Par Wns, durng the season and mantan these values to seasons end, despte mnor varatons n predctons (see Appendx Eq 1). Ths does, on occason, return a result that s not possble. For example, a team wth 4 wns at the completon of round 7, and sttng n 8 th place, yelds a Par Wns of 8.57. Therefore we round to the nearest 0.5, usng 0.25 and 0.75 as the round off ponts. In ths example, we round to 8.5, and ths s nterpreted as team requrng 8.5 wns (mnmum) from the remanng 15 games to make the fnals. Determnaton of the probablty of makng the fnal 8 At the heart of the second stage of the process s the bnomal dstrbuton. A number of other methods were consdered, such as smulatng the remander of the season usng success probabltes for each team usng p = 0.5, or varyng p; also averagng the number of wns of all teams and forward multplyng to determne the number of wns needed to make the fnal 8. Ultmately, t was both smplcty and a reducton of varablty that settled our choce. We defne the probablty of team at the completon of round r makng the fnal 8 as Pr (F8/r). Usng B (x, n, p) (the cumulatve bnomal dstrbuton functon wth x = number of successes, n = number of trals and p = probablty of success), we have Equaton 2 (see Appendx Eq 2). Notably, one must consder the value of p. We have chosen to look at two methods, the frst, and predomnant choce n our results, s the classc con toss model p =0.05. The second method uses the wnnng rato (p = TW /r). One could be tempted to use successful predcton probabltes such as those determned by Stefan and Clarke (1992); or the smpler wnnng rato. However, we wanted the system to be as smple as possble, and the ntroducton of a nested probablty model may complcate ths dea. A bref treatment of ths s gven n the dscusson secton. The unmportance of a match We defne the Importance for team at the end of round r, or I (r), as Equaton 3 (see Appendx Eq 3). Now we unpack the two components of Importance (see Appendx Eq 4 and 5). By usng the bnomal cumulatve densty functon to model the probablty of makng the fnals based on wnnng or losng the next match, we can, n turn, calculate the Unmportance of a match. Through some neat cancellaton of terms we obtaned a smple result for the Unmportance (see Appendx Eq 6). Allocaton of Draft Pont Reward (DPR) The allocaton of DPR s smply the Unmportance probablty multpled by the probablty of not makng the fnal 8 at round r. In ths way, the Unmportance s tempered by the lkelhood of makng the fnal 8. Teams that cannot make the fnal 8 receve the hghest weght possble (1), that s, the full Unmportance probablty, as long as they wn the match. The allocaton of DPR for team at round r s gven by the followng Equaton 7 and 8 (see Appendx Eq 7 and 8). Usng the DScore for the natonal pre-season draft The use of the DScore towards draft selectons encompasses some parts of the AFL s latest polcy on prorty pcks. For our DScore system, the teams are ranked 1 through 16, wth the hghest DScore attractng pck 1, and the lowest pck 16. Ths orderng remans for the subsequent teratons of the draft wth one excepton. Current AFL polcy dctates that a team that wns less than or equal to 4 matches n a season receves a prorty pck n the second round of the draft. As a method of protectng teams that may never wn another match after round 6, we employ a smlar prorty pck system,

512 Probablstc AFL draft model whereby a team that wns less than or equal to 5 matches n a season receves a prorty pck at the start of the second round of the draft. Ths s a lttle more generous than the AFL system, however the bottom sde wll not necessarly end up wth the frst draft pck under the DScore model. RESULTS We begn by examnng how the system operated for 2005 n fner detal. We then cover some nterestng scenaros, and nvestgate the mplcatons of the model. The 2005 season For season 2005, a number of teams remaned n contenton for the fnal 8 rght through to the last round. The fnal round saw fve teams competng for three fnals places. One wn separated 6 th through 10 th at seasons end. Notably, half a wn separated last (16 th ) from 14 th and all three bottom sdes receved a reward from the AFL for wnnng less than or equal to 5 matches. Table 1 outlnes the fnal results of three draft systems; frst the varable success DScore model, then the 50-50 DScore model, and fnally the AFL system (Fgure 1). Note that there s lttle varaton when usng a team s wn rato to determne Pr (F8/r) nstead of the smpler p = 0.5, and henceforth we wll only consder the equal success probablty model. Table 1. Draft pck comparson for DScore and actual draft system for the 2005 AFL season. Team DScore AFL DScore p = Draft p TW /r =.05 System Carlton 9 9 4 Collngwood 7 7 5 Hawthorn 5 3 6 Essendon 2 2 7 Rchmond 11 11 8 Brsbane 4 4 9 Fremantle 6 6 10 Western Bulldogs 1 1 11 Port Adelade 3 5 12 Melbourne 14 13 13 Geelong 15 14 14 Kangaroos 13 15 15 St Klda 10 10 16 Sydney 8 8 17 West Coast 16 16 18 Adelade 12 12 19 Prorty 17 17 1 Prorty 18 18 2 Prorty 19 19 3 Carlton, Collngwood and Hawthorn receved prorty pcks 1, 2 and 3 respectvely under both the AFL model and our model, although ours comes nto effect n round 2 of the draft. Varatons n the 2005 season round-by-round results are gven n Fgure 1. 6.00 5.00 Western Bulldogs, 5.28 Essendon, 4.67 DScore 4.00 3.00 Hawthorn, 3.55 Brsbane, 3.54 Port Adelade, 3.40 Fremantle, 3.25 Collngwood, 2.89 2.00 1.00 0.00 Sydney, 2.18 Carlton, 2.00 St Klda, 1.52 Rchmond, 1.22 Adelade, 0.87 Melbourne, 0.86 Geelong, 0.82 Kangaroos, 0.53 West Coast, 0.10 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Round Fgure 1. DScore by Round by teams for 2005.

Bedford and Schembr 513 Varaton of the DScore throughout the season s evdent; wth the number 1 pck changng teams 11 tmes durng the season - twce n the last three rounds. Also, pcks 3 to 7 provded extremely close results n the fnal round, gven that f Collngwood had won ts last match aganst the Western Bulldogs they could have secured pck 3 (nstead of 7) and cost the Western Bulldogs frst pck. So a wn to Collngwood under the DScore model would see a rse to pck 3, however a wn under the AFL model would have seen a drop to pck 5. An evaluaton of the ncentve of the DScore model Ideally the DScore model should evdence hgh DPR contnuously for low placed teams, gven they wn. Table 2 outlnes the number of teams n contenton for the number one pck n the last round, and three rounds before the end of the home and away season under the DScore model. The frst overall draft pck changed teams n the last round durng seasons 2001, 2005, and n the last 3 rounds durng seasons 1997, 1999, 2001, 2003, 2004, 2005. There was a blowout n the DScore n 2000 and thus, the race for the top draft pck was over by round 20. However, sx teams fought t out for pcks 2 to 7. Of course, these matches were not played wth the DScore ncentve and therefore mposng t retrospectvely s hypothetcal. Table 2. Number of teams n contenton for the number one draft pck gong nto the fnal round of seasons 1997 to 2005. Season Number of teams n contenton for Pck 1, Round 22 Teams n contenton for Pck 1, Round 19 2005 2 5 2004 2 3 2003 3 5 2002 2 5 2001 3 4 2000 1 5 1999 2 5 1998 1 6 1997 4 6 Asde from the 1998 and 2000 seasons, the race for the number one draft pck would have remaned alve and well pror to the fnal round. Importance Of nterest to us was when the maxmum value of mportance occurs for each team. We then sorted the teams by fnal ladder poston (FLP) and calculated the mean and standard devaton of the round, as gven n Fgure 2. Round 21 19 17 15 13 11 9 7 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1415 16 Fnal Ladder Poston Fgure 2. Error bars of maxmum round of mportance by Fnal Ladder Poston (1997-05). As shown n Fgure 2, the teams fnshng n the top 2 and bottom 3 have ther most mportant games generally n the early rounds of the season (note that we have only consdered round 6 onwards). All other teams headng towards the mddle of the ladder have maxmal mportant matches later n the season. As one would expect, the 8 th FLP has the maxmal mportance match n the last three rounds. DISCUSSION It s somewhat dffcult to measure the effect of our model on past results as we are mplementng our method retrospectvely. As a consequence, where players would end up under our model would be dfferent to realty and therefore team success may change. Even so, the fndngs are stll an eyeopener, and ndeed motvate poorer teams toward success. As was shown n the results secton, for the fnal round of 1998, the 1 st and 2 nd draft pck had been decded. However, 11 teams could stll be playng n expectaton of a change n ther draft pck wth a vctory. The deal advocate of our system was the fnal round of 2003. Geelong played St Klda, and t could be argued they were playng for nothng, sttng 10 th and 13 th on the ladder - no fnals place or prorty pck at stake. Under our DScore system the wnner of that match would take 1 st pck and the loser potentally 3 rd. The match played on Saturday had Geelong preval by 19 ponts, snatchng 1 st pck. Remarkably, the result was not yet settled, wth the Sunday encounter between Hawthorn (9 th ) and Rchmond (10 th ), (agan two sdes wth nothng to play for), pvotal n the DScore outcome. Hawthorn won by 4 ponts, wnnng ther fourth game n a row, snatchng the number 1 pck on the last game of the home and away season!

514 Probablstc AFL draft model Alternatves A crtcsm that may be leveled at the DScore system s that teams whch contnually lose are never rewarded. A possble way of assstng teams that consstently lose may be to reward a gallant defeat. Calculatng an expected and actual margn, then smoothng the dfference, s an approach used n other areas of sport analyss, such as tenns as n Bedford and Clarke (2000). They used ther model to predct and mprove upon ATP ratngs n tenns based on margn of vctory rather than wn or loss. Once agan, a team may play so poorly as to never get wthn the expected margn, and the same problem arses. We beleve the prorty crtera s a reasonable approach to combat ths, and we can only hope teams would try harder to wn to obtan better draft pcks, and n turn, enhance ther future chances, rather than le down and be rewarded for defeat. A pont of nterest rased n the methods secton was the possble ncluson of a team s relatve skll nto the system, ether usng probabltes such as those poneered by Stefan and Clarke (1992), or more arbtrary measures such as a wn rato to weght the DPR. The use of an opponent weght would see some rather unattractve scenaros. Specfcally, the use of a probablty based multpler on the DPR ntroduces only occasonal need for lowly placed teams to wn, as they need only defeat one successful team and reap a hgh DPR, thereby obtanng a hgh draft pck. Ths s a clear dsncentve as the DScore system s desgned to encourage teams to wn every game possble. CONCLUSIONS In ths paper, we have developed a unque system for player allocaton n the AFL draft usng probablstc prncples desgned to encourage success. Whlst the AFL system was not desgned to encourage teams to lose, t does reward teams that only wn a small amount of games. Our model, known as the DScore model, unquely encourages teams to strve for vctory wth a hgh draft pck as the prze, especally when the game (and ther season) s - n terms of the fnals - Unmportant. Utlzng ths prncple of unmportance, we cted exctng and motvatng cases whereby otherwse meanngless encounters become a battle for hgh draft pcks. The DScore model may also have a broad appeal, wth potental outcomes easly publshable n daly newspapers and on the nternet, wth the relevant draft permutatons provdng a motvator not only for the club, but for the supporters alke. REFERENCES Bedford, A. and Clarke, S. (2000) A comparson of the ATP ratngs wth a smoothng method for match predcton. In: Proceedngs of the Ffth Australan conference on Mathematcs and Computers n Sport. Eds: Cohen G. and Langtry T. Sydney: Unversty of Technology Sydney. 43-51. Grer, K. and Tollson R. (1994) The rooke draft and compettve balance: The case of professonal football. Journal of Economc Behavor and Organzaton 25, 293-298. Morrs, C. (1977) The most mportant ponts n tenns. In: Optmal strateges n sport. Eds: Ladany, S.P and Machol, R.E. Amsterdam: North-Holland Publshng Company. 131-140. Spurr, S. (2000) The baseball draft: A study of the ablty to fnd talent. Journal of Sports Economcs 1, 66-85. Stefan, R. and Clarke, S. (1992) Predctons and home advantage for Australan rules football. Journal of Appled Statstcs 19, 251-261. Taylor, B.and Trogdon, J. (2002) Losng to wn: Tournament ncentves n the Natonal Basketball Assocaton. Journal of Labor Economcs 20, 23-41. KEY POINTS Draft choces are allocated usng a probablstc approach that rewards teams for wnnng unmportant matches. The method s based upon Carl Morrs Importance and probablstc calculatons of makng the fnals. The mportance of a match s calculated probablstcally to arrve at a DScore. Hgher DScores are weghted towards teams wnnng unmportant matches whch n turn lead to hgher draft selectons. Provdes an alternatve to current draft systems that are based on losng to wn. AUTHORS BIOGRAPHY Anthony BEDFORD Employment Senor Lecturer n Statstcs Degree Ph.D. Research nterests Sport Statstcs, Queueng Theory, Bostatstcs, Smulaton. E-mal: anthony.bedford@rmt.edu.au

Bedford and Schembr 515 Adran SCHEMBRI Employment Honours Student Degree Bachelors Degree Research nterests Psychology and Statstcs n Sport E-mal: s3020239@student.rmt.edu.au Dr Anthony Bedford School of Mathematcal and Geospatal Scences, RMIT Unversty, Bundoora, VIC, Australa

516 Probablstc AFL draft model APPENDIX Equatons (Eq): ( TW ( r) ) 22 Eq 1: () 8 ranked team Par wns = max th r TW ( r), 0 r We formally defne the number of wns requred after round r for team as Par wns ( r) ; and the total number of wns for team at the completon of round r as TW ( r). Usng the 8 th ranked team at any round r as the deal Par proporton n determnng the wns requred to make the fnals. Eq 2: Pr ( F8 r) = 1{ ParWns ( r ) = } + 1 ( ParWns ( r ) > 0) ( ParWns ( r) 22 r ) { }[ 1 B( Par wns( r) 1;22 r, p )] 0 where 1 {} a s the ndcator functon takng value 1 f condton a s true and 0 f false. Eq 3: I () r = Pr ( Make F8 Wn match r + 1) Pr ( Make F8 Lose match r + 1) Pr ( Make F8 Wn match r + 1) = Eq 4: 1{ ParWns r) = 0} + 1 ( ParWns ( r ) > 0) ( ParWns ( r) 22 r ) { }[ 1 B( ParWns ( r) 2;22 ( r 1 ), p) ] ( + Pr ( Make F8 Lose match r + 1) = Eq 5: 1{ ParWns r) = 0} + 1 ( ParWns ( r ) > 0) ( ParWns ( r ) 22 r ) { }[ 1 B( ParWns ( r) 1;22 ( r 1 ), p) ] ( + () U ()=1 r I r =1+ Pr ( MakeF 8 Losematch r +1) Pr ( MakeF 8 Wn match r +1) =1+ [ 1 Bx;n, ( p) ] [ 1 Bx 1; ( n, p) ] Eq 6: x =1+ 1 bk;n, ( p) k = 0 x 1 1 bk;n, ( p ) k= 0 =1+ [ 1 ( b(0;n, p) + b(1; n, p) +L+ b(x;n, p) )] 1 b(0;n, p) + b(1;n, p) +L+ b(x 1;n, p) =1 b(x;n, p) So, ( ParWns ;22 ( r 1 p) U ( r) = 1 b + ), [ ( )] Notng b ( x; n, p) (the dscrete bnomal dstrbuton functon wth x = number of successes, n = number of trals and p = probablty of success) n the fnal result, Unmportance s smple to evaluate, relyng on a dscrete rather than contnuous result, and gven the values of Par Wns, can be easly computed usng a scentfc calculator. Eq 7: DPR ( r) 1{ } 1{ 6} U ( r) ( 1 Pr ( F8 r) ) = wns matchr r> where 1 {} a s the ndcator functon takng value 1 f condton a s true and 0 f false. The Draft Score, or DScore, for team at round r s smply the sum of the DPR: Eq 8: DScore () r DPR () k, r { 7,..,22} r = k= 7