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.
[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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