Measurement of Competitive Balance in Conference and Divisional Tournament Design # Liam J. A. Lenten * School of Economics and Finance

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1 Measuremet of Competitive Balace i Coferece ad Divisioal Touramet Desig # Liam J. A. Lete * School of Ecoomics ad Fiace La Trobe Uiversity Abstract The coferece ad divisioal system has log bee a staple part of touramet desig i the major pro-sports leagues of North America. This popular but highlyrigid system determies o how may occasios all bilateral pairigs of teams play each other durig the seaso. Despite the virtues of this system, it ecessitates removig the biases it geerates i the set of wi-ratios from the regular seaso stadigs prior to calculatig withi-seaso measures of competitive balace. This paper applies a modified versio of a recet model, a extesio that is geeralizable to ay ubalaced schedule desig i professioal sports leagues worldwide, to correct for this iheret bias for the NFL over the seasos , the results of which suggest the NFL is eve more competitively balaced tha thought previously. JEL Classificatio Number: C4, L83 Keywords: Competitive Balace, Measuremet Methods # Research assistace o this paper was provided by La Nguye. A earlier versio of this paper was preseted at a semiar at Temple Uiversity, Philadelphia, 3 September 200. The author would like to thak the participats of the semiar for their commets ad suggestios, especially Michael Bogao. The author would also like to thak Damie Eldridge, Stefa Késee ad Markus Lag for their iput. * Cotact details: School of Ecoomics ad Fiace, La Trobe Uiversity, Victoria, 3086, AUSTRALIA. Tel: , Fax:

2 . Itroductio The coferece ad divisioal system has log bee cosidered the default touramet desig i the Natioal Basketball Associatio (NBA), Natioal Football League (NFL), Major League Baseball (MLB) ad Natioal Hockey League (NHL), referred to heceforth as the major leagues. The practice is also stadard i other less popular North America leagues, such as Major League Soccer (MLS) ad the Caadia Football League (CFL). Fudametally, the coferece ad divisioal system is a maifestatio of ubalaced schedules, whereby ot all bilateral pairigs of teams play each other a equal umber of times. Other forms of ubalaced schedules exist elsewhere, like where the ubalacedess is largely radomized, for example, the Australia Football League (AFL). Aother alterative is power-matchig like i Scottish Premier League (SPL) soccer, where teams play a fourth time oly teams that fiished i the same half of the stadigs followig three full roud-robis. Other forms are eve more radical such as the Belgia League format from 200, ivolvig may playoff combiatios after two full roud-robis, producig a seaso whereby all teams do ot play the same umber of games (a ueve schedule). Despite these isolated examples, ubalaced schedules are rare i Europea soccer see Cai ad Haddock (2005) for a historical accout of Tras-Atlatic sports idustry differeces, icludig touramet desig. There are umerous motivatios i favor of the system. Logistically, it ca be used to co-ordiate ad miimize travel distaces for teams. Ecoomically, the system fosters local rivalries ad arguably icreases aggregate demad, measured typically by The year refers geerally to the caledar year i which the regular seaso cocludes. 2

3 televisio ratigs or attedaces. Paul (2003) ad Paul, Weibach ad Melvi (2004) demostrate this poit, although the MLB results of Butler (2002) suggest that fas are kee to see their team agaist other teams that they have ot played agaist i past seasos. Notwithstadig these merits, it is far from perfect may fas do ot like seeig their team miss out o a playoff spot to a team i aother divisio with fewer wis. There is also the pereial dilemma of teams havig already cliched a coferece seedig or playoff spot losig games they should otherwise wi due to purposely restig key players i the last few regular seaso games, idirectly ifluecig the playoff destiy of others (see Blodget, 200). More problematically, ubalaced schedules cause biases i fial regular seaso team stadigs, sice they do ot cotrol for the stregth of schedule, rederig ivalid stadard measures that sports ecoomists use to characterize competitive balace i sports leagues (see Lete, 2008 ad 200, for the SPL ad AFL, respectively). This study applies a modified adjustmet procedure to idetify the extet to which stregth of schedule affects competitive balace measures i coferece ad divisioal systems, based o a simple matrix algebra method. This issue is aalogous to the ogoig idustrial orgaizatio debate o the appropriateess of stadard quatitative measures of market cocetratio to accurately reflect true competitive structure. The focus is o NFL data specifically, ad raises a umber of empirical questios that pertai to the uaces of the coferece ad divisioal system as a ubalaced schedule. Firstly, this is a critical ogoig policy issue for major league touramet desig the NHL altered the compositio of its schedule i advace of the 200 seaso, ad as the remaiig leagues cosider expasio from 30 to 32 teams. More geerally, it is difficult to speculate for major league data whether 3

4 adjustig the league stadigs will affect reported competitive balace measures sigificatly. We are also iterested i whether the team-quality equalizatio aspect of the NFL fixture, uique to the major leagues, is successful i makig the touramet more eve. There is also the possibility that some teams (or eve divisios) fare better or worse tha others i the fixture systematically because of the fixed allocatio of frachises to cofereces ad divisios over seasos. The remaider of the paper proceeds accordig to the followig structure: sectio 2 summarizes the relevat literature, while sectio 3 illustrates the stregth of schedule adjustmet methodology. Sectio 4 outlies alterative ubalaced schedule formats used i the various major leagues. The results are preseted ad the fidigs discussed extesively i sectio 5, followed by a brief coclusio i sectio Literature Review The uiversal applicatio of the coferece ad divisioal system across the major leagues has made it the attetio of substatial previous research, though ot quite to the extet imagied. Give the largely North America cotext of the problem, much of this work ceters o team ratigs i ubalaced schedules Faimesser, Fershtma ad Gadal (2009) is a recet example with respect to the highly idiosycratic NCAA College Football. Lete (200) lists various studies utilizig a rage of statistical methods for this purpose, as well as other studies that relate to the more geeral ecoomic implicatios of touramet desig for the professioal sports idustry. Of uderlyig relevace to sports ecoomists ad statisticias is the competitive balace paradigm, which exteds back to Rotteberg s (956) baseball players labor market study. Competitive balace is defied as the degree of parity i sports leagues, ad 4

5 helps describe the ucertaity of outcome hypothesis a corerstoe of sports ecoomics. Sice demad for sport is expected to be greater whe the league is competitively balaced, this represets a rare example i the ecoomics disciplie i which ucertaity is associated with higher utility. Cosequetly, i sports ecoomics there is a large volume of literature comparig leagues (major ad other), such as Quirk ad Fort (992), ad Vrooma (995 ad 2009), to assess relative effectiveess of labor market ad reveue-sharig policies used by leagues to maitai/improve competitive balace. Sigle-league time-series studies, such as Lee ad Fort (2005) ad Fort ad Lee (2007), are also useful for the same reaso. Lee (200) follows a similar directio for the NFL specifically, yieldig the coclusio (recetly topical agai) that the 993 collective bargaiig agreemet (CBA) improved competitive balace. Examiig whether adjustig for ubalaced schedules affects competitive balace measures is motivated largely by its possible implicatios for these studies, sice they ivariably use these commo measures for the purposes of testig. Therefore, there is a questio of how sesitive their coclusios are to ay possible biases i these measures. O these coclusios, umerous ifluetial modelig studies o policy effectiveess, such as Fort ad Quirk (995) ad Vrooma (995), argue that the Coasia ivariace priciple holds i leagues comprised of profit-maximizig teams. That is, free agecy competitive balace is ivariat to the itroductio or removal of labor market ad reveue-sharig policies. However, other otable studies preset a differet view Depke (999) fids some (albeit mixed) empirical evidece that the removal of the reserve clause i MLB (freeig up the players labor market), resulted i a 5

6 deterioratio i competitive balace. Meawhile, it is well-kow that these itervetioist policies ted to be effective i leagues of wi-maximizig teams. Sice the major leagues do ot ted to be modeled i this way, several factors (other tha team objectives) will collectively determie the (i)effectiveess of such policies. I the case of reveue sharig, these factors iclude the: (i) specificatio of team reveue fuctios (icludig or excludig absolute quality of competitio); (ii) sharig arragemet (at-the-gate or pooled; equal or performace-related); ad (iii) equilibrium coditio (fixed-supply Walras or Nash cojectures). O these lies, Késee (2000) cocludes that reveue sharig ca improve competitive balace eve if teams are profit maximizers, while Szymaski ad Késee (2004) demostrate that the cotrary could occur uder very differet coditios. Moreover, Dietl, Lag ad Werer (2009), show that reveue sharig improves competitive balace i certai types of leagues cotaiig (heterogeeous) teams with mixed objectives. 3. Methodology We use a modified versio of the Lete (2008) -team ubalaced schedule league model where team i is scheduled to play some other teams k times ad all remaiig teams k + times. This model has the advatage of treatig oly the ubalacedess of the schedule as edogeous, while other factors typically icluded i team ratig models are exogeous. This allows us to idetify purely the effect arisig from the policy uder ivestigatio. However, the model is iadequate to deal with the flexible rage of coferece ad divisioal systems used throughout the various major leagues, a shortcomig we aim to overcome. Therefore, the seaso structure is relaxed such that teams ca play other teams o ay umber of occasios, by allowig m differet values of k (ot just k or k + ). I this settig, a iitial schedule 6

7 (symmetric) matrix for seaso t is specified similarly to Jech (983: p. 247), i which elemet x, deotes the umber of times team i plays team j durig the seaso, as i j x, x,2 L x, = x2, x2,2 L x2, X () M M O M xi, j j = x, x,2 L x, where x { k k k }, i j ; k + = { k Z : k 0}, l{ m}, 2,..., i, j m, Z ; = 0, i l l l,..., x i, i ad xi, j = x j, i, i j. Oe ca similarly defie aother matrix as beig a full set of head-to-head ex-post decimalized wi-loss records, W, such that: w, w,2 L w, = w2, w2,2 L w2, W (2) M M O M w, w,2 L w, where kl w i, j 0,,...,,, kl > 0 (3) kl kl ad where wi, j = w j, i, i j ; ad w i, i = 2, i. 2 For k l = 0, we ca assume w i, j = 2 for ow. 3 The iterpretatio of the diagoal elemets is that team i, if hypothetically playig agaist itself (despite beig impossible ituitively), is expected to have a equal probability to wi or lose. The costrait that all teams play a equal seaso legth is imposed to make formal represetatio easier, that is: 2 The set of possible values of w i, j i equatio (3) igores ties uusual i the NFL sice teams have to be level after regulatio time, with o further score i overtime. This occurred oly twice i the sample (Pittsburgh Steelers v Atlata Falcos, Week 0, 2002; ad Ciciati Begals v Philadelphia Eagles, Week, 2008). Assumig equal allocatio of competitio poits for a tie (as i the NFL), the 2kl set of possible values ca istead be geeralized as: w i, j 0,,...,,, kl > 0. 2kl 2kl 3 This is possible oly because we are iterested purely i the diagoal elemets from the product matrix, ad these elemets are quatitatively uaffected by the presece of this assumptio. Nevertheless, this poit is revisited i the latter stages of the appedix. 7

8 j= m xi, j = kl k, i (where l k is the umber of teams that team i plays k l l times), l= though it is ot essetial to the validity of the procedure. This also esures that det [ X] is fixed for all i. I assessig withi-seaso competitive balace, the usual startig poit is costructig a colum vector of observed team stregths, to resemble a league table (a cocept familiar to fas). Stregth is measured simply as the wi-ratio (games wo divided by games played), represetig the actual fial seaso record of orderig of all teams i the league from all games played. For simplicity, the associated vector, V, ca be expressed as [ w w K ] ' V = 2 w (4) where w i = wi, j x j, i j=. These elemets are idetical to the diagoal elemets of the product of the matrices from () ad (2). While valid for ay orderig of teams, it is easiest to visualize if the orderig is idetical to the ex-post rakig of teams. If it were assumed that this is idetical to the (uobservable) ex-ate orderig of teams, the such a orderig also allows us to defie both extremes agaist which observed competitive balace measures ca be deflated. This could be either: (i) a perfectly ubalaced league (closest outcome possible to a moopoly) as oe i which W is a upper triagular matrix where w i j <, =, i j ; or (ii) a perfectly balaced league, whereby w i, j, = 2, i j. A third possible bechmark is that whereby match outcomes are accordig to the biomial distributio: Pr ( i wis) = Pr( j wis) = 0. 5 which case, for ay k > 0 i (3) ad igorig ties, ca be geeralized as l, i 8

9 where {,2 } z Pr w, z k l z kl z i, j = wi j k = Pr = = l k (5) l z z,...,k l is a idex. The distributioal properties of the dispersio of the resultig set of w i i (4) ca the be calculated (or if impractical, simulated). I order to correct the ubalaced schedule bias i the ed-of-seaso stadigs, a logical way to proceed is calculatig implied wi-probabilities (usig oly actual stadigs) of theoretical uplayed matches required to reset each ( i j) max { } x i, j x i j, equal to. The simple logit-style rule used by Lete (2008) produced a solutio cotaiig a umber of attractive umerical properties, such as i= w i = w~ t{,..., T} (6), t i= i, t where w ~ i is the adjusted wi-ratio i seaso t, which is to say that the adjustmet procedure should ot affect the sum of wis across all teams i ay give seaso. However, a more direct way to accout for the wi-ratio bias is to compare the stregths of team i s actual full-seaso schedule to that from a hypothetical schedule i which every game were agaist a oppoet of average stregth of all teams (excludig itself). The former, A, is simply a colum vector of sum products of the umber of games agaist each oppoet ad the correspodig wi-ratios, hece A = x i, j XV (7) Meawhile, the latter, Y (also a colum vector), ca similarly be computed as j= xi, j ' j= Y = w w2 K wn (8) 9

10 Usig oly these two pieces of iformatio to determie the adjustmet meas that the distributio of idividual head-to-head records do ot matter, which is also cosidered to be a desirable property. 4 I vector form, takig equatio (4) as a base, the adjusted set of wi-ratios arisig from equatios (7) ad (8) for all teams ca be expressed as follows ~ V ' = [ w~ w~ K w~ ] 2 ' = V + j= x i, j [ A ] ' Y ' (9) where each elemet i V ~, for ay sigle team, i, ca alteratively be calculated accordig to the followig fuctio relatig the adjusted wi-ratio to the uadjusted wi-ratio directly, expressed i scalar form as m k l = = = w i w~ 2 i = wi + w k k xi j x l l, i, j (0) l k j l j= for ay orderig of iequalities for k,...,, k2 km. Equatio (0) has the additioal desirable property, oted by Lete (200), that it is ivariat to the assumed umber of full roud-robis, which is crucial for NFL data, sice it cotais zeros ad oes. 4. Touramet Formats Correctio for ubalaced schedules is idetical irrespective of the ature of the touramet desig; however, the applicatio of the proposed procedure is potetially profoudly differet. A quick glace at table demostrates the similarities ad differeces betwee the major league seaso structures. The middle two colums, which reveal how may times team i plays all other teams, are ordered vertically such that k < < k2 <.. < k m km. The curret NFL system was itroduced upo expasio 4 The appedix presets a alterative procedure i which these head-to-head records do matter. 0

11 to 32 teams (6 i both cofereces ad 4 i each divisio). This system sees each team playig teams i their ow divisio twice per seaso, automatically allocatig oppoets i 6 out of 6 matches. They also play all teams i oe other divisio withi their ow coferece, plus all teams from oe divisio i the other coferece, allocatig a further 8 matches. The other two matches i the seaso are allocated accordig to a (lagged) power-matchig rule they are agaist teams i the two divisios i the same coferece they would ot play otherwise, that fiished i the same divisioal rak-positio i the previous seaso. To formalize the earlier otatio, this meas collectively that uder the curret system, = 32, 0 k ( ) = k = 8, = 2 k ( ) k 2 = 0, ad 3 = 2 k ( ) k 3 = 3. From 2002, the idea was for a 8-year rotatioal system, i which each team would play every other team at least twice oce home, oce away (eve iter-coferece teams). This full rotatioal period defies the sample, extedig over the regular seasos from This ivolves the placig ad aggregatio of the pairig of teams ad match results from 2,048 regular seaso matches ito the costructio of 8 separate matrix observatios for both X ad W. A look at the remaider of table reveals the varyig ature of touramet desigs, mostly o the basis of seaso legth, as all other listed examples are based o 30-team leagues. I a comparative sese, the NFL touramet desig was similar, except with slightly greater weightig o the itra-divisioal ad power-balacig aspects. Despite a idetical divisioal compositio ( 6 5) ad seaso legth of 82 games, the NHL ad NBA differ markedly i terms of schedule cocetratio, with 5 From 200, a ew rotatioal system applies, similar to the previous system, though altered slightly to reduce the overall travel burde o East Coast teams.

12 NHL teams playig a sigificatly higher proportioal umber of games agaist itracoferece teams tha i the NBA. The previous ( ) NHL system was eve more highly cocetrated, with values of k l ragig from 0 to 8. The MLB system is the most difficult to describe, with geeral differeces betwee divisios, depedig o how may teams i both that divisio ad the correspodig coferece ( league i MLB termiology), ad further differeces still o a team-specific basis. A geeral way to describe the cocetratio of schedules is to use the Herfidahl idex from idustrial orgaizatio. The results are outlied for each of the major leagues i table 2. Iitially, the idex is deflated by the idex from a theoretical idealized schedule, ivolvig the most eve allocatio of games (betwee oppoets) possible holdig both ad seaso legth fixed. O this basis, the MLB schedule is the most cocetrated, due to the log seaso ad there beig little iterleague play. As examples from the 2009 seaso, the idex ratio was 2.5 for the Bosto Red Sox (AL) ad 2.02 for the Philadelphia Phillies (NL). However, if the idex is istead deflated by the bechmark idex from a truly balaced schedule, the the NFL provides the most cocetrated schedule out of the major leagues. While this is somewhat geerated merely by the short seaso, it demostrates that the NFL presets the most iterestig test case for the adjustmet procedure, as it provides the greatest scope for idetificatio of large adjustmets should they be preset. While applicable to ay type of ubalaced schedule, the NFL is easily the most suitable cadidate of the major leagues for this techique. I additio to the cocetrated schedule, it also overcomes the other major leagues for umerous compellig reasos. Firstly, the allocatio of competitio poits is equal for all games 2

13 ulike the NHL, where the total allocatio of poits is greater if the match exteds to overtime, creatig drawbacks for some withi-seaso competitive balace measures. Aother problem with the NHL is discotiuity of the sample arisig from the 2005 seaso cacellatio due to the players strike. Secodly, MLB is the least suitable cadidate, owig to the o-uiform ature of schedules betwee teams discussed earlier combied with the scarcity of iter-league play (oly about 6% of matches, compared to 25% i the NFL). Fially, the comparative aalysis is more powerful whe the compositio of teams is idetical throughout the sample. This is the case for the NFL; with the same 32 teams sice 2002 (eve the NBA has had two relocatios sice 2005). 5. Results The descriptive statistics of the raw wi-ratios for all 32 teams over the sample are displayed i table 3. As show, there is cosiderable turover of stadigs over the sample (idicative of betwee-seaso competitive balace), with oly the Detroit Lios failig to have a wiig seaso. Iversely, the New Eglad Patriots ad Idiaapolis Colts were the oly teams that did ot have a losig seaso. These three teams, ad the Oaklad Raiders, are also the oly teams to have a mea wi-ratio over the eight seasos outside the bouds of 3 ad 2 3. The team-specific stadard error of wi-ratios over the sample useful as a betwee-seaso measure of competitive balace accordig to Humphreys (2002) supports this cotetio. A look at the middle colum of table 4 shows that the mea value of the team-specific stadard errors is higher tha the aalogous figures for the other major leagues. 3

14 As for withi-seaso, oe may use a rage of measures to describe NFL competitive balace. Firstly, we cosider the most widely-used measure the ratio of the actual stadard deviatio to the idealized (biomial distributio of match results as i (5)) stadard deviatio (Noll, 988; Scully, 989), represeted as 2 ( wi 0.5) 2 ASD/ISD = x () i= The right-had colum of table 4 shows that the mea dispersio of NFL wi-ratios is the lowest (most balaced) of the major leagues, accordig to ASD/ISD. Recallig the sectio 2 debate o labor market ad reveue-sharig restrictios ifluecig competitive balace, high NFL balacedess (both withi- ad betwee seasos) is due arguably to the cosiderable equalizig effect of the restrictive suite of policies (relative to other major leagues) used by the NFL. However, this has recetly become far less so, with the abolitio of the salary cap i March 200. j= i, j As a meas of robustifyig the results to alterative measures, three others are also reported, as formalized i equatios (2)-(4). Respectively, these measures are: (i) the Herfidahl idex of competitive balace (HICB), similar to the baselie Herfidahl idex, except allowig to be time-varyig; (ii) the cocetratio idex of competitive balace, C2ICB, measurig the proportio of total wis by the top twelve teams i that seaso the umber that qualifies for the playoffs; 6 ad (iii) the stadard Gii coefficiet. 7 For (ii) ad (iii), w i are assumed rak-ordered. 6 The twelve teams assiged by simple rak-order is typically ot the precise set of teams that make the playoffs. This is aother distictive feature of the coferece ad divisioal system. 7 While GINI is variat to chages i ad x j = i, j over time, either chages durig the sample, circumvetig this crucial cocer. Otherwise we would be uable to apply the Utt ad Fort (2002) methodology, as it depeds o the (uobservable but assumed) ex-ate rak-order of teams uder NFL possible combiatios. touramet desig, of which there are 32! ( ) 4

15 2 HICB = 4 w (2) C2ICB = i= 2 i= i w 6 (3) h 2 GINI = 4 wi (4) h= i= Iitially of curiosity are the magitudial chages to the wi-ratios arisig from the adjustmet. It is show i table 5 that the outliers are quite large, because of the high schedule cocetratio oted previously. Out of 256 adjustmets (for 32 teams over 8 seasos), a total of are greater i absolute magitude tha the value of a wi (0.0625), of which 6 are egative ad 5 positive, ad a further 58 are greater tha the absolute value of half-a-wi. The largest sigle adjustmets are (Seattle Seahawks i 2007) ad (Clevelad Brows i 2004). The adjustmets are also scatter plotted agaist the origial wi-ratios i figure. Fortuately, there appears to be little systematic bias, providig techical backig for the methodology. i Normally of acute cocer is the extet to which hypothetical adjustmets create qualitative differeces, i terms of chages to team rak-order. This is ot quite as much of a issue i the coferece ad divisioal system, sice it is recogized that ot ecessarily the best teams will always make the playoffs. Nevertheless, uder the criteria specified for playoff qualificatio, it is still of iterest to see whether differet teams would have qualified otioally uder adjusted wi-ratios. Specifically, there are a total of oly ie cases where this occurs over the sample, eight of which ivolve AFC teams. Oe case amazigly ivolves three teams (AFC East i 2002), i which New York Jets fall from first to third, promotig New 5

16 Eglad Patriots from secod to the divisio title. The eight remaiig cases ivolve oly two teams swappig positios. Of these, four have o bearig o playoff positios, while a further two (AFC North i 2005 ad NFC East i 2009) ivolve top-two teams i cases where secod ears a wildcard, affectig oly seedigs. The two remaiig cases would have altered playoff qualificatio, with the Dever Brocos beatig Kasas City Chiefs for a AFC wildcard i 2006 ad New Eglad Patriots toppig Miami Dolphis for the AFC East 2008 divisio title. I additio, there are three further cases (agai all AFC) i which adjusted stadigs would have altered wildcard places, with Dever Brocos replacig Clevelad Brows i 2002, ad Miami Dolphis (2003) ad Baltimore Raves (2004) both replacig the Brocos. The latter case is the oly oe (out of twelve) i which the combied adjustmets of the two teams ivolved overcome the value of a wi. 8 I most cases ivolvig teams tied o wis (4 out of 22), the adjustmet assigs a orderig ot icosistet with that derived from the secodary ad tertiary criteria specified curretly by the NFL to separate teams tied o wis. Aspects of the adjustmets at a aggregate level are also of ituitive appeal to fas. Table 6 provides a profile of possible favoritism i the fixture. Begiig with the right-had colum, it is see that the mea adjustmet for AFC teams is , a sizable figure over 28 observatios. This results from AFC teams wiig the majority of iter-coferece games over the sample. Furthermore, the divisioal adjustmets are positive across the board i the AFC, alog with NFC East. By team, Teessee Titas are the oly AFC team with a easier tha average schedule over the 8 seasos, whereas oly 4 NFC teams have positive mea adjustmets, three of 8 I umerous other cases, the adjustmet chages playoff seedigs, however, this is of less cocer. 6

17 them i East divisio. The mea adjustmet is a descriptive outlier greater tha two stadard errors usig mea adjustmets from all teams for Oaklad Raiders (positive) ad Seattle Seahawks (egative), with the differece betwee them of greater tha oe full wi per seaso. The Seahawks case demostrates icely the outcomes arisig from the procedure. While their raw wi-ratio of is slightly above average over the sample, they play a disproportioate umber of games agaist three teams (the other NFC West teams) that have performed weaker tha average the adjusted wi-ratio (0.4876) takes this ito accout. Returig mometarily to figure ad the lie of best fit, further ispectio also reveals a oticeable egative correlatio ( ρ = ), idicatig that teams achievig a higher wi-ratio o average received more favorable fixtures, though this factor accouts for a small fractio of the variatio. This result is totally ituitive, sice a team with a fixed level of talet ad performace will (all else equal) achieve a higher wi-ratio whe allocated a easier schedule. Whe mea adjustmets by divisioal place fiish are computed, as i table 7, it is foud that the meas are icreasig motoically i rak-order, as expected. If, however, the adjustmets are based o rak-order from the previous seaso (oe seaso of observatios lost), the correlatio chages sig ( ρ = 0.02), so do the sigs of all the mea adjustmets i table 7, though all are close to zero. The iterpretatio is that the power-balacig aspect to the fixture eves up the competitio, though to a lesser degree tha iteded sice chages i stadigs from oe seaso to the ext are difficult to predict. Most crucially, the calculated uadjusted measures of competitive balace are show i the top portio of table 8. It is observed (o the basis of ASD/ISD) that the

18 seaso was comfortably the most competitively balaced of the sample (i fact, most eve sice 995), ad while 2006 was also more eve tha average, all other seasos produced a similar competitive balace levels. This fidig is reiforced by the other various measures reported i table 8. The adjusted competitive balace measures ad adjustmets are also computed ad exhibited i the middle ad bottom portios of table 8, respectively. The iformatio from table 8 is replicated graphically i figure 2. The most strikig result is that these adjustmets are egative for all seasos across all measures, although i some years the magitude of the adjustmets are relatively large (most oticeably i 2003), ad close to zero i other years (2004, 2006 ad 2007). These figures idicate i a quatitative sese that the NFL is cosiderably more competitively balaced whe the ubalaced schedule is accouted for. Naturally, the issue of statistical sigificace of differeces betwee uadjusted ad adjusted arises. While ay basic oparametric sig test or rak-sum test would defiitively reject the ull of o statistical differece, we seek further support or otherwise to substatiate this. To this ed, the more powerful parametric differece i meas t-test for paired samples is applied. Sice there is o a priori reaso to presume that the adjusted measures should be lower, the two-tailed versio is used. The t- statistics for ASD/ISD ad HICB of ad respectively are both easily above the 5% critical value of , ad while the values of for C2ICB ad 3.39 for GINI are also sigificat at 5%, they are isigificat at %. Despite this apparetly strog result, cautio is exercised for two reasos: (i) small sample size; ad (ii) the distributio of competitive balace measures may be o-ormal. Oe possible reaso for our result may be that the level of ability of a team coverges o the level of other teams that they play more ofte. However, eve if true, it is difficult 8

19 to coclude outright that the coferece ad divisioal touramet desig plays ay part i geeratig that result. The same applicatio to data from the CFL a league with oly 8 teams but with a similar (albeit reduced-form) coferece-style ubalaced schedule, produces a differece of meas t-test statistic for the ASD/ISD measure of merely , makig it comfortably isigificat Coclusio This paper has preseted a modified ubalaced schedule model that ca be geeralized to ay league i which teams play other teams ay umber of times. The model was the applied to NFL data over the regular seasos. The major coclusio is that, quatitatively, the adjusted competitive balace metrics imply the league to be more competitively balaced tha idicated by uadjusted measures i every seaso, ad sigificatly so for the sample meas. Qualitatively, the fidigs, while cosistet, are less categorical. The logically followig questio of whether adjusted or uadjusted measures more accurately reflect the ifluece of ucertaity of outcome o demad for America football is oe that is difficult to aswer usig average seaso attedaces, sice virtually all NFL games are sell-outs. Regardless, adjusted measures are still of much cosequece to sports ecoomists. Especially so for ecoomists usig microecoomic models to make ifereces about the effectiveess of itervetioist league policies, ad usig withi-seaso competitive balace metrics from differet sports leagues as a meas of comparative aalysis to test those ifereces empirically. This is because the use of adjusted measures could alter coclusios about which leagues are more competitively balaced. 9 The CFL is more comparable to the NFL tha the other major leagues. Not oly does it ivolve virtually the same sport ad a almost idetical seaso legth (8 games), but data from seasos ad is used, esurig idetical sample legth. To agai guaratee the same set of teams through the sample, seasos are excluded, whe the ill-fated Ottawa Reegades also competed. The sample ASD/ISD meas are.404 (uadjusted) ad.4233 (adjusted). 9

20 There was also a appealig story behid the adjustmets geerated from the procedure. At a aggregate level, i terms of both by team ad by rak-order i divisio stadigs, the meas were sufficietly large to raise cocers about the high cocetratio of the schedule. While it is improbable that the league s costituet teams would collectively favor overhaulig the NFL touramet desig radically i the short ru, they would likely be ope to the idea of at least debatig the issue upo beig made aware of this story. The fidigs preseted ad discussed here have crucial implicatios for the various major league commissios, i terms of their ogoig reforms to their labor market ad reveue-sharig policies, as well as touramet desig itself. 20

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