Classification of the teams in the Brazi

Soccer & Society
Vol. 11, No. 3, May 2010, 261–276

Classification of the teams in the Brazilian Soccer Championship by
probabilistic criteria composition
Annibal Parracho Sant’Anna,* Eduardo Uchoa Barboza and João Carlos Correia
Baptista Soares de Mello
Department of Engenharia de Produção, Universidade Federal Fluminense, Niterói-RJ, Brazil
AnnibalParracho
1466-0970
Soccer
10.1080/14660971003619560
FSAS_A_462465.sgm
Original
Taylor
302010
11
tppaps@vm.uff.br;
00000May
and
&

Article
Francis
(print)/1743-9590
Francis
Society
2010Sant’Anna
annibal.parracho@gmail.com
(online)

The essay discusses alternative rules for the classification of the teams in the
Brazilian Soccer Championship. Three explanatory variables for the final
classification are explored: the total number of goals scored and taken by each
team and the results of their matches against each opponent. Two teams are placed
in different classes only if their relative ranks according to a combination of these
three criteria do not contradict the official ranks. The essay studies the rules for
building the joint classification based on the probabilities of dominance according
to these three criteria. Finally, it presents the results of the application of different
rules to the data of 2006 and 2007.

Introduction

The Brazilian Football Confederation (CBF) employed during the 2007 season a
championship framework applied for some decades in the European championships.
The championship had two half-seasons, each formed by matches among all teams,
without playoffs. The teams were ranked by the sum of points, counting three points
for a victory and one for a tie. By the end of each season, four clubs at each end of the
final official standings were selected for relegation and for promotion or indication for
an international tournament.
This model, we argue, generates a long championship, with important decisions in
matches of the last rounds, subject to the influence of lack of motivation of teams
already relegated by then. In this context, the results of games played in different
moments against a same third team, may determine relegation of a team that wins all
games against another that is not relegated. Different aspects of this model, like the
asymmetry in the valuation of wins and losses,1 or the two half-seasons round-robin
framework,2 have been the object of extensive discussion.
In 2007, the Brazilian champion of 2005 was relegated to the second division
through the result of a match in the last round. To obtain a higher number of points,
another team, facing at home an opponent without any further motivation, had the
decisive goal scored in the third attempt of a penalty kick. The two previous unsuccessful kicks were declared invalid due to foul moves of the goalkeeper detected by
the referee. This is an evidence of the phenomenon of ‘home advantage’.3 Specifically,
the contribution of the referees to home advantage is a well studied subject.4

The present essay proposes parallel classification rules designed to assure clearer
cut points in the official ranking. With the addition of these rules, small differences in
*Corresponding Author. Email: tppaps@vm.uff.br
ISSN 1466-0970 print/ISSN 1743-9590 online
© 2010 Taylor & Francis
DOI: 10.1080/14660971003619560
http://www.informaworld.com

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A.P. Sant’Anna et al.

the official number of points contradicted by the performance according to other criteria are erased. If the number of clubs to be relegated or, at the other end, selected for
higher level tournaments, has to be fixed, playoff games between tied teams may be
scheduled. Untying rules may also be based on rankings that take into account a larger
series of results, like the CBF’s permanent ranking of clubs.
Three factors are considered to generate the alternative classifications: number of
goals scored, number of goals taken and the balance of goals in the direct confrontation of each pair of teams. This last factor does not determine a relation of order in the
set of teams, because it lacks the transitive property. In fact, for any three teams, X, Y
and Z, it is frequent that X beats Y in their direct confrontation, Y beats Z and Z beats

X. But an approach based on pairwise comparison according to all factors allows for
taking into account criteria without the transitive property.
Other factors are suggested in the relevant literature.5 They may be employed if a
more precise representation is required. The criteria composition must then be formulated in such a way as to avoid that the increase in the number of factors might reduce
excessively the number of resulting classes.
Different forms of composition are studied here. The composition algorithms
employ, as scores for the joint ranking, probabilities of simultaneous preference
according to the whole set of criteria or to at least one criterion.6 The determination
of such probabilities may be done in different ways. If it is possible to assign weights
to the criteria, weighted means may be employed. Alternatively, as will be done here,
joint probabilities will be employed. The computation of these probabilities will be
discussed in the next section while the peculiarities of the application of this approach
to the classification of teams in a championship and the results of its application to the
data of first and second divisions of the Brazilian Soccer Championships of 2006 and
2007 are discussed in the two following sections. In 2006, the above referred model
of championship was applied for the first time simultaneously for the two divisions in
Brazil.
Probabilistic criteria composition
The probabilistic composition enables to objectively combine the criteria in different
ways. If it is possible to numerically evaluate preferences among the criteria, the preference according to each criterion may be dealt conditionally on the choice of the

criterion. But preferences among criteria are difficult to quantify. Probabilistic evaluation of the preferences allows for the combination of the criteria without the previous
assignment of numerical weights to them. For instance, adopting an optimistic point
of view, we may decide on the basis of the probability of preference according to at
least one of the available criteria. Alternatively, we may use the probability of preference according to all the criteria together. Another possible choice is between basing
the decision on the vector of probabilities of being the best choice among all or deciding, conservatively, on the basis of the probabilities of not being the worst.
Thus, in general, four composition approaches may be taken: compose by the
probability of beating the opponents according to all the criteria; by the probability of
beating the opponents according to at least one criterion; of not being beaten according to all the criteria; of not being beaten according to at least one criterion. In sports
competitions, the natural reference is the best performance. Sports spirit points to the
goal of beating all opponents according to every criterion. But other points of view
may be taken, for instance when the problem is picking clubs for relegation. In that

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263

case, it is distance to the frontier of worst performance that must count and a good
performance according to one criterion only may be enough to satisfy the fans in the
stands and justify the presence in the first division.
Another aspect that the probabilistic composition permits to objectively take into

account is the possibility of dependence relations between the criteria. If correlation
between criteria can be quantified, it may be introduced in the computation of joint
probabilities. In cases which lack ways of direct evaluation, extreme instances of
dependence may be assumed. To simplify, it might be taken as constant the correlation
between the indicators of preference. But this hypothesis is unrealistic because, dealing with indicators, if the probability of the events indicated is high, it is low in its
probability of further increase in such a way that the correlation coefficient must be
small for indicators of events of high probability and, on the contrary, large for indicators of events of low probability. In numerical terms, the covariance between indicators is a difference between a minuend bounded above by 1 and a subtrahend that is
the product of the isolated probabilities; if these are large, the difference must be
small.
Taking into account this inversion effect, the instance of extreme dependence that
will be considered here assumes covariances proportional to the probabilities of
unfavourable evaluations. Following this principle, the difference between the preference probability according to a pair of criteria together and the product of the preference probabilities according to each of them separately is proportional to the product
of the probabilities of not being preferred according to each of them separately.
While the hypothesis of independence, by the other end, computing the joint probability simply by the product of the probabilities of preference according to each criterion, results in small values, especially if there are many criteria, this hypothesis of
covariance proportional to the rejection probabilities leads, as proven by Sant’Anna,7
to the highest possible value to the joint probability, given by the minimum of the
preference probabilities according to the different criteria. It is interesting to notice
that the composition by the minimum or the maximum constitutes the basis of the
concepts of possibility and necessity of Fuzzy Logic.8
The composition by the minimum or by the maximum has the disadvantage of not

taking into account the exact estimates of the probabilities of preference according to
the majority of the criteria. Due to that, even if there are sound reasons to assume
correlation, it is interesting to compute classifications by the products and their
complements as well as the classifications by minima and maxima.
Strategies for joint evaluation of the teams
In this section, the forms of composition described in the previous section will be
employed to develop a structure able to classify the teams by the end of a championship. It is assumed that the teams face each other at least once along the tournament.
The decision variable is the official number of points earned by each team in the final
standings. The main goal of the analysis is to offer classifications that might be joined
to that derived from the official counting of points to determine classes for relegation
or promotion. The confirmation of the cutting points by alternative rules based on a
set of relevant criteria reinforces the legitimacy of relegation and promotion decisions.
Three explanatory factors are considered: total number of goals scored, total
number of goals taken and the balance of goals in the direct confrontation against each
opponent. The sum of earned points, measuring the probability of a team’s win, must

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A.P. Sant’Anna et al.

be somehow explained by its offensive and defensive abilities. Offensive power is
naturally evaluated by the total of goals scored. Analogously, the total number of
goals taken is a good evaluation of defensive capacity. Results of direct confrontation
complete the information in these two attributes, by bringing under consideration
aspects of the teams that affect their matches against each opponent. Playing at home,
for instance,9 may differently influence games against more popular or unpopular
opponents.
How to use the balance of direct confrontations? It is easy to see that it does not
determine a transitive relation, so that it may not be represented by a cardinal variable in the domain of teams that might be employed, for instance, in a regression
model. The exit taken here, instead of requiring of each criterion an evaluation for
each team, employs the results of the pairwise comparisons of the teams according
to the criterion. Factors like the result of direct confrontations can then be represented by a variable in the domain of ordered pairs of teams, associating to the pair
(X,Y) the difference between the number of goals scored by X and the number of
goals scored by Y in the matches between these two teams during the championship.
For factors like the total of goals scored or taken along the championship, to the pair
of teams (X,Y) will be associated the difference between the total of X and the total
of Y. Any other factor, like the number of attack moves or the time of ball possession during the games,10 may similarly enter the analysis, whether on total or on
individual basis.
In a previous analysis of the data of 2006, Sant’Anna et al. evaluated the effect of
the addition of two other explanatory variables,11 total number of wins and total

number of losses. Including such variables, which naturally present a higher correlation with the number of points earned, results in better fit. Nevertheless, the three
variables employed in the present analysis are more suitable to the aim of providing
relegation and promotion rules capable of keeping the teams motivated until the last
round. In fact, the classification according to goals scored and taken and balances in
direct confrontation may be changed by efforts of any team in moments when, by the
other side, the number of wins or losses could not be explored in a last try to contradict
the classification by points. Thus the probabilistic composition of these criteria makes
every game important, even though changes resulting from a particular final game are
difficult to happen.
Before combining the criteria, the distribution of probabilities of preference
according to each criterion must be determined. To estimate these probabilities, it is
here assumed a Pareto law, relating, to the positive value x of the difference between
the teams X and Y, the probability pYX(x) = 1/(x+1)/(x+2) of preference of Y over X
and the probability pXY(x) = 1 - pYX(x) of preference of X over Y. In the case of a tie,
that means, of x = 0, the probability is pXY(0) = pYX(0) = 1/2 for each alternative.
The Pareto distribution has been chosen due to its form of slow decay. This property allows for giving more importance to higher differences than that allowed by
distributions of lighter tails. With that, the compositions considered in the present case
will assign less importance to direct confrontation balances, which present, in general,
smaller values than the comparisons by the totals of goals, without the need of assigning weights to the criteria. This suitably fits the fact that the total values provide more
stable information.

Taking into account the different points of view and dependence assumptions
discussed in the previous section, ten different composition forms are studied here. In
the first approach taken, progressive and pessimistic, the probabilistic composition

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265

employs as global preference score the probability of being preferred according to
all the criteria. That means, each team is evaluated by the probability of, when
compared to each opponent, beating it on every criterion. Assuming independence,
this score is the product of the probabilities of beating each opponent according to
each criterion.
In a progressive and optimistic approach, the evaluation is based on the probability
of, when compared to each opponent, beating it on at least one criterion. This probability is evaluated, assuming independence, by the product, along all opponents, of the
probabilities of the complements of the events of being beaten on every criterion. The
probability of each of these last events is calculated, assuming independence again, by
the product of the probabilities of being beaten according to each criterion.
The third classification examined takes into account the possibility of positive
correlation. The global score is then given by the minimum of the probabilities of

beating each opponent, this minimum evaluated along all the opponents and along
all the criteria. With that, this score may be strongly affected by one isolated bad
confrontation.
The fourth composition considered is hybrid. It consists of assuming, by one side,
independence between criteria, and, on the other, correlation between the comparisons
with different opponents. The probability of a team beating any opponent is given by
the product of the probabilities of beating it according to each criterion. These probabilities of beating particular opponents are combined in a global score given by the
minimum along the whole set of opponents.
Finally, a last score envisaging the upper frontier consists of assuming independence between comparisons with different opponents and dependence between the
criteria. In the present application, this hybrid alternative may be more realistic than
the previous one, since random disturbances affecting the development of matches
against the same opponent may affect simultaneously the three explanatory variables,
while it is more conceivable that matches against different opponents become subject
to independent disturbances. In this approach, the global score is the product, along
the opponents, of the minimum of the probabilities of beating each opponent according to the different criteria.
Similarly, from the conservative point of view, five measures of inverted preference are considered. The first one is the product of the probabilities of being beaten
by all the opponents according to all the criteria. The second is the product along the
set of all opponents of the rest to one of the probabilities of beating such opponents
according to every criterion. This second measure estimates, assuming independence, the probability of, when compared with each opponent, being beaten according to at least one criterion. The third is the maximum maximorum of the
probabilities of being beaten. The first hybrid measure is the maximum, along the set
of all opponents, of the products of the probabilities of being beaten according to the
three criteria. A simple computation shows that this measure is also given, complementarily, by the minimum, along all the opponents, of the probabilities of not being
beaten by at least one criterion. Finally, the last global score is the product of
the three minima of the probabilities of being beaten, varying the opponents while
the criteria remain fixed.
Formally, these measures are described in Table 1, where i denotes the team being
evaluated, j varies along the set of all opponents, k varies along the set of criteria, pkij
denotes the probability of preference for i against j according to k, π denotes the product operator, min denotes the minimum and max, the maximum.

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A.P. Sant’Anna et al.

Table 1. Probabilistic compositions.
Code

Formula

Approach

UAI
USI
UAD
UAO
UAC
LAI
LSI
LAD
LAO
LAC

πjπkpkij
πj(1-πkpkji)
minjminkpkij
minjπkpkij
πjminkpkij
1 - πjπkpkji
1- πj(1-πkpkij)
1 - maxjmaxkpkji
1-maxjπkpkji
1-πjminkpkji

upper bound, all, independent
upper bound, some, independent
upper bound, all, dependent
upper bound, all, dependent objects
upper bound, all, dependent criteria
lower bound, all, independent
lower bound, some, independent
lower bound, all, dependent
lower bound, all, dependent objects
lower bound, some, dependent criteria

Data analysis
Tables 2 to 5 present, for the years of 2006 and 2007, the final official total of points
of the teams in the 1st and the 2nd divisions of the Brazilin Soccer Championship as
well as their ranks according to the above referred strategies of composition of the
explanatory factors. The official total of points is obtained from the sum of three
points per win and one point per tie. The values of the explanatory variables, extracted
from CBF,12 are presented in the Appendices.
The strongest contradictions appear in Table 5, suggesting the existence of a
higher homogeneity of the 20 teams in the 2nd division of 2007. A clear illustration
Table 2. 2006 First Division ranking.
POINTS UAI USI UAD UAO UAC LAI LSI LAD LAO LAC
S. Paulo (SP)
Internacional (RS)
Gremio (RS)
Santos (SP)
Parana (PR)
Vasco (RJ)
Figueirense (SC)
Goias (GO)
Corinthians (SP)
Cruzeiro (MG)
Flamengo (RJ)
Botafogo (RJ)
Atletico (PR)
Juventude (RS)
Fluminense (RJ)
Palmeiras (SP)
Ponte Preta (SP)
Fortaleza (CE)
S. Caetano (SP)
Santa Cruz (PE)

78
69
67
64
60
59
57
55
53
53
52
51
48
47
45
44
39
38
36
28

1
5
3
2
9
8
7
4
6
13
12
10
11
16
15
14
18
19
17
20

1
4
5
2
7
9
3
6
13
10
14
12
8
16
15
11
19
18
17
20

1
5
3
2
8
9.5
5
7
13
5
11.5
9.5
16.5
11.5
14
19
18
16.5
15
20

1
3
4
2
8
12
5
6.5
9
14
11
13
6.5
19
15
10
18
16
17
20

1
5
3
2
8
7
9
4
15
6
11
10
12
13
14
19
17
18
16
20

1
4
3
2
8
9
6
5
7
14
12
11
10
16
15
13
18
19
17
20

1
3
2
4
6
9
11
5
14
8
12
7
10
15
13
17
16
19
18
20

1
2.5
5.5
2.5
9.5
11.5
4
9.5
5.5
7
8
11.5
13
15
17
16
19
18
14
20

1
3
5
2
6
16
4
9
11
10
14
15
8
19
13
7
18
17
12
20

1
4
3
2
9
8
6
5
13
10
14
12
7
16
15
11
18
19
17
20

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267

Table 3. 2006 Second Division ranking.
POINTS UAI USI UAD UAO UAC LAI LSI LAD LAO LAC
Atletico (MG)
Nautico (PE)
Sport (PE)
America (RN)
Paulista (SP)
Coritiba (PR)
Santo André (SP)
Brasiliense (DF)
Ituano (SP)
Marilia (SP)
Gama (DF)
Guarani (SP)
Avai (SC)
Remo (PA)
Ceara (CE)
Portuguesa (SP)
CRB (AL)
Paysandu (PA)
S. Raimundo (AM)
Vila Nova (GO)

71
64
64
61
61
59
56
53
50
50
48
47
46
46
45
45
44
44
43
42

1
2
4
7
3
6
9
5
8
10
15
13
17
14
12
16
11
19
18
20

1
4
2
5
3
8
7
6
9
10
16
17
11
15
14
12
13
19
18
20

1
2
5
5
5
5
12.5
5
9
8
15
12.5
20
10
12.5
12.5
17
19
16
18

1
4
3
6
2
7
9
5
8
10
11
15
14
12
18
13
16
20
19
17

1
2
4
7
3
5
10
6
9
8
13
11
20
12
14
15
16
18
17
19

1
4
2
7
3
6
8
5
9
10
15
12
14
16
13
17
11
18
19
20

1
4
2
7
3
5
8
6
9
10
12
11
19
13
16
15
14
17
18
20

2.5
5.5
2.5
9
1
5.5
7.5
4
10.5
10.5
14
13
12
16.5
16.5
18
7.5
15
19
20

1
4
2
5
3
8
7
6
9
11
13
15
14
10
17
12
19
20
18
16

1
4
2
8
3
7
6
5
9
11
15
13
12
16
14
17
10
18
19
20

of how the risk of artificial cuts may be reduced by combining two classifications is
shown by the contradictions between the ranks of S. Caetano in this table. Due to a
small number of goals suffered, which is, among the attributes considered, that with
the largest amplitude in the 2nd division of 2007, this team reaches the best position
in the LAD composition, based on the maxima. Notwithstanding, due to its bad offensive performance, it is classified by various other compositions in the worst quartile,
even reaching the second worst rank by LSI.
Another noticeable example is given by the contradiction between the official rank
of CRB and its ranks in the alternative classifications, all of them much lower. This
club had been classified among the four with fewer points in 2006 and was not relegated
because an administrative development determined the relegation of Guarani instead.
Another indication of the homogeneity of the 2nd division in 2007 is the relegation
of Paulista. This team was close to promotion to the 1st division at the end of 2006.
Even though its relegation in 2007 was by a difference of five points, it is better
ranked than CRB and Avai by all the alternative classifications.
The analysis of the concordances between the classifications shows that, in
general, whatever the form of composition and the frontier of reference chosen, it is
easy to separate one team at the upper end, the champion, and another team at the
lower end. So, in the 1st division, in 2007 as well as in 2006, all the classifications
agree about the first and about the last ranked. In the 2nd division, in 2006, nine out
of the ten aggregate measures agree with the official classification about the champion
and six agree about the last one. Even in the 2nd division in 2007, whose homogeneity
has been already commented above, although the variations reach the upper end, there
is full agreement about the last position.

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A.P. Sant’Anna et al.

Table 4. 2007 First Division ranking.
POINTS UAI USI UAD UAO UAC LAI LSI LAD LAO LAC
S. Paulo (SP)
Santos (SP)
Flamengo (RJ)
Fluminense (RJ)
Cruzeiro (MG)
Gremio (RS)
Palmeiras (SP)
Atletico (MG)
Botafogo (RJ)
Atletico (PR)
Internacional (RS)
Vasco (RJ)
Figueirense (SC)
Sport (PE)
Nautico (PE)
Goias (GO)
Corinthians (SP)
Juventude (RS)
Parana (PR)
America (RN)

77
62
61
61
60
58
58
55
55
54
54
54
53
51
49
45
44
41
41
17

1
4
6
2
7
14
12
5
9
10
8
3
11
13
15
17
16
19
18
20

1
7
10
3
2
9
11
4
6
14
13
5
12
15
8
17
16
19
18
20

1
4.5
8
2
14.5
7
4.5
10
14.5
9
3
6
13
12
17
16
11
19
18
20

1
5
4
3
6
17
14
2
7
11
15
8
9
12
10
16
13
18
19
20

1
4
6
2
9
14
10
5
13
8
7
3
11
12
15
16
17
18
19
20

1
6
10
2
3
11
12
4
9
14
8
5
13
15
7
17
16
18
19
20

1
4
6
2
10
15
12
5
14
13
8
3
7
11
9
16
18
17
19
20

1
11
13
4
2
7
11
5
6
15
8
9
11
15
3
17
15
18
19
20

1
7
8
3
2
15
9
4
5
11
16
6
10
13
12
17
14
18
19
20

1
9
11
3
2
8
12
4
6
14
10
7
13
15
5
17
16
18
19
20

Table 5. 2007 Second Division ranking.
POINTS UAI USI UAD UAO UAC LAI LSI LAD LAO LAC
Coritiba (PR)
Ipatinga (MG)
Portuguesa (SP)
Vitoria (BA)
Fortaleza (CE)
Brasiliense (DF)
CRB (AL)
Criciuma (SC)
Marilia (SP)
S. Caetano (SP)
Ponte Preta (SP)
Avai (SC)
Barueri (SP)
Gama (DF)
S. André (SP)
Ceara (CE)
Paulista (SP)
S. Cruz (PE)
Remo (PA)
Ituano (SP)

69
67
63
59
56
53
53
53
53
53
52
51
51
51
51
50
45
42
36
33

4
1
2
3
8
5
16
6
9
12
7
14
17
15
11
10
13
19
18
20

5
1
3
2
7
9
16
6
8
4
12
15
13
17
14
11
10
19
18
20

5
2
1
3
10
4
15
9
13
16
7
6
19
12
11
8
14
17
18
20

5
3
1
2
7
8
17
4
9
6
14
13
16
18
10
11
12
19
15
20

4
1
2
3
9
5
15
8
11
16
6
12
19
14
10
7
13
17
18
20

4
2
3
1
10
9
17
7
5
6
8
14
16
15
12
11
13
19
18
20

4
1
3
2
15
8
11
10
7
19
6
13
17
14
9
5
12
16
18
20

2.5
2.5
5
7
6
10
17
4
8
1
12
12
16
12
9
14
15
19
18
20

6
1
3
2
7
8
16
5
9
4
17
12
14
19
11
13
10
20
15
18

5
2
4
1
8
9
17
7
6
3
10
15
14
16
12
11
13
19
18
20

Soccer & Society

269

As a consequence of higher homogeneity of the teams in the middle of the tables,
trying to identify extreme classes of larger size is much harder. Nevertheless, to
confirm the upper class of size 4, only in the 1st division of 2007 there is difficulty
with all the alternative measures. Five measures agree with the official classification
in the 1st division in 2006 and four in the 2nd division in 2007. The 2nd division of
2006 presents a very interesting pattern on that matter, due to a tie in the number of
points earned that required the application of a secondary untying criterion, the
number of wins, to choose the 4th club selected for promotion to the 1st division.
America and Paulista got the same total of points. While the larger number of wins
determined promotion of America, all the alternative measures would place Paulista
among the best four.
As for the other clubs promoted to the 1st division at the end of the 2006 season,
there is no doubt. Besides agreeing with respect to the champion, all of them place its
official runner-up and all but one place the third ranked in the official classification
among the best four. Looking at the following season, we find that these three clubs
delivered a pretty satisfactory performance in the 1st division of 2007, while America
soon distanced from the others in the last position, its return to the 2nd division being
early unavoidable. This seems to strengthen the case for a revision of the asymmetric
valuation of wins and losses presently adopted in soccer rules.
At the lower end, eight measures agree with the official class of four teams in the
1st division of 2006, only one in the 2nd division, two in the 1st division of 2007 and
none in the 2nd division. The concordances about relegation of three teams are, on the
contrary, very high in every case. In the 1st division of 2007, the 10 measures agree.
It is here interesting to notice that the 4th relegated was Corinthians, the champion of
2005. In general, the results obtained suggest that it would be sounder to select three
clubs, instead of four, for promotion and for relegation.
The analysis of Tables 2 to 5 reveals also that, with few exceptions, there is
considerable agreement between the alternative classifications. That means that, once
chosen as explanatory variables, the total number of goals scored and taken and the
balance of goals in the direct confrontation, the form of composition does not much
matter. This agreement is confirmed by the high correlations registered in Table 6.
This table presents the ranks correlation (Spearman correlation coefficient) between
the official classification and each of the 10 alternative vectors of scores, for both
years and both divisions.
For the comparison between the hypotheses of independence and of maximal
correlation, it is noticeable the proximity of the correlations in Table 6 related to
classifications employing different hypotheses. This suggests that other intermediary
assumptions would also lead to similar rankings. The vectors of scores derived from
the hypothesis of independence presented a slightly higher correlation with the classification by points than those involving dependence. But, even if the significance of
Table 6. Ranks correlations with the official classification.
DIVISION

UAI

USI

UAD

UAO

UAC

LAI

LSI

LAD

LSO

LAC

2006 1a
2006 2a
2007 1a
2007 2a

0.927
0.914
0.852
0.843

0.903
0.917
0.819
0.846

0.905
0.883
0.774
0.706

0.856
0.948
0.772
0.813

0.917
0.934
0.830
0.751

0.930
0.928
0.802
0.846

0.922
0.952
0.744
0.652

0.898
0.851
0.686
0.841

0.754
0.939
0.812
0.786

0.902
0.912
0.765
0.846

270

A.P. Sant’Anna et al.

this difference is confirmed, such result might be related to the fact that the multiplicative algorithm explores the exact numeric values of all differences, most of which
are irrelevant for the computations involving maxima and minima.
As for the choice of one only alternative measure, a high correlation with the official
classification is a good property, since it should generate more cutting points. Table 6
shows that the composition that is closer to the classification by points is UAI, which
is also among those of easier interpretation. But, the correlations are close, in such a
way that in different groups other measures present the highest value. For instance, in
the 1st division of 2007, that highest value is of UAI, which in the 1st division of 2006
presents also a correlation very close to the highest, which is that of LAI. In the 2nd
division of 2007, where are more divergences, five classifications present very close
results: UAI, LAI, USI, LAD and LAC. Finally, in the 2nd division of 2006, the highest
correlations with the official classification are those of UAO and LSI.
Among the six measures assuming dependence, those assuming dependence only
between the criteria seem to present higher correlation with the official classification
than the other four. These measures have an advantage of simplifying computation
and interpretation, mainly if a large number of criteria are to be taken into account.
Given the small variations, this concern with simplicity of computation may turn even
UAD and LAD best choices.
Counting on a variety of measures slightly different from each other, a strategy to
keep all teams motivated until the end of the championship may consist of holding the
determination of the measure to be applied until the last round. This may be done by
establishing that such measure will be chosen among those in a list by applying a preassigned rule. For instance, the alternative classification to be applied maybe that
generating a vector of scores with the highest Spearman correlation coefficient with
the Final Official Ranking. The results here obtained indicate, however, that such a
complex rule is in fact unnecessary. To assure that the final decision will not be anticipated it is enough to apply a combination of a few relevant attributes, as any of those
above studied.
Finally, to assure the highest importance of the last matches for both teams, it may
also be advisable to base the rules applied to relegation on composition approaches
different from those of the rules applied for promotion. For instance, to keep the teams
in good standings motivated until the end, their composition approach would be
referred to the frontier of best performances while the approach in the rules applied
for relegation would involve probabilities of not being the worst.
Final comments
This work developed and tested alternative approaches to compose classification
decision rules that may enable more soundly founded decisions on promotion and
relegation of clubs by the end of a soccer season. These approaches can be applied in
the same way to other sports and to different kinds of selection strategies based on
ranking. The different forms of composition studied were found to be able to approximate the classification on points, allowing for the determination of clear cutting points
for relegation and promotion.
A particular feature of the framework employed is its ability to deal with intransitive relations. Allowing for the inclusion of the results of direct confrontations
between the teams among the factors to be combined is one advantage resulting from
that.

Soccer & Society

271

A last comment may be interesting to add. Since the win in a soccer match is ordinarily determined by a very small difference of goals scored, errors of the referees
may be more important in soccer than in other sports. These errors may become
systematic as the influence of attendance pressures, tradition of a club, press evaluations and other factors not related to the performance of the teams playing may affect
the instantaneous and hardly revocable decisions of the referees. By the end of a long
championship, discrepancies between classifications based on wins and losses and
classifications by technical attributes that should determine such wins and losses may
sign the presence of such extra-field influences.
The biggest discrepancy observed in the classifications analysed here is between
Flamengo and Vasco in 2007. These two teams are ranked, respectively, third and
tenth in the Final Official Standings, while the second is much better ranked than the
other in all but one of the alternative classifications. They are two clubs of Rio de
Janeiro, of strong rivalry. Flamengo has a much larger support of public and press and
Vasco has a tradition of forming better teams that by the end of the season rarely reach
compatible results.
Besides, in the last few years preceding 2007, the President of Vasco developed a
violent animosity against the Brazilian sports press, which, for instance, after
denouncing irregularities in the election in 2006 when he was re-elected as President,
created for him the epithet of Interim President. This may explain the bad practical
rank of the team. On the other side, Flamengo counted on a powerful favourable mobilization that resulted in successive records of attendance in its home games. Many of
these games ended in epic and sometimes surprising victories.

Notes
1. Bloyce and Murphy, ‘Sports Administration’.
2. Courneya and Carron, ‘The Home Advantage’.
3. Smith, ‘The Home Advantage’; Thomas, Reeves and Davies, ‘An Analysis’; Page and Page,

‘The Second Leg’; Jacklin, ‘Temporal Changes’; Pollard, ‘Worldwide Regional Variations’.
4. Groot, ‘Referees’; Boyko, Boyco and Boyco, ‘Referee Bias’; Dawson et al., ‘Are Football

Referees?’; Downward and Jones, ‘Effects of Crowd Size’.
5. Carmichael, Thomas and Ward, ‘Team Performance’; Hoffman, Lee and Ramasamy,
6.
7.
8.
9.
10.
11.
12.

‘The Socioeconomic Determinants’; Haas, ‘Productive Efficiency’; Barros and Leach,
‘Performance Evaluation’; Goddard, ‘Who Wins the Football?’
Sant’Anna, ‘Data Envelopment Analysis’.
Sant’Anna, ‘Probabilistic Majority Rules’.
Zadeh, ‘Fuzzy Sets’, and Zadeh, ‘Fuzzy Sets as the Basis’.
Leifer, ‘Perverse Effects’; Madrigal and James, ‘Team Quality’; Nevill, Balmer and
Williams, ‘The Influence of Crowd Noise’.
Espidia-Escuer and Garcia-Cebian, ‘Measuring the Efficiency’.
Sant’Anna, Uchoa Barboza and Soares de Mello, ‘A Framework’, 180–5.
CBF, Site dos Campeonatos Brasileiros das series A, B e C, 2007. http://200.159.15.35/
seriea/Table.aspx.

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Boyko, R.H., A.R. Boyko, and M.G. Boyko. ‘Referee Bias Contributes to Home Advantage in
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Cup Competitions’. Journal of Sports Sciences 25 (2007): 1547–56.
Pollard, R. ‘Worldwide Regional Variations in Home Advantage in Association Football’.
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Preferences’. International Journal of Industrial Systems Engineering 4 (2009): 32–45.
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the Brazilian Soccer Championship Classifications’. In Proceedings of the IMA Mathematics in Sports Conference, Salford, UK, 2007.
Smith, D.R. ‘The Home Advantage Revisited: Winning and Crowd Support in an Era of
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Zadeh, L.A. ‘Fuzzy Sets as the Basis for a Theory of Possibility’. Fuzzy Sets and Systems 1
(1978): 3–28.

Appendix 1.

2007 1a
PT

GP GT

Team

A-M A-P Bot Cor Cru Fig Fla Flu Goi Gre Int
17
55
54
55
44
60
53
61
61
45
58
54
41
49
58
41
62
77
51
54

24
63
51
62
40
73
57
55
57
49
44
49
43
66
48
42
57
55
54
58

80
51
50
58
50
58
56
49
39
62
43
44
65
63
47
64
47
19
55
47

−4

−1
−1

−2
−2
0

−2
3
0
0

−3
−3
0
2
0

−3
2
3
0
1
−2

−3
−1
0
0
−1
0
1

−3
3
−2
−1
0
2
−2
−1

−3
2
−2
−3
1
1
0
4
1

−6
−1
2
0
1
4
2
1
−2
−1

−3
0
0
1
−3
0
−1
−3
6
0
3

Juv

Nau

Pal

Par

San

S. P

Spo

Vas

−6
4
4
2
1
−2
3
4
1
0
3
1

−8
2
−5
−1
−4
3
0
0
1
−1
3
2
−3

−2
1
3
0
−2
7
−2
−3
−2
0
−2
1
1
−2

2
2
1
1
−1
−1
5
0
−2
1
−1
0
−3
2
2

−2
−1
−3
−4
2
−4
−1
−2
5
−2
1
0
−3
0
0
−3

−4
1
−1
−2
1
−2
−2
1
1
−2
−3
−2
0
−4
−1
−7
−3

0
0
−1
2
−2
1
−1
0
5
−3
−1
4
−2
−1
−3
−1
−1
3

−3
−3
0
3
−3
4
0
1
0
−4
−2
−1
3
−3
1
−3
−3
4
−2

Soccer & Society

America
Atletico-MG
Atletico-PR
Botafogo
Corinthians
Cruzeiro
Figueirense
Flamengo
Fluminense
Goias
Gremio
Internacional
Juventude
Nautico
Palmeiras
Parana
Santos
S. Paulo
Sport
Vasco

Direct confrontation

273

2007 2a

GP

GT

Team
Avai
Barueri
Brasiliense
Ceara
Coritiba
CRB
Criciuma
Fortaleza
Gama
Ipatinga
Ituano
Marilia
Paulista
Ponte Preta
Portuguesa
Remo
Santa Cruz
Santo André
S. Caetano
Vitoria

Direct confrontation
Bar Bra Cea

51
51
53
50
69
53
53
56
51
67
33
53
45
52
63
36
42
51
53
59

52
57
57
57
54
54
51
51
50
60
41
66
58
58
63
53
47
51
45
68

55
71
53
56
41
62
44
46
55
41
74
61
61
55
45
69
65
50
39
50

0

0
0

1
0
3

Cor

crb

Cri

For

Ga

Ipa

Itu

Ma

0
−4
−2
−2

0
−4
−2
5
0

0
−1
−1
0
−1
−1

−2
−1
−1
−2
−2
0
0

−1
−4
1
2
−1
−1
5
−3

1
2
−2
−1
0
−4
1
0
−5

4
4
1
4
3
3
3
1
2
4

−1
0
1
0
1
−1
0
3
−1
0
−1

Pau Pon
3
−6
1
1
1
0
−3
0
−2
5
3
2

−1
1
1
3
1
0
−3
−2
−1
−2
−1
2
0

Por

Re

Sat

S.A

S.C

Vit

−2
−3
0
1
0
1
−2
1
−3
2
−1
−3
0
−3

−1
2
−3
−1
2
1
1
2
2
2
−1
3
1
1
0

0
1
2
3
3
−3
1
0
1
0
−3
0
−1
6
0
1

−2
0
1
−4
1
1
−2
2
−1
1
−3
3
1
3
3
−2
−4

1
−2
−1
−1
−1
−3
0
−2
2
2
1
3
−3
1
1
−3
0
0

−3
1
5
−3
−2
−2
2
−5
−1
−3
2
−4
4
−7
2
−1
−3
−1
1

A.P. Sant’ Anna et al.

PT

274

Appendix 2.

Appendix 3.

2006 1a
PT

GP

GT

Team

A-P Bot Cor Cru Fig Fla Flu For Goi Gre
48
51
53
53
57
52
45
38
55
67
69
47
44
60
39
28
64
36
78
59

61
52
41
52
52
44
48
39
63
64
52
44
58
56
45
41
58
37
66
57

62
50
46
45
44
48
58
62
49
45
36
54
70
49
65
76
36
53
32
50

−1

0
−1

0
−1
−1

−3
0
−2
2

0
−1
−5
0
−2

0
1
1
−2
−1
2

1
2
−4
2
0
1
1

−1
0
−3
1
−1
2
−1
−3

−3
−4
−4
1
3
−2
−1
−2
2

−1
−1
−1
1
3
−2
−3
−4
3
−1

Int

Juv

Pal

Par

Pon

Sat

San S. P

Vas

0
0
2
0
0
1
1
3
2
2
−1

2
−3
0
1
5
−1
−3
−3
0
2
3
1

3
4
2
−1
−1
−1
0
−2
−2
−2
0
2
0

3
2
0
5
−2
−3
−3
3
2
4
4
−3
1
4

2
3
0
−3
2
−3
1
−2
1
4
3
2
0
5
2

−1
1
1
−2
0
−3
1
−2
−1
0
−1
0
−5
−1
−1
−2

1
1
2
2
−2
3
0
1
2
2
1
3
0
3
−1
−5
1

1
3
2
0
−2
−1
0
−2
2
−1
−1
0
−1
−1
−1
−3
−2
0
4

0
−3
−2
−2
1
−1
−2
1
−3
−1
0
−5
−1
−1
−2
−6
3
−2

Soccer & Society

Atletico (PR)
Botafogo
Corinthians
Cruzeiro
Figueirense
Flamengo
Fluminense
Fortaleza
Goias
Gremio
Internacional
Juventude
Palmeiras
Parana
Ponte Preta
S. Cruz
Santos
S. Caetano
S. Paulo
Vasco

Direct confrontation

275

2006 2a

GP

GT

Team
America
Atletico-MG
Avai
Brasiliense
Ceara
Coritiba
CRB
Gama
Guarani
Ituano
Marilia
Nautico
Paulista
Paysandu
Portuguesa
Remo
Santo Andre
S. Raimundo
Sport
Vila Nova

Direct confrontation
Ame A-M

61
71
46
53
45
59
44
48
47
50
50
64
61
44
45
46
56
43
64
42

59
70
36
66
47
64
61
52
53
49
58
64
72
51
47
50
47
42
57
45

51
39
51
51
56
51
67
62
61
48
58
48
51
70
58
60
45
59
36
68

1

2
1

Av
3
0
−2

Cea Cor crb
0
4
−1
−1

−4
2
−3
5
2

−1
6
−3
2
0
1

Ga

Gu

Itu

Ma

Na

3
1
−1
−3
2
1
−1

3
3
1
1
−1
−1
−1
−2

0
0
2
1
−3
1
0
0
0

0
2
2
1
0
3
−1
1
−4
1

1
−1
−3
0
−1
0
−2
−3
−3
−2
3

Pau Pay Por
2
2
2
−1
−6
−1
−1
−1
1
0
−3
−1

−2
3
1
3
2
3
3
0
0
−1
0
0
9

−2
1
0
0
1
3
0
1
3
1
2
1
0
2

Re

S.A S.C S.R

Vil

1
1
−1
2
0
2
0
−1
−3
5
0
2
2
0
3

−2
1
−1
4
0
−1
2
−1
−1
−1
1
2
−1
2
−1
−3

1
4
−1
2
0
1
2
−1
2
0
0
4
−2
1
−2
2
2

2
0
−1
1
−2
4
−3
3
5
1
0
−2
3
0
2
2
2
0
3

0
2
−3
−1
−1
−2
1
−4
−5
−2
0
−1
3
−3
1
0
−1
−2

A.P. Sant’ Anna et al.

PT

276

Appendix 4.

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