Towards a New Dynamic Measure of Competitive Balance: A Study Applied to Australia s Two Major Professional Football Leagues *

Transcription

1 Economic Analysis & Policy, Vol. 39 No. 3, december 2009 Towards a New Dynamic Measure of Compeiive Balance: A Sudy Applied o Ausralia s Two Major Professional Fooball Leagues * Liam J. A. Lenen School of Economics and Finance, La Trobe Universiy, Vicoria, 3086, Ausralia ( l.lenen@larobe.edu.au) Absrac: A new measure for compeiive balance beween seasons is proposed, which akes he form of a mobiliy gain funcion, based on each eam s win raios from he curren and previous seasons. This dynamic funcion measures compeiive balance wihin a oneperiod change framework. While i is no suggesed ha his measure replace useful exising wihin-season measures, such as he widely used acual-o-idealised sandard deviaion (ASD/ISD) raio, his measure does overcome one of he shorcomings of wihin-season measures ha is, he abiliy o pick up uncerainy of oucome from season o season, raher han merely from round-o-round. Hence, i is suggesed ha his measure could be used in conjuncion wih wihin-season measures in ime-series analysis. An applicaion o Ausralia s Ausralian Fooball League (AFL) and Naional Rugby League (NRL) over a cenury of daa reveals numerous ineresing comparisons. I. INTRODUCTION Compeiive balance (CB) refers simply o he degree of evenness in spors leagues. However, he idea of measuring and quanifying CB is far from clear-cu, given he diversiy of defining i precisely. Mos noably, complicaions arise when one considers he disincion beween he hree ofen-cied dimensions of CB. Firsly, here is he noion of uncerainy of oucome of any single mach/cones, o which he uncerainy of oucome hypohesis referred originally. Secondly, we have he concep of pariy or oherwise in erms of he disribuion of wins beween eams in any given season, or wihin-season CB. Finally, he idea of an * Earlier versions of his paper were presened a: (i) he Saff Developmenal Workshop, Deparmen of Economics and Finance, La Trobe Universiy, 28 Sepember 2006; (ii) he Seminar Series, Cenre for Operaions Research and Applied Saisics, Salford Universiy, UK, Ocober 2006; (iii) he Seminar Series, Deparmen of Economics, BI Norwegian School of Managemen, Norway, 30 May 2007; and (iv) he Ausralasian Meeing of he Economerics Sociey, Universiy of Queensland, Brisbane, 3-6 July The auhor would like o hank he various paricipans of he workshop, seminar and conference for heir commens and suggesions, especially David Prenice and David Forres, as well as Suzanne Sommer, Ishaq Bhai and Andrew Raponi for some preliminary inpu. 407

2 Towards a New Dynamic Measure of Compeiive Balance: A Sudy Applied o Ausralia s Two Major Professional Fooball Leagues equal disribuion of premierships (iles) over he medium-o-long run (or beween-season CB), is also of imporance. The former is ofen proxied in cross-secional sudies by he use of being marke daa; see Owen and Weaherson (2004) as a recen example for daa from he Super 4 SANZAR provincial league in rugby union. However, his sudy is more concerned wih comparing and conrasing he experiences of wihin-season CB and beween-season CB in Ausralia s wo larges professional spors leagues. These leagues are: (i) he Ausralian Fooball League (AFL); and (ii) he Naional Rugby League (NRL). As will be demonsraed, he hisorical comparison beween he AFL and NRL wih respec o CB is very differen depending on which mode of CB is he basis for evaluaion. Subsequenly, an aemp o reconcile his apparen conundrum is made via he proposal of a new CB measure, which akes he form of a gain funcion of each eam s sanding in a given season relaive o he previous season, and aggregaing hese individual eam gain funcions league-wide. The gain funcion will ake a higher value when here is greaer insabiliy and cenral endency from season-o-season in he win percenages of eams in he league. A horough empirical invesigaion of his measure is underaken in his paper using all available hisorical daa in boh leagues, and he resuls are analysed. The moives for invesigaing hese specific issues using he AFL and NRL specifically as case examples are srong. Boh compeiions have long and illusrious hisories. The AFL (formed in Melbourne as he Vicorian Fooball League in 896) and he NRL (formed in Sydney as he New Souh Wales Rugby League in 907) have a cenury-long radiion of league compeiion, uninerruped even by World Wars I and II. 2 Furher, he respecive radiional hearlands of boh codes of fooball are almos muually exclusive geographically, wih Ausralian Rules fooball he dominan winer spor in Vicoria, Souh Ausralia, Wesern Ausralia, Tasmania and pars of Souhern New Souh Wales. Meanwhile, Rugby League is mos influenial in mos of New Souh Wales, he Ausralian Capial Terriory and Queensland. Moreover, unil he commencemen of naional expansion of boh leagues in he early 980s, here was virually no encroachmen by eiher compeiion on he oher s erriory, meaning ha boh leagues had a near-monopoly saus in heir respecive radiional markes. 3 Even in he curren day, hese leagues sill wield a considerable amoun of monopoly power in hose radiional markes. Furhermore, hese leagues exhibi a number of ineresing characerisics ha are highly unique o Ausralian professional spors. However, i is worh noing ha he curren sudy could be applied o oher professional spors, hough in he Ausralian conex, here is an insufficien sample lengh for oher spors, as oher naional leagues are a relaively new phenomenon. Despie he geographical divide, here is sill some degree of compeiion beween he leagues (and o an exen wih oher, smaller leagues). While he AFL is he larger of he wo leagues by any meaningful measure, he NRL is iself sill significanly bigger han he nex bigges of he oher professional spors leagues in Ausralia. Neverheless, despie he compeiion, each of hese leagues has always kep a keen wach over developmens in he oher, in an aemp o In heory, his measure has oher possible applicaions in professional spors, such as he disribuion of revenues beween eams, or he disribuion of income beween ahlees in individualisic spors. 2 In fac, he NRL suffered neiher a reducion in he number of eams nor he number of rounds during eiher of he World Wars. 3 This is also rue when comparing hese leagues o oher spors, as here was no ruly Ausralian naional league in any spor unil fooball (soccer) and baskeball in 977 and 979, respecively. 408

3 Liam J. A. Lenen learn from he successes and failures of is apparen rival. On his heme, one noable difference beween he wo leagues is ha he AFL has mainained remarkable consisency wih respec o he se of paricipaing eams, wih only one eam disconinued, one relocaed and one merged is enire hisory. Comparaively, he NRL has hisorically experienced a significan churn of eams hroughou is hisory, culminaing in he raionalisaion ha followed he Super League (SL) War wih he hen Ausralian Rugby League (ARL) of he mid-o-lae-990s. The remainder of his sudy proceeds as follows: he nex secion oulines he naure of exising CB measures and wha hey have o reveal abou he AFL and NRL hisorical daa. This is followed by a shor exposiion on he problems associaed wih hese measures. In secion 4, a new CB meric is oulined, which provides a possible soluion o he problems oulined earlier. An empirical analysis of all seasons in he hisory of he AFL and NRL using he new measure is hen underaken in secion 5. Secion 6 concludes on a very general noe. II. BACKGROUND ON CB MEASURES 2.. The Baseline Meric Subsequen o he discussion in he previous secion on he differen dimensions of CB, i is worh poining ou ha wihin a ime-series framework; ofen wihin-season measures are required so ha an annual (one season per year) series can be produced for he purposes of empirical modelling. While ha limiaion should, in heory, simplify he analysis, here is sill significan diversiy of wihin-season CB measuremen mehods. In he spiri of his heme, here are several measures uilised commonly in he spors economics lieraure (refer o Michie and Oughon, 2004, for a comprehensive lising). The mos popular measure wihin his framework is he ASD/ISD raio, which is ofen aribued o Noll (988) and Scully (989). This raio is defined simply as he quoien of A he acual sandard deviaion of win raios of all eams in he compeiion in season, σ, o he idealised sandard deviaion ha would be expeced in a league (wih r rounds) if he I resul of each mach were purely random, σ. This measure is represened formally by he following equaion A σ I σ = N i= wi 0.5 r 0.5 r 2 N where in season, w i denoes he number of games won by eam i, r is he number of games A played (rounds), and he number of eams in he league is denoed by N. The numeraor σ is no affeced by changes in N over ime, since i uilises he mean squared deviaion of he I win raio from is mean (0.5). The denominaor σ is allowed o change over ime in he even ha r changes. This is no a rivial maer given he significan number of changes o boh N and r over ime during he respecive hisories of boh he AFL and NRL. A full annual imeseries reproducion of he evoluion of he values of N and r in boh compeiions is provided in Table, from which an appreciaion can be gleaned as o he number of hese changes. () 409

4 Towards a New Dynamic Measure of Compeiive Balance: A Sudy Applied o Ausralia s Two Major Professional Fooball Leagues Table : Annual Series of N and r in Boh Leagues AFL NRL AFL NRL Year Teams Rounds Teams Rounds Year Teams Rounds Teams Rounds / / / / / / /0 22/ Rounds figures in 908, 920, 925, 928, 936 and 942 indicae uneven disribuion of byes. In he 943 AFL season, boom-placed S.Kilda was eliminaed afer maches. NRL figures for 997 indicae ARL/SL. 40

5 Liam J. A. Lenen 2.2. A Hisorical Comparison In an aemp o compare he relaive hisories of wihin-season CB using his measure, Figure plos boh he original series daing back o he commencemen of boh compeiions, as well as a HP filer (Hodrick and Presco, 997) for each series, merely as a way of geing an idea of he casual rend. The HP filer is obained by finding a soluion o he opimisaion problem where X is he acual value of he series; Z is he rend (or growh ) componen; and λ is he smoohing parameer, se o he sandard value of 00 in line wih he suggesion of Hodrick and Presco (997) for annual daa. Figure : Original (Thin Line) and HP Trend (Bold Line) Daa for he ASD/ISD Raio in Boh Leagues (2) 2.4 AFL (Solid) and NRL (Dashed) Figure reveals ha he ASD/ISD raio is reasonably volaile as would be expeced, since one season can be very differen o he nex, bu ha hisorically, using he HP rends, he NRL has hisorically been he more compeiively balanced compeiion. The only hree periods where his was he excepion were: (i) during and immediaely following World War I, a period when he hen recen inroducion of Melbourne meropolian zoning had succeeded iniially in balancing he AFL somewha; (ii) he lae 950s and early 960s, a period ha coincided wih he dominaion of S. George in he NRL, which may be considered an oulier; and (iii) he mid-o-lae 990s, which may be explained by he surge of inequaliy in he NRL ha arose from he SL War. In fac, beween 908 and 2006, he raw ASD/ISD raio has been lower for 4

6 Towards a New Dynamic Measure of Compeiive Balance: A Sudy Applied o Ausralia s Two Major Professional Fooball Leagues he AFL han he NRL in only 32 of hose 99 seasons. See Booh (2004) for a more deailed analysis of he hisorical AFL sory. This supposiion is suppored by he figures in Table 2, which repors he summary saisics in boh leagues for he ASD/ISD raio, as well as a column for he AFL since 908 (he year in which he NRL commenced) o equae he sample period lenghs. As is shown clearly, he NRL is easily he more balanced compeiion hisorically by any crierion according o his measure. This finding is a lile difficul o explain, since hisorically, he AFL has always used a sricer and wider range of labour marke and revenue-sharing devices han he NRL, in an aemp o ensure compeiive balance. Furhermore, here appears o be no obvious explanaion ha can be used o accoun for his finding. Table 2: Summary Saisics for he ASD/ISD Raio for Boh Compeiions AFL AFL (908) NRL Maximum Minimum Mean Median Sandard Deviaion < > Alernaive CB Measures In any case, oher measures of CB are also popular, wih a view o capuring differen elemens of he disribuion of wins wihin a season. One such alernaive measure is he Herfindahl Index of CB (HICB), based on he radiional Herfindahl Index of marke concenraion, expressed as N 2 4 wi HICB = (3) N i= r Also, he Concenraion Index of CB of he op x eams, C(x)ICB, specified as C( x)icb = 2 x w j (4) x r j= where j =,2,..., x now refers specifically o he op x eams in rank-order. Ye anoher measure is he Gini Coefficien, represened as Gini = 4 N i i= j= w r j N 2 2 N j= w r j While no highly sensiive o changes in N and r, i could be adjused according o he U and For (2002) mehodology. Newer CB measures seem o arise on an ongoing basis, wih one (5) 42

7 Liam J. A. Lenen of he more recen conribuions being he Index of Dissimilariy (ID) as applied by Mizak, Sair and Rossi (2005), which resembles a discree version of he Gini Coefficien ID = N i= ' 2wi ( r $ % " & Nr # 2 Anoher is he surprise index of Groo and Groo (2003), which is no applied in he empirical secion here. Some of hese measures can also be applied o beween-season CB, where he number of premierships over he period under examinaion is he disribued variable. However, he deficiency wih his approach is ha only a single observaion for he enire sample period, raher han a complee ime-series will resul. 4 Even he ANOVA mehodology used by Eckard (998 and 200) involves he spliing of he sample period ino subsamples, each of several years lengh. Moreover, i is problemaic o underake such an exercise over a period where eams have enered and/or exied he league Comparison wih Beween-Season In a casual aemp o gain an idea of hisorical beween-season CB in he wo leagues, Tables 3 and 4 have been consruced. Table 3 repors he frequency of successive premiership runs caegorised by lengh for boh leagues. Since a more balanced compeiion should have less successive premiership runs, he AFL easily sands ou as he more balanced compeiion in his respec, according o he weighed sum (number of runs by lengh muliplied by heir individual lenghs). Table 3: Frequency of Runs of Premierships by Lengh of Years for Boh Compeiions Run AFL AFL (908) NRL > * Weighed Sum *Five successive premierships were won by Souhs ( ) and by S. George ( ). An analogous way of looking a he same problem is repored in Table 4, which reveals he number of (non-overlapping) runs of years whereby he previous k premierships have all been won by differen eams. Looking a he middle column, i can be seen ha since 908, here have runs of four successive seasons wih differen premiers in he AFL, bu only 5 such runs in he NRL. There has been one single run of five years wih differen premiers in each league. The longes runs in each compeiion are six years in he AFL ( ) and seven in he NRL ( ). Since a more balanced compeiion should have more of hese 4 See Leeds and von Allmen (2005), appendix 5A (pp ) for a demonsraion of how his evenuaes. (6) 43

8 Towards a New Dynamic Measure of Compeiive Balance: A Sudy Applied o Ausralia s Two Major Professional Fooball Leagues runs, he AFL sands ou again as he more balanced compeiion according o he weighed average. Table 4: Non-Overlapping Runs of Years (by Lengh) whereby Previous k Premierships had been Won by k Differen Teams for Boh Compeiions # Run AFL AFL (908) NRL >5 Weighed Sum # In he case of overlapping runs of differen lenghs, preference is given naurally o he longer run. A puzzle arises from he previous comparaive analysis on CB beween he AFL and NRL, which is ha he evidence of CB is so significanly in favour of one league (NRL) in a wihinseason framework, ye so significanly in favour of he oher league (AFL) in a beween-season framework. While i is conceded ha he wo modes are obviously no equivalen, hey are cerainly no unrelaed eiher a eam ha is (as an exreme case) many imes more powerful han any oher eam in a league will no only dominae over many seasons, bu will also dominae wihin any given season as well. Wih his puzzle in mind, le us proceed o look a he measuremen of CB in greaer deail. III. A POSSIBLE PROBLEM WITH THESE MEASURES 3.. Effec of Ouliers The pracice of using he ASD/ISD raio as a measure of compeiive balance is validaed by, iner alia, Humphreys (2002), who describes he ASD/ISD raio as a useful measure, and finds ha i does a beer job a explaining compeiive balance han he alernaive measures ha he evaluaes. The specificaion of he ASD/ISD raio as exposed in equaion (), however, brings o ligh one of is possible idiosyncrasies since i is a relaive sandard deviaion measure, i is highly sensiive o he occasional oulier. Recen examples include he highly dominan eams (one-season basis) such as Carlon (995) or Essendon (2000) in he AFL, as well as Parramaa (200) or Canerbury (2002, nowihsanding he salary cap breach penaly) in he NRL. Equivalenly, very poor eams such as Fizroy in heir final year of 996 or Fremanle (200) in he AFL, as well as Wess in heir final year as an independen eam of 999 or Souhs (2003) in he NRL, will also conribue heavily o he measure. As a nice numerical illusraion of he possible effec of a single oulier on he ASD/ISD raio, if you were o ake he final fooball (soccer) ladder (league able) from he inaugural season (2006) of he fledgling Naional League (A-League), by assigning a win value of 0.5 for a draw, he ASD/ISD raio is calculaed o be However, if he las-placed New Zealand Knighs had hypoheically won one exra mach a he expense of each of he oher eams in he compeiion, he ASD/ISD raio would have fallen o jus significanly 44

9 Liam J. A. Lenen more compeiive han even a random disribuion of wins. 5 This idiosyncrasy highlighs a possible shorcoming of he ASD/ISD raio, insofar ha i could be argued ha an exraordinarily good or bad eam in a given season is no such a bad hing, so long as hey rever owards he pack in he following season Shor-run or Long-run Dominance The ASD/ISD raio is good a picking up only wihin-season effecs, no mobiliy of eams in erms of pecking-order on he ladder. Therefore, he main disadvanage wih he ASD/ISD raio is ha lack of CB, as refleced by eams occupying a similar place in he pecking order of he compeiion over a number of years (i.e. over ime), will no be picked up very well. Wha may have more profound implicaions on CB is he scenario whereby a eam finishes in he same (or a leas very similar) posiion from year o year, wihou being eiher oally dominan or weak, like Richmond in he AFL from (excluding 200 and 2004). As a recen AFL example of his disincion, one could ake he Brisbane side ha won hree premierships in succession, and hen los he Grand Final in he following year ( ). This was arguably derimenal o CB insofar ha he compeiion gained a degree of predicabiliy over ha period. However, his effec will no be picked up very well by he ASD/ISD raio measure, since hey did no win he minor premiership (i.e. finished on op of he able a he conclusion of he regular or home-and-away season) in any given year, hus no being he bigges oulier hence, no conribuing mos heavily o he ASD. The closes equivalen (bu more daed) example in he NRL would be he Parramaa side ha in a sixyear period (98-986) won four iles and one runners-up, ye achieved a win raio of beer han 0.7 only once during ha period (heir exremely dominan 982 season). The measure for CB proposed here, however, will pick hose effecs up, as i compares he win raios in any given season o hose of he previous season, o idenify if he compeiion has predicabiliy via lack of variaion in erms of he win raios from season-o-season. IV. A NEW (DYNAMIC) CB MEASURE 4.. Basics The (albei limied) dynamics in his CB measure quanifies direcly he gains from eams ending cenrally in a given season vis-à-vis he previous season. This one-season change operaor approach provides he advanage ha he measure incurs a loss of only one degree of freedom, which is paricularly appealing wihin a ime-series framework. This measure has some concepual similariies o he Markov-chain analysis approach, which can be applied in a similar way o ha of Hadley, Ciecka and Kraumann (2005) for Major League Baseball daa. However, he measure advocaed here is more clearly defined, since here is a large number of alernaive ways o define Hadley e. al. s mehodology of caegorising eams as winners, conenders and losers. The measure also has concepual similariies wih he 5 In line wih Cain and Haddock (2006), i could be argued ha he 3 poins for a win and for a draw sysem implies ha a draw should be assigned a win value of only /3. However, if his was he case, hen he previous analysis does no change much, since he implied ASD/ISD raio changes from.5562 o

10 Towards a New Dynamic Measure of Compeiive Balance: A Sudy Applied o Ausralia s Two Major Professional Fooball Leagues more convenional Spearman rank-order correlaion marix, bu is coninuous (raher han discree) in naure, and is also asymmeric in erms of direcion of movemen up or down he able, hus overcoming wo major problems associaed wih ha measure. Prior o defining he meric in quesion, a clarifying poin needs o be made. In he consrucion of he proposed measure of CB, i mus be remembered ha CB is he objecive funcion, and herefore, he use of he erm gain funcion henceforh refers o gains in he objecive funcion. This poin is made o clear up any possible confusion wih gains o individual eams, since when a dominan eam comes back o he pack, he value of he gain funcion increases, even hough ha paricular eam is worse off Formal Expression For noaional purposes, le y i, be known as he CB gain funcion for eam i in season, and c can be used o represen he league-aggregaed CB gain funcion in season. Using his noaion, we can hen formulae a couple of simplifying relaions wi, Wi, = (7) r i, α 0.5 (8) = W i, where W i, can be hough of simply as eam i s win raio in he curren season, and α is simply a sensiiviy parameer in he gain funcion ha performs hree funcions: (i) i ses he maximum value of he gain funcion for each eam, depending on he observed value of W i, ; (ii) i ensures a monoonic ransformaion of y i, as W i, changes; and (iii) i ensures symmery of gains beween boh dominan eams and sruggling eams, if hey end cenrally he following season. We are now in a posiion o define he proposed compeiive balance measure, he mobiliy gain funcion (MGF), henceforh denoed by c, as N yi, c = (9) i= N where y $, # = & " i ' $ Wi, & = 0.5 = 0 : if eiher# Wi, % Wi, & < 0.5 " 0.5 < Wi, & % Wi, $ Wi, & < Wi, < 0.5, i + ' : if eiher# " 0.5 < Wi, < Wi, & $ Wi, % 0.5 < W&, i = ' : if eiher# " Wi, & < 0.5 % Wi, 2 ( W & 0.5) (0) 46

11 Liam J. A. Lenen where y i, = 0. 5 and y = 0 (he upper and lower bounds of y i, i,, respecively). In beween he wo bounds, he funcion akes a simple quadraic form. The firs case (y i, = 0: if W i, = 0.5) is simply a compleing condiion, acknowledging ha if eam i wins exacly half of heir maches in season, hen here can be no furher compeiive balance gains for ha eam in season +. Equaion (0) may appear o be raher convolued, however, y i, can sill be represened quie sensibly in graphical form, as in figure 2. For eams and 2, imagine ha W, = 0. 4 and W 2, = For eam, which is near mid-able anyway, only low compeiive balance gains are possible if hey converge on 0.5. If W, > 0. 5, hen he gains are reained a α (i.e. hey canno decline), because wha we are really rying o capure here is mobiliy of able posiions of eams from one season o he nex. Therefore, even if W,, eam i has sill managed o work is way up o he dominan eam from he boom half of he able he year before, hence an occurrence ha was highly unpredicable a -, which is why he maximum score is sill reained. However, he gains canno exceed α, because once he eam crosses 0.5, hen compeiive balance is resored fully (according o he definiion being applied here). Figure 2: Illusraive Example of Gain Funcion for Two Teams y i, W i For eam 2, however, which was he dominan eam in season -, he poenial for compeiive balance gains in season are quie significan. The reason for he quadraic specificaion lays in ha he larges marginal increase in he gain funcion comes when eam 2 comes back o he field slighly, increasing he uncerainy of oucome. However, he diminishing marginal gains occur because as W 2, declines o (say) 0.6, he oucome of games becomes highly uncerain anyway. Finally, for he sake of simpliciy, i is decided ha y 0. This is jusified on he grounds ha if W 2, > 0. 9, hen season is hardly any less ineresing han season - anyway. Wi, < Wi, < 0.5 A furher jusificaion for he quadraic naure of i lies i, y, in he region 0.5 < W < W i, i, 47

12 Towards a New Dynamic Measure of Compeiive Balance: A Sudy Applied o Ausralia s Two Major Professional Fooball Leagues in he evidence presened in Table 5 for AFL hisorical daa. If we iniially impose a definiion of wha consiues a dominan eam as one ha wins a leas 90 per cen of is maches in he home-and-away season, i can be seen ha no one has ever failed o win he premiership in ha season, hough admiedly he sample is quie small. 6 Then, if he definiion is eased such ha any eam ha wins a leas 85 per cen of is maches is deemed o be dominan, hen i is seen ha home-and-away season dominance is no longer necessarily even a near-guaranee of premiership success. In fac, he probabiliy of winning he premiership falls o jus over wo-hirds, alhough in one case (920), wo eams were dominan, and hus i was impossible for boh of hese eams o win he premiership. Addiionally, if he definiion of dominance is weakened even furher o a minimum 80 per cen winning record, hen he probabiliy of premiership success declines even furher o jus over one-half, alhough he frequency of muliple dominan eams in he same season increases markedly. Finally, for eams winning a leas 60 per cen of heir home-and-away maches, bu less han 80 per cen (for which he erm conender may be preferred o dominan ), he probabiliy of winning he premiership predicably plummes, o less han one-sixh. Table 5: Frequency Disribuion of Dominan Teams in he AFL and Likelihood of Premiership Success Crieria Toal Frequency Premiers % Frequency (i > ) W i, W i, W i, * 0.60 W i, < N/A * However, his has occurred only wice since 939, and no a all since 972. On one of hese occasions (935), hree eams were dominan a W i, Analogously, i becomes exremely rare for a eam o win a premiership as W i, 0. 5 (from above). An ineresing case sudy o demonsrae his poin is he 997 season, when Adelaide won he premiership, despie winning only 3 ou of 22 home-and-away games (59. per cen). For good measure, hey wen back-o-back in 998 wih an idenical homeand-away record. The former insance represened he firs occasion whereby he evenual premier had won less han 60 per cen of heir home-and-away maches since he infamous war-affeced 96 season in which only four eams paricipaed and Fizroy amazingly won he wooden spoon and premiership in he same season Incidenally, i had been achieved once before by Melbourne in 900, bu wih he aid of an exremely idiosyncraic finals sysem. This hisorical evidence demonsraes boh he high desired reurns o cenral endency (i.e. forcing W i, owards 0.5) in season when W i, and he diminishing marginal reurns o 6 This kind of exercise is paricularly fascinaing in he conex of ournamen design of Ausralian professional spors leagues, because unlike he radiional European firs pas he pos sysem, leagues in mos Ausralian spors have a finals (playoff) series. Furhermore, unlike mos Norh-American spors, he finals series is no purely knockou-syle, raher he finals series is consruced specifically o give he eams ha finished higher on he ladder a he end of he home-and-away season an easier pah hrough o he (Grand) Final, and hus an inheren advanage in he finals series. This characerisic makes he AFL and NRL somewha unique in professional spors in his respec. 48

13 Liam J. A. Lenen cenral endency as W i, Furhermore, i could be argued ha here is no reason for he gain funcion o be asymmeric when W i, 0 and W i, 0. 5 (from below). As an alernaive form, equaion (0) could also be re-wrien o specify a linear (raher L han quadraic) gain funcion, y i,, should i be he case ha simpliciy were a huge issue, alhough in he spiri in which his gain funcion was consruced (and he jusificaion provided previously), i would be less preferable o do so. Neverheless, in proceeding o do so, one would use he following derived funcion in place of equaions (9) and (0), in order o calculae he L linearised mobiliy gain funcion, henceforh referred o as MGFL and denoed as c N L y L i, c = () i= N where W i, = 0.5 = 0 : if eiher W i, W i, < 0.5 (2) 0.5 < W i, W i, L y i, = W i, W i, : if { W i, < W i, < 0.5 = W i, W i, : if { 0.5 < W i, < W i, = α : if eiher W i, 0.5 < W i, W i, < 0.5 W i, A quick inspecion of Table 6, which oulines he same dominan eams analysis for he NRL as ha in Table 5, may provide some jusificaion for such a linear represenaion of he gain funcion. Table 6 reveals ha wih dominan eams being classified a W i, 0.90, he likelihood of winning a premiership is approximaely wo-hirds, alhough wo eams were dominan on wo separae occasions (928 and 995). While his probabiliy may be subsanially lower han he corresponding figure for he AFL, a W i, 0.85 however, he probabiliy of success hardly declines any furher. Nor does he likelihood of premiership success decline furher sill a W i, 0.80, remaining above 60 per cen and by hen higher han he corresponding AFL figure. Collecively, hese figures hrow some cauion on he asserion ha high iniial (bu diminishing) marginal reurns o cenral endency are desired. Table 6: Frequency Disribuion of Dominan Teams in he NRL and Likelihood of Premiership Success Crieria Toal Frequency Premiers % Frequency (i > ) W i, W i, W i, * 0.60 W i, < N/A Couning boh seasons and iles won by Brisbane and Newcasle in he Super League season (997). *However, his has occurred only hree imes since 934 and only once since

14 Towards a New Dynamic Measure of Compeiive Balance: A Sudy Applied o Ausralia s Two Major Professional Fooball Leagues Analogously, he premiership likelihood for conenders (0.60 W i, < 0.80) as displayed in Table 6 is higher (8. per cen) han he equivalen AFL figure. Incredibly, Wess Tigers became he firs NRL eam o win a premiership wih less han a 60 per cen home-and-away record in he exraordinarily compeiive 2005 season, a fea ha was asoundingly repeaed by Brisbane he following year. 7 Given he mixed evidence from Tables 5 and 6, boh he MGF and MGFL series are calculaed for boh leagues in he nex secion. In any case, having defined he CB mobiliy merics, le us advance now o some empirical evidence from hisorical AFL and NRL daa. V. AN EMPIRICAL INVESTIGATION OF THE HISTORICAL DATA 5.. A Simulaion Iniially, as a way of demonsraing he uniqueness of hese measures, ake he (alernaive) hypoheical wo-league example of Humphreys (2002). Here, hese hypoheical leagues, boh involving 5 eams and 4 rounds over 5 seasons, are perfecly unbalanced by any wihinseason measure in each of hose five seasons (see Table, p. 35). The primary difference is ha in league 2, each eam experiences a full roaion of ladder posiions over he 5-season period, whereas in league, he rank order of he eams is idenical in each season. Wihin his example, i can be shown easily ha for league c = L c = 0 ( = 2,3,4,5) whereas for league 2 c = 0.225, L = 0.2 ( = 2,3,4,5) c This example clearly demonsraes he abiliy of hese measures o pick up beween-season effecs in CB in cases where wihin-season effecs are idenical beween he wo leagues Mobiliy Gain Funcion Resuls over Time A ime-series represenaion of boh he MGF and he MGFL is provided in Figure 3, wih he former depiced by a solid line, he laer wih a dashed line. Figure 3 is decomposed ino six labelled sub-samples. These sub-samples correspond o Booh s (2004) six-period hisorical analysis of he various combinaions of labour marke and revenue-sharing devices used by he AFL. These resuls are supplemened by Table 7, which splis he sample ino hese six aforemenioned periods. Table 7 includes he means and sandard errors (for boh MGF and MGFL) for each period, as well as he difference beween he wo means in each period. Among Booh s mos subsanial findings was ha (uilising he ASD/ISD raio) CB 7 I should be noed, however, ha in he formaive years of boh compeiions, i was much harder for NRL eams o win he premiership wih a moderae home-and-away record because of differences in he respecive finals sysems. In comparison o he AFL (where he finals series never really consised of less han he op four eams), he NRL seasons of 92-95, 97-92, 925 and 937 did no have a finals series a all (i.e. a firs-pas-he-pos sysem), while in seasons 90, 96 and , only a Grand Final was played (i.e. op-wo sysem). 420

15 Liam J. A. Lenen Figure 3: Original (Thin Line) and HP Trend (Bold Line) AFL Daa for boh MGF Measures MGF Quadraic (Solid) and Linear (Dashed) deerioraed owards he end of (almos) each period, as clubs figured ou ways o ge around he sysem, a which ime he AFL decided ha i had o change is CB policy mix. Firs of all, i can be clearly observed (concenraing primarily on he MGF) ha here is a considerable amoun of volailiy in he series over ime. The same is also rue of he MGFL. This ype of behaviour implies ha one paricular season can be subsanially less or more ineresing han he immediaely preceding one, which is cerainly plausible. Because of his volailiy, a HP filer is again applied (for boh he MGF and MGFL) in order o gain some insigh ino he underlying rends over ime. Recalling a his poin ha, unlike he oher CB measures discussed in secion 2, an increase in MGF indicaes improved CB; Figure 3 reveals an ineresing sory. Specifically, he resuls do no conform o Booh (2004) findings. Period 2 (94-929, whereby in addiion o free agency, Melbourne meropolian zoning was used as a alen disribuion device) is he only obvious period in which he MGF appears o be declining srucurally owards he laer sages of he sub-sample (a ime a which Collingwood won heir record four successive premierships), as would be expeced. Furhermore, period 6 (985-presen, he era of he naional player draf, salary cap and league-revenue sharing), in sark conras o he ASD/ISD raio, appears o be he period wih he mos diminished level of CB (according o he mean), alhough he volailiy of he series also appears o be a is lowes during his period a finding reinforced by Table 7. This may indicae ha a cerain level of mobiliy has become more predicable in period 6, mos likely due o he equalising effecs of he (reverse-order) draf, no employed previously. 8 In oher 8 As measures of he mean, he decline in he sandard deviaion of boh MGF and MGFL may also simply be reflecive of he larger number of eams in he compeiion in period 6. If his were o be he primary reason for his, hen i would be ineresing o produce he corresponding series for he four big Norh-American professional leagues, in which N is subsanially higher sill. 42

16 Towards a New Dynamic Measure of Compeiive Balance: A Sudy Applied o Ausralia s Two Major Professional Fooball Leagues words, he reverse-order naure of he draf has increased he propensiy of sruggling clubs o acquire he mos alened youhs, build a side over he following seasons and become srong. Then, afer a few seasons of high finishes, hey cease having access o he mos highly alened young players coming ino he league and go ino decline again. Under previous regimes, his regulariy caused by he draf was no presen. Table 7: Relevan MGF and MGFL Saisics wihin he AFL Hisorical Six-Period Framework Period TGF (σ TGF ) TGFL (σ TGFL ) TGF TGFL (0.0394) (0.0360) (0.0277) (0.0247) (0.033) (0.0284) (0.0363) (0.0306) (0.0248) (0.092) (0.0207) (0.095) The conras beween he ASD/ISD raio resuls and hose obained here illusrae he sparseness beween he concepual bases of he common wihin-season measures and he MGF. I also illusraes ha, aking Booh s (2004) resuls, in assessing CB he AFL is more likely o reac o a deerioraion in CB when i becomes obvious hrough a lopsided ladder in a given season, as opposed o when i occurs hrough he same eams finishing in heir cusomary rankings from season-o-season. In recen years, wih non-vicorian eams sysemaically ouperforming heir Vicorian counerpars, he laer may require aenion. Also from Figure 3, as would be prediced, MGFL is exremely srongly posiively correlaed wih MGF, as can be seen clearly from boh he original series and he HP rend series. However, he gap beween he wo HP rend series appears o narrow slighly in he laer par of he sample. This is indicaive of here being more eams in recen imes conribuing eiher 0 or L α o MGF in any given year (he only wo cases where y i, = yi, ), raher han 0 < y i, < α, however, he exac reasons for his are no immediaely clear. The MGF and MGFL series for he full NRL sample period are displayed in Figure 4, which is consruced equivalenly o Figure 3. Here, one hing is immediaely noiceable he HP rend for boh series hovers around he same level from he commencemen of he league unil he 950s, hen here is a significan srucural decade-long decline, wih sagnaion once more around he same (lower) level since around 960. This finding is consisen wih he increasing HP rend of he ASD/ISD raio from Figure during ha period. While i is difficul o find informaion regarding labour marke and revenue-sharing devices used in he NRL prior o World War II, in he years leading up o 960, an anecdoally ineffecive series of residenial zoning rules was used as he primary alen-allocaing device 422

17 Liam J. A. Lenen Figure 4: Original (Thin Line) and HP Trend (Bold Linr) NRL Daa for boh MGF Measures 0.22 MGF Quadraic (Solid) and Linear (Dashed) (Dabscheck, 993). This is consisen wih he observaion ha declining CB forces he league o ac and o make changes. While he NRL has since used several labour marke devices (see Daly and Kawaguchi, 2004, p. 25) for a full lis), CB has no appeared o change much in ha ime according o he HP rends of hese measures. 9 To furher subsaniae he sark conras beween he wo halves of he hisorical NRL sample, refer o he las wo columns of Table 8, which provide he summary saisics of he TFG and TFGL for boh leagues. In hese las wo columns, he NRL sample is spli ino wo equally-sized sub-samples, and There is srong evidence from his comparison ha CB has declined markedly in he las half a cenury in he NRL across he board of summary saisics, irrespecive of wheher MGF or MGFL is he focus of aenion Iner-League Comparison More generally, however, we are ulimaely ineresed in he comparison beween he AFL and NRL since 909, o see how he wo leagues compare overall. This can be achieved be looking a he second and hird columns of Table 8. Concenraing iniially on he MGF saisics, i is seen ha he mean is slighly higher for he AFL han he NRL, hough he difference is quaniaively negligible. The median for he AFL is again higher han for he NRL, hough he gap is sill small, despie being wider han for he mean. For he MGFL, however, he mean is slighly higher for he NRL han for he AFL, whereas he median is acually higher for he AFL, alhough he quaniaive difference in boh cases is again negligible. Though 9 Mos of he changes since 960 appear o have had lile o do wih CB-relaed moives. 423

18 Towards a New Dynamic Measure of Compeiive Balance: A Sudy Applied o Ausralia s Two Major Professional Fooball Leagues revealing, hese resuls unforunaely fail o shed much ligh on resolving he puzzle menioned in secion II.4 regarding he conflicing evidence on CB beween he wo leagues. While he MGF and MGFL measures capure elemens of boh wihin-season CB (favouring he NRL) and beween-season CB (favouring he AFL), here appears o be lile evidence o favour eiher league according o hese measures. In oher words, his cones adds a draw o he one previous vicory o each league. Table 8: Summary Saisics for MGF and MGFL for Boh Compeiions MGF AFL (898) AFL (909) NRL (909) Maximum Minimum Mean Median Sandard Deviaion MGFL AFL (898) AFL (909) NRL (909) Maximum Minimum Mean Median Sandard Deviaion Comparison o Oher CB Measures One final exercise ha may be considered useful is o consruc a correlaion coefficien marix of he various CB measures discussed hus far. To his end, Tables 9 (AFL) and 0 (NRL) show he full-sample correlaions for he following CB measures: he ASD/ISD raio; wo concenraion indexes of CB, he C3ICB and C5ICB; ID; HICB; Gini; as well as he Range beween he win raios of he op and boom eams, expressed formally as 0 Range = (5) W, WN, The various correlaions of hese wihin-season CB measures are exhibied in he firs seven rows and columns of Tables 9 and 0. Looking a Table 9 iniially, all of hese correlaions are posiive in sign (as would be expeced) and quie srong. In fac, ou of he 2 bilaeral correlaions beween hese seven measures, hree have a magniude of greaer han 0.95, and all bu one have a magniude greaer han 0.6. The corresponding analysis for he NRL is also compelling, bu no quie as definiive wo of he pairings have a magniude of correlaion greaer han 0.95, and all bu four have a magniude greaer han 0.6 (all of hese involving C5ICB). 0 There is no C5ICB observaion for he AFL in 96, since N=4. 424

19 Liam J. A. Lenen Table 9: Correlaion Marix of Common Alernaive Measures of Compeiive Balance over he Full AFL Sample ASD/ISD C3ICB C5ICB ID HICB Gini Range MGF MGFL ASD/ISD C3ICB C5ICB ID HICB Gini Range MGF MGFL.0000 Table 0: Correlaion Marix of Common Alernaive Measures of Compeiive Balance over he Full NRL Sample ASD/ISD C3ICB C5ICB ID HICB Gini Range MGF MGFL ASD/ ISD C3ICB C5ICB ID HICB Gini Range MGF MGFL.0000 Two similariies emerge beween Tables 9 and 0. Firsly, he sronges correlaions in boh leagues appear o be beween he riumvirae of ID, HICB and Gini. The reason for his may no be immediaely obvious, especially given ha equaions (3), (5) and (6) do no appear o be srikingly similar. However, his finding is no so surprising, when he concepual similariies beween hese measures are considered. Secondly, he ASD/ISD raio is he bes all purpose measure, wih he weakes correlaion wih any oher measure in eiher league sill almos 0.7. Finally, he correlaions beween hese seven measures on one hand and boh MGF and MGFL on he oher hand, are shown in bold in he final wo columns of Tables 9 and 0. Immediaely obvious is ha he sign of hese correlaions is negaive in all cases, which is expeced due o he specificaion of boh measures and how hey relae o he oher measures inversely. Wih respec o Table 9 iniially, wha is more ineresing is he relaively narrow range of magniudes of he correlaions for he AFL daa, all falling beween 0.3 and These magniudes demonsrae ha ha MGF measures are picking up a very differen se of CB effecs compared wih all of he oher measures, ye hey sill conain some common elemens o hose measures a desirable aribue. The evidence from he NRL sample is similar, bu no quie as conclusive, as he range of correlaion magniudes is wider, varying 425

20 Towards a New Dynamic Measure of Compeiive Balance: A Sudy Applied o Ausralia s Two Major Professional Fooball Leagues beween 0. and The similariies beween he wo ses of MGF and MGFL correlaions (wih he oher measures) in boh ables are no surprising, since he correlaion coefficien beween MGF and MGFL is 0.97 in boh leagues, demonsraing he obvious similariies beween he wo measures. A comparison of he HP rends (reducing noise) of he ASD/ISD raio from Figure for boh leagues and he respecive HP rends of he MGF from Figures 3 (AFL) and 4 (NRL), also ells an ineresing sory. In he early par of he sample, he HP filers of he ASD/ISD raio and MGF for he AFL daa are exremely highly negaively correlaed, indicaing ha he wo measures are picking up very similar effecs. However, hey are highly posiively correlaed in he laer par of he sample, which suggess ha he measures are picking up an enirely differen se of effecs. By conras, for he NRL daa, he HP filers of he wo measures are very highly posiively correlaed in he formaive par of he sample, and quie highly negaively correlaed in he laer par. An implicaion of his finding is ha for he periods of high posiive correlaion (modern AFL hisory, earlier NRL hisory), in siuaions when he ASD/ISD raio fails o offer an adequae explanaion of he enire underlying sory, he MGF (and by implicaion MGFL) may offer a beer alernaive for he purposes of analysis. An example of his asserion is eviden for he period from in he AFL whereby six differen eams won he premiership in successive years, indicaing improved compeiive balance. Concurrenly, he HP rend of he MGF is rising, which is consisen wih his, whereas he HP rend for he ASD/ISD raio is also increasing during his period, which is an inconsisency. VI. CONCLUSION An exhausive comparison of compeiive balance in Ausralia s wo bigges professional fooball leagues, he AFL and he NRL, has exposed a mos fascinaing case sudy. A hisorical comparison of he ASD/ISD raio for boh leagues reveals he NRL o be he more compeiive league in a wihin-season framework; however, a casual analysis of runs wih respec o he disribuion of premierships shows he AFL o be he more compeiive league in a beweenseason framework. However, here are some obvious problems wih looking only a wihinseason CB, and measuremen issues are obviously problemaic when beween-season analysis is chosen for invesigaion. Some of hese problems were discussed a lengh. Wih his in mind, a new measure of compeiive balance was proposed, uilising he win raios of all eams in he league in successive seasons, producing a quadraic one-period change meric ha akes he form of a mobiliy gain funcion. This dynamic specificaion picks up movemens of eams up and down he ladder over ime, and while i does no solve he problems associaed wih measuring compeiive balance discussed previously, i does offer a useful bridge linking he wihin- and beween-season analyses. Furher, some of he specifics of he measure were discussed, and a linearised varian of he meric was also offered. Using hese derived measures (which are highly posiively correlaed wih each oher), he hisorical evidence is inconsisen o some degree wih ha from uilising he ASD/ISD Acual figures for hese correlaion coefficiens are as follows: AFL: ( ); 0.62 ( ); NRL: (9-944); ( ). 426

21 Liam J. A. Lenen raio, as posied by Booh (2004). I is found according o hese measures ha while here appears o have been no large srucural change in compeiive balance in he AFL over he sample (hough he sandard deviaion has declined), here was a decade-long srucural decline in compeiive balance ha occurred in he NRL during he 950s. Ulimaely, by comparing he wo leagues in erms of he level of compeiive balance over heir respecive hisories, i is very difficul o separae he wo via he means and medians of hese measures. Unforunaely, his resul does no help o reconcile he apparen superioriy of he NRL compeiive balance over he AFL in erms of wihin-season, and he conrary resul in erms of beween-season. Finally, a correlaion marix analysis involving several common wihin-season measures confirms ha hey are all picking up similar effecs, whereas hese wo new measures are picking up a subsanially differen se of facors. Therefore, while i is no being suggesed ha hese new measures ake he place of he exising wihin-season measures, hey do help overcome some of he problems of he wihin-season measures, and hence could be considered o be useful complemens in any ime-series hisorical or empirical sudy involving professional spors leagues. REFERENCES Booh, D.R. (2004). The Economics of Achieving Compeiive Balance in he Ausralian Fooball League, , Economic Papers. 23: Cain, L.P. and D.D. Haddock (2006). Tying ino he Idealized Sandard Deviaion, Journal of Spors Economics. 7: Dabscheck, B. (993). Rugby League and he Union Game, Journal of Indusrial Relaions. 35: Daly, A.E. and A. Kawaguchi (2004). Compeiive Balance in Ausralian and Japanese Spor, Oemon Journal of Ausralian Sudies. 30: Eckard, E.W. (998). The NCAA Carel and Compeiive Balance in College Fooball, Review of Indusrial Organizaion. 3: Eckard, E.W. (200). Baseball s Blue Ribbon Economic Repor: Soluions in Search of a Problem. Journal of Spors Economics. 2: Groo, J. and L. Groo (2003). The Compeiive Balance of French Fooball, Économie Appliquée. 56: 9-3. Hadley, L., J. Ciecka, and A. C. Kraumann (2005). Compeiive Balance in he Afermah of he 994 Players Srike. Journal of Spors Economics. 6: Hodrick, R.J. and E.C. Presco (997). Poswar U.S. Business Cycles: An Empirical Invesigaion. Journal of Money, Credi and Banking. 29: -6. Humphreys, B.R. (2002). Alernaive Measures of Compeiive Balance in Spors Leagues, Journal of Spors Economics. 3: Leeds, M.A. and P. von Allmen (2005). The Economics of Spors. 2 nd ed. Boson: Addison Wesley. Michie, J. and C. Oughon (2004). Compeiive Balance: Trends and Effecs. Spors Nexus, London. Mizak, D., A. Sair, and A. Rossi (2005). Assessing Alernaive Compeiive Balance Measures for Spors Leagues: A Theoreical Examinaion of Sandard Deviaions, Gini Coefficiens, he Index of Dissimilariy, Economics Bullein. 2: -. Noll, R.G. (988). Professional Baskeball. Sanford Universiy Sudies in Indusrial Economics: