The traditional methodology to calculate the efficiency
of decision making units is to estimate a frontier either using nonparametric
techniques (e.g., DEA) or parametric techniques (e.g., stochastic frontier
models). In order to do this the output and inputs need to be established. If
the purpose is to calculate the efficiency of teams/managers in a sport league
there is a consensus to use the number of points or winnings as output and
quality measures of the squad as inputs. Once it is estimated the frontier to
calculate the efficiency index is straightforward by diving observed output by
the frontier output given the inputs.
I have recently proposed (http://footballperspectives.org/rankingfootballmanagersbig5leagues201112season) an alternative way to calculate the efficiency of
managers by using odds. The idea is quite simple, first it has to be computed
the probability for the teams of getting a certain amount of points at the end
of the league given the odds[1], that
means that it is calculated the density function of the points at the end of
the league. Thereafter, it can be computed the probability of the cumulative
distribution function at the actual number of points. In other words, it is
computed the probability that a certain team would have done less points. This
figure can be interpreted as an efficiency index given that it is bounded
between zero and one and that the greater the value the greater the efficiency.
Next, I am going to compare the efficiencies that arises
from estimating a production frontier for the coaches in the Liga BBVA at the
season 20112012 (http://footballperspectives.org/efficiencymanagersspanishfootballleague201112season) and the efficiency of the teams derived from the
odds. To estimate the production function it was used as output the ratio
between points obtained and the total possible points (i.e., 3 x the number of
matches) and as input the value of the most valuable goalkeeper, 6 defenders, 6
defenders, and 3 forwards from http://www.transfermarkt.co.uk.
Table 1 shows such comparison.
Team

Squad €

Points

TE frontier

Rank frontier

TE odds

Rank odds

Rank diff.

Levante U.D.

2.6E+07

55

100.0%

1

93.0%

2

1

Real Madrid C.F.

4.6E+08

100

100.0%

2

96.2%

1

1

C.A. Osasuna

3.0E+07

54

95.0%

3

87.9%

3

0

F.C. Barcelona

5.5E+08

91

88.0%

4

38.2%

14

10

R.C.D. Mallorca

4.4E+07

51

83.0%

5

85.7%

4

1

Real Betis Balonpié

3.4E+07

47

81.0%

6

48.4%

11

5

Rayo Vallecano

2.2E+07

43

81.0%

7

47.4%

12

5

Valencia C.F.

1.3E+08

61

80.0%

8

49.0%

10

2

Málaga C.F.

1.0E+08

58

79.0%

9

56.3%

9

0

Getafe C.F.

5.2E+07

47

74.0%

10

59.8%

8

2

R.C.D. Espanyol

4.7E+07

46

74.0%

11

45.5%

13

2

Real Sociedad

6.0E+07

47

72.0%

12

72.3%

5

7

Atlético de Madrid

1.5E+08

56

70.9%

13

30.8%

16

3

Real Zaragoza

4.4E+07

43

70.3%

14

63.4%

7

7

Granada C.F.

4.3E+07

42

68.5%

15

66.8%

6

9

Athletic de Bilbao

1.1E+08

49

67.0%

16

26.9%

17

1

Sevilla F.C.

1.2E+08

50

66.0%

17

19.4%

18

1

Sporting de Gijón

3.9E+07

36

59.9%

18

37.0%

15

3

Villarreal C.F.

1.5E+08

41

51.9%

19

11.6%

19

0

Racing de Santander

2.8E+07

27

48.4%

20

6.8%

20

0

Mean



75.5%


52.1%



SD

0.14

0.26


Corr TE frontiersquad

0.34


Corr TE oddssquad

0.02







There is one team that is really benefited from
obtained the efficiency using the production instead by using the odds
methodology, FC Barcelona. Why? To answer this question is worthy to analyze
the following picture that helps to explain how it works the production
function methodology.
Note: The
red line indicates the estimated production function
FC Barcelona with a bit better squad than Real Madrid earned 93 points
instead of 100 of Real Madrid. Real Madrid is on the frontier, thus the
efficiency index of FC Barcelona is calculated dividing 93 by a figure a bit
greater than 100. The result is that the efficiency from the production
function was 0.88. The interpretation is that to be fully efficient FC
Barcelona would have to gain 106 points. 0.88 is a high efficiency index, the
fourth in the ranking, but the league from FC Barcelona was so good?
According to the odds in order to make a season on the
average (0.52) FC Barcelona would had to gain 95 points (the red line) but it did
91 points (the green line). Now let us assume that FC Barcelona would have
gained 80 points, ceteris paribus. In the production frontier the efficiency
would be close to 0.8, so a high efficiency but in the odds methodology the
efficiency would be around 0.05, so a very bad season that is a much more
sensible efficiency index.
On the other hand, Real Sociedad, Real Zaragoza and
Granada were considered quite inefficient in the production function approach
(i.e., 12, 14 and 15 respectively in the rank) but they were considered quite
efficient in the odds approach (i.e., 5, 7, 6 respectively in the rank). Why do
arise these huge differences? The answer is the overperforming of Levante.
Levante with a close squad quality to these teams performed a really good
season, thus the frontier for these teams is defined by the Levante. Thus, even
though they have done a really good season according to the expectations from
the odds they were not considered such good in the production function. So,
once again the odds methodology seems to be appropriate than the production
function in this framework since the efficiency of a team does not depend from
a overperforming of other team.
Last but no least the coefficient of correlation
between the efficiency from the production function and the squad value was
0.34 whereas the coefficient of correlation between the efficiency from odds
and the squad value was 0.02. That is, the production frontier methodology is
not able to produce an efficiency index not related with the quality of the
teams but the efficiencies using the odds methodology are not related at all
with the quality of teams which is an adequate property for the efficiencies.
Thus, the efficiencies of managers/clubs derived from
the odds look to be a good alternative to the wellestablished production
function approach.
* I acknowledge the valuable assistance in recording
the data from Fernando del Corral, Raúl Laguna and Jesús GómezRoso.
[1] In doing so the odds are converted into probabilities
and subsequently it is used the formula that tells us that the joint
probability of two independent events (e.g., a victory of the same team in two
different football matches) equals the
product of their probabilities. Using this simple formula for all
possible combinations of match results of each team, the probability of each
team within a league obtaining a certain amount of points can be computed. The
total points ranges between zero (i.e., the team loses all matches) and the
product of the number of matches and three (i.e., the team wins all matches). In
particular, we use the betting odds from CODERE APUESTAS.
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