The Extent to Which Unbalanced Schedules Cause Distortions in Sports League Tables#

Liam J. A. Lenten*

School of Economics and Finance

La Trobe University

Abstract

The Australian Football League (AFL) has operated its fixture on the basis of an unbalanced schedule since the league expanded from 12 to 14 teams in 1987. This system contains a number of factors (some random) determining the set of bilateral combinations of teams that play each other on an extra occasion during the course of the season, not least of all maximising attendances. While the status quo may be unavoidable to some extent (it is also a bone of contention to many fans), its implications for within-season measures of competitive balance are nonetheless obvious. This is because biases are created in the end-of-season league table as a result of the unbalanced schedule. This paper uses a modified model to correct for this inherent bias over the seasons 1997-2008, and the results are discussed in intricate detail. The model is also generalisable to many unbalanced schedule designs observed in professional sports leagues worldwide.

JEL Classification Number: C14, L83

Keywords: Competitive Balance, Measurement Methods

#Current version prepared for the Research Seminar Series, Department of Economics, Maquarie University, 20 March 2009. Research assistance on this paper was provided by Lan Nguyen. The author would also like to thank Damien Eldridge for some useful comments. *Contact details: Department of Economics and Finance, La Trobe University, Victoria, 3086, AUSTRALIA. Tel: + 61 3 9479 3607, Fax: + 61 3 9479 1654. E-mail: l.lenten@latrobe.edu.au

1

1. Introduction

An unbalanced schedule is the practice in some professional sports leagues, whereby

not all bilateral pairings of teams play each other on an equal number of occasions in

a season. Historically, unbalanced schedules are extremely rare in the major football

leagues of Europe, where the common practice has always been to have each team

play each other once home and away. However, they have long been commonplace in

the major professional sports leagues in North America, such as the National

Basketball Association (NBA), National Football League (NFL), Major League

Baseball (MLB) and National Hockey League (NHL). In these leagues, the basis of

the unbalancedness of the fixture is determined within the conference and divisional

system, which varies highly in terms of specific detail in each of these leagues.

In the last two decades or so, the same has applied also in Australia’s two biggest

leagues, Australian Rules Football’s AFL and Rugby League’s National Rugby

League (NRL), as these leagues expanded during the 1980s in their drive towards

becoming national competitions. In doing so, the number of teams increased, whereas

the length of the season remained constant (trading off economic factors with

concerns over player fatigue), creating unbalanced schedules in both competitions. In

the AFL, this status quo is unpopular with many fans, especially since there are a

number of high-drawing fixtures that are guaranteed to be scheduled twice in every

regular season (as opposed to balancing the schedule out over a number of years) for

the purposes of maximising attendances, rather than on the basis of team-quality

equalisation. Table 1 displays the number of times each bilateral pairing of teams has

played each other over the period 1997-2008. As is shown, there are a number of

bilateral pairings that have been fixtured the maximum 24 times over the period, such

2

as the two Perth teams, the two Adelaide teams, and the triumvirate of large-market

Melbourne teams: Carlton, Collingwood and Essendon.1 Meanwhile, combinations

such as Collingwood-Geelong and West Coast-Carlton have played each other only

13 times - close to the possible minimum.

In spite of the practicalities of this system, the problem it creates for sports

economists is that it renders team standings from the end-of-season league table

biased, due to the raw placings not taking the strength of schedule into account. Thus,

standard metrics of competitive balance used by sports economists are potentially

invalid. While this result was found by Lenten (2008) with respect to Scottish

Premier League (SPL) data, the correction technique proposed was itself based on a

biased estimator.

Therefore, the purpose of this study is to apply a modified version of that adjustment

procedure to investigate to what extent accounting for the strength of schedule affects

reported competitive balance measures for AFL data. This exercise has intuitive

appeal for numerous reasons, relating mainly to the differences in the basis on which

the schedule is unbalanced. For AFL data, it is more uncertain than the SPL case

whether the adjusted league table will produce reported competitive balance outcomes

significantly different to the unadjusted table. Additionally, of more concern is

whether some teams are systematically better or worse off than others, merely

because of the fixture. Also of interest here is the extent to which (if any) the league

uses the fixture for team-quality equalisation from one season to the next. These

issues are discussed in detail. Finally, this will be a crucial policy issue in the future of

1 It is the latter case that is somewhat controversial, since the other (smaller-market) Melbourne teams would prefer to play these teams more often themselves.

3

tournament design from 2011-2012 as the AFL expands towards an 18-team

competition.

The following sections of the paper are structured hence: section 2 outlines some of

the literature on unbalanced schedules and tournament design. Section 3 summarises

the methodology used to account for the strength of schedule problem inherent in

unbalanced schedules. The results are then presented and discussed in detail in

section 4, along with some (league) policy implications. Finally, a brief summary and

conclusion is presented in section 5.

2. Literature Review

Since unbalanced schedules are common in North American pro-sports leagues, much

of the literature acknowledging their idiosyncrasies has centred on the implications

for these leagues. Weiss (1986) noted the biases that are created by this type of

tournament design, while Jech (1983) provided a formal representation of such

‘incomplete tournaments’. Other work centred on team ratings – mostly with an

application to the NFL or NCAA College Football, where there is the perennial

problem of awarding the national champion. Various statistical methods have been

applied to this problem, including minimisation of (absolute) errors (Bassett, 1997);

the analytical hierarchy approach (Sinuany-Stern, 1988); and maximum likelihood

(Thompson, 1975; Mease, 2003). It is the latter study that most resembles what is

being done here, since Mease’s methodology proposed independence from winning

margins. An additional major reason that administrators utilise unbalanced schedules

is their belief that it increases aggregate attendances. In this light, Paul (2003) and

Paul, Weinbach and Melvin (2004) find supportive empirical NHL and MLB

4

evidence for this, respectively, finding that the role of local derbies is an important

determinant of demand.

The existence of unbalanced schedules has implications for tournament design, which

is a crucial theoretical aspect of economics generally, especially labour economics

(see Sutter, 2006; or Harbring and Irlenbusch, 2003, as recent examples). Szymanski

(2003) provides a thorough exposition on the role and consequences of tournament

design on professional sport generally, with various emphases such as comparative

analyses between sports, revenue-sharing rules, optimal league size, and even club

versus country issues. Meanwhile, Ehenberg and Bognanno (1990) provides a

specific model of tournament design – that of rank-order tournaments – on the

incentives of professional golfers to exert effort on the PGA Tour.

Of ultimate interest to sports economists and statisticians is the measurement of

competitive balance, the notion of which goes back to Rottenberg (1956).

Competitive balance refers to the degree of parity or otherwise in sports leagues, and

is crucial because of the link to the demand for sport via the ‘uncertainty of outcome’

effect. See Sanderson and Siegfried (2003) for a general commentary on these issues.

To this end, there is a desire to make comparisons between leagues, like those of

Quirk and Fort (1992) and Vrooman (1995), for the purpose of comparing the relative

effectiveness of various labour market and revenue-sharing league policies that are

used to enhance competitive balance. Therefore, one of the primary motivating

factors behind investigating whether adjusting for unbalanced schedules affects

competitive balance measures is that in the affirmative case, it then begs the question

of whether the results of such comparative studies would themselves be different. For

5

example, would Booth’s (2005) findings comparing AFL competitive balance with

the NRL and Australian National Basketball League (NBL) be different, since the

latter has a balanced schedule?

3. Methodology

As a simple illustrative example, imagine a league ( )I comprised of a set of four

teams ordered completely, reflexively and transitively in terms

of team quality (and market size), this ordering observable ex-ante. Suppose league

officials face an imposed four-game season length, in which there is one full round-

robin of matches, plus one extra round of matches - in which is pitted purposely

against because it is considered a premium fixture (in which case plays in the

same round). While this would arguably maximise league-wide aggregate fan

interest, the payoffs from the final league table may be perversely skewed.

Specifically, assuming every game produces the expected outcome, both i and

finish on two wins from four games, leaving the possibility that could unjustly

finish in second place (depending on the secondary criterion for separating teams tied

on wins). Furthermore, this problem is accentuated if a single playoff match follows

involving the top two-placed teams from the regular season to determine the official

champion.

{ 4321 iiiiI ppp∈

2i

}

1i

3i

3i

4i

2 3i

For a generalisation of the previous example, we use a modified version of the

unbalanced schedule model proposed by Lenten (2008) that takes a league with a set

of N teams, henceforth denoted as I. The league has an unbalanced schedule in which

every team in the league plays 1−− lN other teams k (any non-negative integer)

6

times during the course of a given season, and the other l teams times (a very

simple unbalanced schedule format quite common outside North America). From this

information, the initial

1+k

( )( )11 +−× kNN

S

( )

) ( )

result matrix for a given season (with

completed balanced schedule), , is constructed as

( ) ( ) ( ) ( )( )

( ( ) ( ) ( )( ) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

+−++−+−+−

−

1

1

N

N

M

}

−

+−++−+−+−

11,11,1,1,1,1,

11,111,11,11,11,11,1

kNNlkNNlkNNkNkNNN

kNlkNlkNkkN

zzyyxx

zzyyxx

LL

MMMOMOM

LL

SL

O

L

(1)

where , ji,∀ { 1,5.0,0, ∈jix

) }k1

{0

is the result in terms of win value for team i in game

, the portion of the season in which the schedule is balanced.({ N,..., −

, ∈jiy

j 1∈

Also,

2

}1,5.0, refers to the analogous result for the portion of the season that is

unbalanced, between the soft and hard partitions ( ) ( ){ }( )lkNkNj +−+−∈ 11

1−− lN

,...,1 .

Finally, is a notional (continuous) win ‘value’ for an extra

games for each team

1, ≤jiz0 ≤

( ) ( )( ){ }( )11,...,11− +−++∈ Nj k

jiz ,

) lkN +−1

Nlk required to balance out the

schedule, on the right-hand side of the hard partition, but based purely from

information contained on the left-hand side of the hard partition.3 This is the key

feature that distinguishes this ‘pure’ adjustment factor approach from using one of the

existing plethora of predictive-style procedures to estimate , and arguably the key

improvement of this methodology, as there is no precise consensus on which

predictive model is ‘best’. It is presumed that the ordering of the N teams in equation

(1) corresponds to the actual rank-order from the league table after (

games.4

2 While the incidence of draws is very low in the AFL (0.9 per cent of all matches in this sample), the occurrence of draws is of minimal concern, since the total assignment of competition points is equal (four – two each) for a draw compared to a win for either team. 3 In other words, we want to remove all other exogenous factors (such as injuries, home-ground advantage, bogey teams, etc.) from the analysis. 4 Any casual ordering of teams in S is permissible, however.

7

In order to correct the bias in the end-of-season league table due to the unbalanced

schedule, one needs to calculate implied win-values for each of the unplayed matches

for each team required to balance out the schedule. To do this, one can begin by

calculating an overall season performance vector (actual) , which indicates the win

ratios of all of the teams in the league from the

V

( )kN l+−1 games played as thus

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−=

∑

∑

∑

∑

∑

∑

+−

+−=

+−

+−=

+−

+−=

−

=

−

=

−

=

lkN

kNjjN

lkN

kNjj

lkN

kNjj

kN

jjN

kN

jj

kN

jj

y

y

y

x

x

x

lkN1

11,

1

11,2

1

11,1

1

1,

1

1,2

1

1,1

11

MM

V (2)

The aim is to estimate the theoretical win-ratio that would have applied for each team,

had the schedule been balanced, but by using all of the information contained in the

full season – even in the unbalanced portion. Conceptually, one way to do this would

be to estimate the values, allowing the completion of the vector jiz , R̂ , the best

estimate of the final standings (win-ratios) in the event that each team had actually

played each other times, hence balancing the schedule. 1+k

( )( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )( )

( )

( )( )

( )

( )( )

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−=

∑

∑

∑

∑

∑

∑

∑

∑

∑

+−

++−=

+−

++−=

+−

++−=

+−

+−=

+−

+−=

+−

+−=

−

=

−

=

−

=

11

11,

11

11,2

11

11,1

1

11,

1

11,2

1

11,1

1

1,

1

1,2

1

1,1

ˆ

ˆ

ˆ

111ˆ

kN

lkNjjN

kN

lkNjj

kN

lkNjj

lkN

kNjjN

lkN

kNjj

lkN

kNjj

kN

jjN

kN

jj

kN

jj

z

z

z

y

y

y

x

x

x

kNMMM

R (3)

For each row in , the win ratio is simply the arithmetic mean value of all elements

on the left-hand side of the hard partition. The information contained in V is what is

being used to calculate the implied win-values for all hypothetical matches for

S

8

( ) ( )({ 11,,11 )}+−++−= kNlkNj K . As a method of estimating for all

bilateral combinations of team i playing against an unspecified opponent, , Lenten

(2008) makes use of the logistic relation

jii bz ,'

bi

( )

( )

( )

( )

( )

( ) βyx

yxα

zz

lkN

kNjji

kN

jji

lkN

kNjji

kN

jji

jii

jii

bbb

b +

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+

+=⎟

⎟⎠

⎞⎜⎜⎝

⎛

− ∑∑

∑∑+−

+−=

−

=

+−

+−=

−

=1

11,

1

1,

1

11,

1

1,

,'

,' ˆ1

ln (4)

This simple logit-style rule produced a solution to the SPL unbalanced schedule

problem that contained a number of attractive numerical properties, such as

( )

( )

( )

( )

( )( )( )( )( )112ˆ

1

11

11,

1

11,

1

1, +−=⎟⎟

⎠

⎞⎜⎜⎝

⎛++∑ ∑∑∑

=

+−

++−=

+−

+−=

−

=

kNNzyxN

i

kN

lkNjji

lkN

kNjji

kN

jji (5)

which is to say that the adjustment procedure should not affect the sum of win ratios

across all teams and all seasons. However, in addition to this, the following property

is highly desirable

( )

( )

( )

( )

( )

( )

( )

( )( )

( )

( )( )

( )

( )( )

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

∑

∑

∑

∑

∑

∑

+−

++−=

+−

++−=

+−

++−=

+−

+−=

+−

+−=

+−

+−=

11

11,

11

11,2

11

11,1

1

11,

1

11,2

1

11,1

ˆ

ˆ

ˆ

1

kN

lkNjjN

kN

lkNjj

kN

lkNjj

lkN

kNjjN

lkN

kNjj

lkN

kNjj

z

z

z

l

y

y

y

lNMM

(6)

This property ensures that once the season is ‘scaled up’ to ( )( )11 +− kN games, R̂ is

invariant if instead scaled to any other positive multiple of ( )kN 1− games. However,

equation (4) does not produce this property in R̂ , thus any alternative approach

should contain this property. Figure 1 demonstrates this problem, with the ‘scale-

back’ bias for Lenten’s SPL data plotted against the original win-ratios on the

horizontal axis. This bias is calculated as the difference between the adjusted win-

ratios from scaling the original 38-game win-ratios up to 44 ( )( )( )11 +− kN games on

9

one hand and back to 33 games on the other hand. Figure 1 shows that the

logit-adjustment procedure (by mere virtue of its statistical properties) unfairly

‘rewards’ mediocre teams at the expense of really bad or good teams (albeit to a small

degree), an idiosyncrasy that was not evident in the SPL data, which did not contain

extreme outliers, but nevertheless would be obvious with AFL data.

( )( kN 1− )

To adjust the raw win-ratios (for team i) appropriately in accordance with the desired

properties outlined previously, the procedure should begin by calculating the

difference between the mean win-ratio of all other 1−N

1

teams and the mean win-

ratio of all other teams that team i played on +k occasions. To scale up to

games, this difference should then be weighted by ( )( 11 +− kN )1

1−−−

NlN , the

proportion of one full round-robin of further matches required to balance the schedule

forward. Applying this rule will produce a relation for the adjusted win-ratio for team

i, denoted at this stage as , the ith element of R̂ , as ir̂

( )

( )( ) ( )( )

( ) lkN

lkNlN

w

NlkN

w

Nw

r

kk

kIi

ii

i

i +−

⎟⎟⎟⎟

⎠

⎞

+−−−−

−+−

−−+

=

∑∈

1

11111

ˆ

Nl⎜⎜⎜⎜

⎝

⎛

−2

(7)

where is a set containing IIk ⊂ 1−− lN elements, corresponding to the teams that

team i played only k times during the course of the season. For R̂ to be unbiased in

any way, it must be also possible to scale back to ( )kN 1− games and obtain the

precise same value, that is

10

( )( )( )

( ) lkN

lkNl

w

NlkN

wN

lw

r

kk

kIi

ii

i

i +−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−−

−+−

−−

=

∑++

+∈

1

111

2

ˆ

11

1

(8)

where this time, is the set of teams that team i played II k ⊂+1 1+k times, containing

l elements, such that ∅=+1kIkI I . Furthermore, letting { }1+=′ kII kI U allows the

consideration of any team IIi ′∈ \ .

To complete the picture, it is also possible to isolate the adjustment vector arising

from the procedure for all teams, determined as the following

( ) EVRA +−= ˆ (9)

where is a vector of estimate error terms. The sample period is an annual one,

extending from the 1997 season, following the merger of Brisbane Bears and Fitzroy

and the introduction of Port Adelaide, to 2008 - the time of writing, constituting a

total of twelve vector observations with I identical throughout. The distinctive feature

of 2008 is that the league loosened the restriction that each team had to play every

other team once in the first 15 games, resulting in a scrambling in the order of and

terms in equation (1) if the chronology of matches is important in estimation, due

to time-varying team strength effects.

E

k

jix ,

jiy ,

=N

5 However, this is not the case within this

modelling framework; therefore this issue is not of specific concern. Over the sample

period, the tournament design conditions of the AFL produces the following values:

, and , which is similar to the tournament design in the NRL since

2007, except that

16 1= 7=l

9=l in a 24-game competition.

5 See Clarke (1993) for a predictive model that is specified with time-varying team ratings.

11

4. Results

The starting point is the raw win-ratios (defined as the number of games won by team

i divided by the number played, gwi ) for each of the 16 teams, which are displayed

in table 2. As can be seen, there is considerable turnover of teams over the sample

period, with each team having at least one season where they won less than 40 per

cent of their matches and at least one other season where they won at least 60 per

cent. This is due to the considerable success of the between-season equalising effect

of the suite of labour market devices and revenue-sharing rules used by the league to

enhance competitive balance - see Booth (2004) for more details. There are also a

few extreme values (close to zero or one) present in the data – these are values that

are particularly susceptible to biases created by the non-linearity present in the logistic

adjustment procedure.

As a way of describing the level of competitive balance in the AFL, a range of various

measures are used in an attempt to gain an overall impression. Firstly, the benchmark

ASD/ISD ratio measure, popularly attributed to Noll (1988) and Scully (1989), is

considered, based on the ratio of the standard deviation of win-ratios to that from a

league in which the result of each match were purely random, thus

t

t

N

i t

i

t g

Ngw

5.0

5.0

ASD/ISD1

2

∑= ⎥

⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛

= (10)

where t refers simply to the current season. It can be noted that for any team, i:

( )

( )

( )

( ) lkN

yx

gw

lkN

kNjji

kN

jji

i

+−

+=

∑∑+−

+−=

−

=

1

1

11,

1

1,

. Secondly, the Herfindahl Index of competitive balance

12

(HICB) is reported, based on the original index from IO, but adjusted to allow for

changes in N over time, hence

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

N

i t

i

tt g

wN 1

24HICB (11)

Also included is a concentration index of competitive balance, C4ICB. It measures

the proportion of wins accounted for by the top teams in any given season, selected to

be four in this case, since the top four teams qualify for the ‘double chance’ in the

finals series.6 This is calculated as

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

4

1 2C4ICB

i t

it g

w (12)

where are now assumed to be ordered specifically in terms of end-of-season rank,

as stated on p. 7. Finally, the Gini coefficient is also reported, specified as

iw

14

GINI

1 1

2

−−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

∑∑= =

t

N

h

h

i t

i

tt

Ngw

N (13)

This measure is not totally invariant to changes in and , however, this is not of

concern in this sample anyway, since both variables were constant.

tN tg

7 The unadjusted

measures of competitive balance are shown for all seasons (along with their sample

means) in table 3. It is observed that the first two seasons of the sample (1997 and

1998) were unusually competitively balanced, even by historical standards, whereas

the final year of the sample (2008) was the most uneven season since the introduction

of the national player draft and salary cap in the mid-1980s. All other seasons

produced a similar level of competitive balance, according to these measures. 6 This means that a team can lose in the first week of the finals series and still be in contention to win the title. This is not the case for teams that finish lower (fifth through to eighth). 7 GINI could be adjusted via the methodology of Utt and Fort (2002) to overcome sensitivity to and . Such a procedure is outlined by Larsen, Fenn and Spenner (2006). However, the denominator term is variable to the a priori rank-ordering of teams used to construct the term when the unbalanced schedule is highly randomised in nature, as is the case with the AFL tournament design.

tN

tg

13

Of primary interest is the adjustment procedure and the effect that the procedure has

on the win-ratios, where now,

( )

( )

( )

( )

( )( )

( )( )11

ˆˆ

11

11,

1

11,

1

1,

+−

++=

∑∑∑+−

++−=

+−

+−=

−

=

kN

zyx

gw

kN

lkNjji

lkN

kNjji

kN

jji

i . The adjusted

win-ratios are shown in table 4, listed by actual rank-order finish. Of most concern is

the frequency with which initial placings from the original league table would change

under the hypothetical adjusted placings. From a total of 192 placings (16 teams

multiplied by 12 seasons) where percentage is used as the secondary criterion to

determine initial rank-order, a total of 68 teams finish in a different notional spot.8 In

some cases, teams finish up to three places away from their original rank. Of the 30

multilateral cases where teams swap ranking, 23 involve merely two teams, whereas 4

cases involve 3 teams and 3 cases involve 4 teams. In most cases, order-swapping

involves teams that had an identical win-ratio, however, in 9 cases; rank-order

changes overcome the win value of a draw (0.0227). In the case of 2008, St Kilda

drops from fourth to seventh, swapping order with North Melbourne, despite finishing

with an extra half-a-win. In two cases, the adjustment factor even overcomes one

entire win (Port Adelaide overtakes Hawthorn from thirteenth to twelfth in 2006; and

Adelaide rises from eighth to sixth at the direct expense of Collingwood in 2007). In

various seasons, these rank-order changes affect all major end-of-season outcomes,

including finishing in the finals, double-chance, home-ground advantage in first week

of finals, and the wooden spoon (and by implication, the number one draft pick).

The adjustment vectors for all seasons are also shown in table 5. Over all

observations, the mean absolute adjustment is 0.0133, with a standard deviation of

8 Percentage is calculated simply as total points for divided by points against multiplied by 100.

14

0.0098. No single adjustment is greater than the magnitude value of a win (0.0455),

with the most extreme values being -0.0439 (Hawthorn in 1999) and 0.0422 (Adelaide

in 2003). Most importantly, the adjustments are seemingly not systematically

correlated with the original win-ratios, as can be seen graphically from a casual glance

of figure 3.9

Table 6 displays the adjustments at an aggregate level, helping to identify if there is

any systematic ‘favouritism’ in the fixture. By team, it is seen that the 12-season

means are all lower in magnitude than the mean absolute adjustment, however, the

absolute mean difference between the most favoured team (Hawthorn) and least

favoured team (Adelaide) is nearly half of a win value at 0.0205 – a worryingly high

magnitude for a 12-year sample. Furthermore, from the six non-Victorian teams, five

have mean adjustment values in positive territory. Much of this phenomenon may be

explained by the non-Victorian teams (average win-ratio of 0.5313) having

outperformed the Victorian teams (average win-ratio of 0.4813) on an aggregate level

during the sample period, coupled with the fixturing of all-Victorian and all-non-

Victorian matches (1997-2008 bilateral average of 17.91 and 18.53, respectively)

more often than interstate-pairing matches (bilateral average of 17.13). This result

sounds a warning of the risk of these sorts of structural effects building in season-to-

season inequities in the fixture. It also strengthens the case for discarding the current

system for something that more resembles a rotational system or even a ‘power-

matching’ form of unbalanced schedule – the likes of which is used in the SPL.

9 In comparison to Lenten’s (2008) SPL results, the mean absolute adjustment is lower (compared with 0.0212), although the extreme value is greater in magnitude (compared with only 0.0289). These results are not surprising, given the maximum-bias tournament design of the SPL, coupled with the fact

that the value of { }( ) lkN

lNl+−−−

11,min , indicating the ability for the adjustment procedure to alter the win-

ratio, is over 2.4 times the value of the corresponding value for the SPL.

15

One possible way to do this would be to abandon the guaranteed scheduling of

fixtures involving the large-market Melbourne teams twice, as this would lessen the

bias alluded to in the previous paragraph. While there is little doubt that scheduling

the Perth and Adelaide derby matches twice makes perfect sense, the optimising (in

terms of attendance) effects of the Melbourne ‘blockbusters’ is less clear. One

wonders whether the so-called ‘blockbuster’ premium is overvalued - see Butler

(2002), whose results from Major League Baseball suggests that fans are keen to see

their team against other teams that they have not played against as often in past

seasons. Since the large Melbourne teams would presumably draw large crowds

irrespective of which (Victorian) team they play, the net effect may actually be

negligible overall.

In terms of finish place, the 12-season means are similar in magnitude to the by-team

means. More concerning is that most of the positive (negative) means appear in the

top (bottom) part of the table – this is confirmed by the correlation coefficient of

0.6249 between mean adjustment and finish place. This provides some support for

the contention that teams that would have finished higher on the ladder anyway end

up receiving a net (ex-post) benefit from the strength allocation contained in the

fixture. While of concern, the league obviously cannot observe team quality ex-ante.

Of more policy importance are expectations of team quality when the fixture is

finalised. To this end, the exercise is repeated assuming that the previous season’s

standings had been replicated.10 Table 6 shows that the distribution of positive and

negative mean adjustment values by finish place is more evenly distributed, with a

10 The correlation coefficient between the 16 teams’ standings in seasons t and t+1 has a mean value of 0.40 over the 11 relevant one-season changes.

16

correlation coefficient between mean adjustment and finish place of -0.3156 –

meaning the fixture (as intended) falls on the equalising side of random.

Finally, we would like to use the adjusted win-ratios to re-calculate the four common

competitive balance measures for 1997-2008, to see if there is a significant difference

to the unadjusted measures. Table 7 shows the outcome of this, while table 8 displays

the arithmetic changes of a result of the adjustment. In a stark contrast to Lenten’s

SPL data, there is seemingly no systematic directional effect of the adjustment on the

competitive balance measures. Beginning with the ASD/ISD ratio, the results of

which are shown graphically in figure 2, there are more negative adjustments (that is,

adjustment infers the league to be more even than based on the raw measure) than

positive adjustments, and the 12-season mean is also negative, but very small in

magnitude. The same is true for each of the other three measures. Furthermore, with

the exception of seasons 2001-2003, the four measures produce a consistent

directional change arising from the adjustment procedure in any given season.

To formalise the significance or otherwise in the magnitudinal change from the

adjustment procedure, the nonparametric Wilcoxon rank-sum test is used, since a

casual look at the data casts doubt over the normality of the distribution of these

measures (and hence the differences) over the sample.11 For the record, the test

statistic is , indicating that the rank-sums of the two distributions are precisely

equal. Nevertheless, despite this finding, it could be argued that the use of the

adjusted series is more appropriate that the unadjusted series for use in any time-series

study of the relation between attendances and competitive balance. Such an assertion

0=LT

11 Over the full history of the league from 1897-2008, the distribution of unadjusted ASD/ISD is highly negatively skewed (-0.6182), although it is close to perfectly mesokurtic (0.0135).

17

would be on the basis that fans are not fooled by the distortions created by unbalanced

schedules and base their judgements on the relative quality of teams in a bilateral

contest accordingly.

5. Conclusion

This paper has presented a modified version of the Lenten (2008) unbalanced

schedule model that is generalisable to any league in which teams play all other teams

either k or times. The model was then applied to end-of-season data from the

AFL over the period 1997-2008. The major findings include that, while the

magnitudinal effects of the adjustment procedure were small, the number of notional

positional changes in the league table arising from the adjustment over this period was

indeed considerable. Furthermore, at an aggregate level, in terms of both by team and

by rank-order in the league table, the 12-season means were large enough to be a

cause for concern, prompting questions about optimal scheduling procedures.

Ultimately, however, the adjustment of win-ratios did not cause a significant change

to any of the four common measures of reported competitive balance.

1+k

The findings presented therein have crucial implications for the game’s governing

body, as it looks ahead to further expansion to 18 teams in 2011-2012. One of many

crucial issues that will surface with the entry of the two new proposed teams is that of

tournament design. The league’s constituent clubs may agree collectively that an

alteration to the status quo of scheduling is in order. While far beyond the scope of

the model presented here, the model could be extended further to North-American

pro-sports leagues, where there is a far greater variety of unbalanced schedule

tournament design formats.

18

Table 1: Number of Matches between Each Bilateral Pairing of AFL Teams over the Period 1997-2008

ADL BNE CTN CWD ESS FRM GEE HAW MEL NMB PAD RCH STK SYD WCT WBL

ADL 17 16 18 16 19 17 16 19 15 24 16 16 17 18 20 BNE 17 15 16 17 15 17 21 19 15 19 15 19 21 19 19 CTN 16 15 24 24 17 16 17 18 16 19 18 19 16 13 16 CWD 18 16 24 24 16 13 19 19 14 15 19 16 17 16 18 ESS 16 17 24 24 17 16 16 15 15 16 21 15 17 17 18 FRM 19 15 17 16 17 19 18 14 20 19 18 19 15 24 14 GEE 17 17 16 13 16 19 17 20 19 18 18 20 18 18 18 HAW 16 21 17 19 16 18 17 20 18 17 18 16 17 17 17 MEL 19 19 18 19 15 14 20 20 18 17 16 17 18 17 17 NMB 15 15 16 14 15 20 19 18 18 18 19 17 22 17 21 PAD 24 19 19 15 16 19 18 17 17 18 16 17 16 16 17 RCH 16 15 18 19 21 18 18 18 16 19 16 18 18 16 18 STK 16 19 19 16 15 19 20 16 17 17 17 18 18 19 18 SYD 17 21 16 17 17 15 18 17 18 22 16 18 18 19 15 WCT 18 19 13 16 17 24 18 17 17 17 16 16 19 19 18 WBL 20 19 16 18 18 14 18 17 17 21 17 18 18 15 18

19

Table 2: Final (22-Game) Win Ratios of All Teams in the AFL (1997-2008)

Team 1997 1998 1999 2000 2001 2002 Adelaide 0.5909 0.5909 0.3636 0.4091 0.5455 0.6818 Brisbane 0.4773 0.2500 0.7273 0.5455 0.7727 0.7727 Carlton 0.4545 0.4091 0.5455 0.7273 0.6364 0.1364

Collingwood 0.4545 0.3182 0.1818 0.3182 0.5000 0.5909 Essendon 0.4091 0.5455 0.8182 0.9545 0.7727 0.5682 Fremantle 0.4545 0.3182 0.2273 0.3636 0.0909 0.4091 Geelong 0.6818 0.4091 0.4545 0.5682 0.4091 0.5000

Hawthorn 0.3636 0.3636 0.4773 0.5455 0.5909 0.5000 North Melbourne 0.5455 0.7273 0.7727 0.6364 0.4091 0.5455

Melbourne 0.1818 0.6364 0.2727 0.6364 0.4545 0.5455 Port Adelaide 0.4773 0.4318 0.5455 0.3409 0.7273 0.8182

Richmond 0.4545 0.5455 0.4091 0.5000 0.6818 0.3182 St Kilda 0.6818 0.5909 0.4545 0.1136 0.1818 0.2500 Sydney 0.5455 0.6364 0.5000 0.4545 0.5455 0.4318

West Coast 0.5909 0.5455 0.5455 0.3409 0.2273 0.5000 Western Bulldogs 0.6364 0.6818 0.7045 0.5455 0.4545 0.4318

2003 2004 2005 2006 2007 2008 Adelaide 0.5909 0.3636 0.7727 0.7273 0.5455 0.5909 Brisbane 0.6591 0.7273 0.4545 0.3182 0.4545 0.4545 Carlton 0.1818 0.4545 0.2045 0.1591 0.1818 0.4545

Collingwood 0.6818 0.3636 0.2273 0.6364 0.5909 0.5455 Essendon 0.5909 0.5455 0.3636 0.1591 0.4545 0.3636 Fremantle 0.6364 0.5000 0.5000 0.6818 0.4545 0.2727 Geelong 0.3409 0.6818 0.5455 0.4773 0.8182 0.9545

Hawthorn 0.5455 0.1818 0.2273 0.4091 0.5909 0.7727 North Melbourne 0.5227 0.4545 0.5909 0.3182 0.6364 0.5682

Melbourne 0.2273 0.6364 0.5455 0.6136 0.2273 0.1364 Port Adelaide 0.8182 0.7727 0.5227 0.3636 0.6818 0.3182

Richmond 0.3182 0.1818 0.4545 0.5000 0.1591 0.5227 St Kilda 0.5000 0.7273 0.6364 0.6364 0.5227 0.5909 Sydney 0.6364 0.5909 0.6818 0.6364 0.5682 0.5682

West Coast 0.5909 0.5909 0.7727 0.7727 0.6818 0.1818 Western Bulldogs 0.1591 0.2273 0.5000 0.5909 0.4318 0.7045

20

Table 3: Original Competitive Balance Measures and Summary Statistics (1997-2008)

Season ASD/ISD HICB C4ICB GINI 1997 1.1555 1.0607 1.2955 0.1536 1998 1.3121 1.0783 1.3409 0.1892 1999 1.7139 1.1335 1.5114 0.2600 2000 1.7678 1.1420 1.4773 0.2583 2001 1.8586 1.1570 1.4773 0.2858 2002 1.6096 1.1178 1.4318 0.2345 2003 1.7774 1.1436 1.3977 0.2645 2004 1.7645 1.1415 1.4545 0.2719 2005 1.6114 1.1180 1.4318 0.2383 2006 1.7581 1.1405 1.4091 0.2673 2007 1.6787 1.1281 1.4091 0.2455 2008 1.9511 1.1730 1.5114 0.3019

Mean (1997-2008) 1.6632 1.1278 1.4290 0.2476

21

Table 4: Calculated Adjusted Win Ratios According to Actual Rank-Order Finish from Table 2

Finish Place 1997 1998 1999 2000 2001 2002

1 0.6609 0.7300 0.8125 0.9570 0.7888 0.8218 2 0.6805 0.6764 0.7640 0.7435 0.7702 0.7899 3 0.6248 0.6289 0.6908 0.6155 0.7259 0.6846 4 0.5866 0.6310 0.7006 0.6134 0.6898 0.5567 5 0.5856 0.5897 0.5506 0.5448 0.6207 0.5655 6 0.5506 0.5753 0.5351 0.5258 0.5639 0.5351 7 0.5144 0.5567 0.5351 0.5619 0.5402 0.5382 8 0.4840 0.5361 0.5052 0.5619 0.5444 0.5279 9 0.4882 0.5268 0.4334 0.5176 0.5062 0.5103 10 0.4463 0.4386 0.4732 0.4360 0.4722 0.5083 11 0.4618 0.4206 0.4556 0.3937 0.4453 0.4190 12 0.4577 0.4278 0.4206 0.3783 0.4030 0.4428 13 0.4691 0.3318 0.3917 0.3231 0.4402 0.3989 14 0.4154 0.3247 0.2855 0.3644 0.2174 0.3247 15 0.3804 0.3298 0.2349 0.3494 0.1792 0.2416 16 0.1937 0.2757 0.2112 0.1137 0.0926 0.1348 2003 2004 2005 2006 2007 2008 1 0.8074 0.7517 0.7940 0.7930 0.8187 0.9218 2 0.6774 0.7218 0.7868 0.7538 0.6619 0.7558 3 0.6738 0.7300 0.6960 0.6939 0.6733 0.6986 4 0.6403 0.6629 0.6310 0.5979 0.6341 0.5804 5 0.6351 0.6165 0.5825 0.6124 0.6042 0.5846 6 0.6331 0.6032 0.5598 0.6486 0.5515 0.5810 7 0.6001 0.6094 0.5691 0.6202 0.5572 0.5872 8 0.5639 0.5567 0.5242 0.5918 0.5640 0.5506 9 0.5371 0.5124 0.5124 0.4897 0.5036 0.5253 10 0.5211 0.4433 0.4938 0.4778 0.4732 0.4484 11 0.4638 0.4660 0.4629 0.3731 0.4732 0.4484 12 0.3303 0.3855 0.4371 0.3938 0.4402 0.3649 13 0.3340 0.3566 0.3329 0.3185 0.4541 0.3371 14 0.2298 0.2236 0.2225 0.3371 0.2236 0.2565 15 0.2112 0.1833 0.2246 0.1405 0.1782 0.1968 16 0.1415 0.1771 0.1704 0.1581 0.1890 0.1627

*Percentage was used as the secondary criterion to determine rank-order, as is used in the AFL.

22

Table 5: Difference between Adjusted and Original Win Ratios by Rank-Order for Each Season

Finish Place 1997 1998 1999 2000 2001 2002

1 -0.0209 0.0028 -0.0056 0.0024 0.0161 0.0037 2 -0.0013 -0.0054 -0.0087 0.0162 -0.0025 0.0171 3 -0.0116 -0.0074 -0.0365 -0.0209 -0.0014 0.0028 4 -0.0043 -0.0054 -0.0039 -0.0229 0.0080 -0.0342 5 -0.0053 -0.0012 0.0051 -0.0233 -0.0157 -0.0027 6 0.0051 -0.0156 -0.0104 -0.0197 -0.0270 -0.0104 7 -0.0311 0.0113 -0.0104 0.0165 -0.0052 -0.0073 8 0.0067 -0.0094 0.0052 0.0165 -0.0011 0.0279 9 0.0109 -0.0187 -0.0439 0.0176 0.0062 0.0103 10 -0.0082 0.0068 0.0187 -0.0185 0.0176 0.0083 11 0.0073 0.0115 0.0011 -0.0154 -0.0092 -0.0128 12 0.0032 0.0187 0.0115 0.0147 -0.0061 0.0110 13 0.0145 -0.0318 0.0281 -0.0178 0.0311 -0.0102 14 0.0063 0.0065 0.0127 0.0235 -0.0099 0.0065 15 0.0167 0.0116 0.0076 0.0313 -0.0026 -0.0084 16 0.0118 0.0257 0.0294 0.0001 0.0017 -0.0015 2003 2004 2005 2006 2007 2008 1 -0.0108 -0.0211 0.0213 0.0202 0.0006 -0.0327 2 -0.0044 -0.0055 0.0140 0.0265 -0.0199 -0.0169 3 0.0147 0.0028 0.0142 0.0121 -0.0085 -0.0060 4 0.0039 -0.0189 -0.0054 -0.0384 -0.0023 -0.0105 5 -0.0012 -0.0198 -0.0084 -0.0240 0.0133 -0.0063 6 0.0422 0.0123 0.0144 0.0122 -0.0394 0.0128 7 0.0092 0.0185 0.0237 0.0065 -0.0110 0.0190 8 -0.0270 0.0113 0.0015 0.0009 0.0185 0.0051 9 -0.0083 0.0124 0.0124 -0.0103 -0.0191 0.0025 10 -0.0016 -0.0113 -0.0062 0.0006 0.0187 -0.0061 11 -0.0362 0.0114 0.0083 -0.0360 0.0187 -0.0061 12 -0.0106 0.0219 -0.0175 0.0302 -0.0144 0.0012 13 0.0158 -0.0070 -0.0308 0.0003 0.0223 0.0189 14 0.0025 -0.0037 -0.0048 0.0189 -0.0037 -0.0162 15 0.0294 0.0015 -0.0027 -0.0186 -0.0037 0.0149 16 -0.0176 -0.0047 -0.0342 -0.0010 0.0300 0.0264

*Percentage was used as the secondary criterion to determine rank-order, as is used in the AFL.

23

Table 6: Average Difference between Adjusted and Original Win Ratios (1997-2008) by Team and Finish Place, and Based on Previous Season Finish

Team Finish Place Season Lag

Adelaide 0.0111 1 -0.0020 0.0016 Brisbane 0.0018 2 0.0008 0.0065 Carlton 0.0003 3 -0.0038 0.0026

Collingwood -0.0035 4 -0.0112 -0.0039 Essendon -0.0059 5 -0.0075 -0.0064 Fremantle 0.0040 6 -0.0020 0.0018 Geelong -0.0039 7 0.0033 0.0084

Hawthorn -0.0094 8 0.0047 0.0027 North Melbourne 0.0012 9 -0.0023 -0.0004

Melbourne -0.0018 10 0.0016 -0.0062 Port Adelaide 0.0027 11 -0.0048 -0.0033

Richmond 0.0046 12 0.0053 0.0058 St Kilda -0.0071 13 0.0028 0.0041 Sydney -0.0033 14 0.0032 0.0023

West Coast 0.0066 15 0.0064 -0.0101 Western Bulldogs 0.0027 16 0.0055 -0.0054

Table 7: Adjusted Competitive Balance Measures (1997-2008)

Season ASD/ISD HICB C4ICB GINI 1997 1.0860 1.0536 1.2764 0.1428 1998 1.2637 1.0726 1.3332 0.1815 1999 1.6145 1.1185 1.4840 0.2408 2000 1.7498 1.1392 1.4647 0.2517 2001 1.8634 1.1578 1.4874 0.2853 2002 1.6311 1.1209 1.4265 0.2359 2003 1.7711 1.1426 1.3994 0.2666 2004 1.7401 1.1376 1.4332 0.2656 2005 1.7153 1.1337 1.4539 0.2588 2006 1.7994 1.1472 1.4193 0.2768 2007 1.6257 1.1201 1.3940 0.2340 2008 1.8598 1.1572 1.4783 0.2850

Mean (1997-2008) 1.6433 1.1251 1.4209 0.2437 *Ordering for C4ICB was determined by pre-adjustment rankings. This made a difference in only 3 years out of 12 in the sample, and the difference was small in each case.

24

Table 8: Arithmetic Change Arising from the Adjustment Procedure for Each of the Competitive Balance Measures

Season ASD/ISD HICB C4ICB GINI 1997 -0.0696 -0.0071 -0.0190 -0.0108 1998 -0.0484 -0.0057 -0.0077 -0.0077 1999 -0.0994 -0.0150 -0.0274 -0.0191 2000 -0.0179 -0.0029 -0.0126 -0.0065 2001 0.0048 0.0008 0.0101 -0.0006 2002 0.0215 0.0032 -0.0053 0.0013 2003 -0.0062 -0.0010 0.0017 0.0021 2004 -0.0245 -0.0039 -0.0213 -0.0063 2005 0.1039 0.0157 0.0221 0.0205 2006 0.0413 0.0067 0.0102 0.0095 2007 -0.0531 -0.0080 -0.0151 -0.0114 2008 -0.0913 -0.0158 -0.0331 -0.0169

Mean (1997-2008) -0.0199 -0.0027 -0.0081 -0.0038

Figure 1: SPL 'Scale-Back' Bias from 44 to 33 Games in Lenten (2008)

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95

25

Figure 2: Original (Thick Line) and Adjusted (Dashed Line) ASD/ISD Ratio

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Figure 3: Adjustment Factor Against Original Win-Ratio

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

26

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