Managers’ Efficiency in the English Premier League in the season 2012-2013

Odds for football matches offered in the betting market can be reconverted into probabilities for each possible result (home win, draw, away win). If the betting market were efficient, these probabilities would reflect the true probabilities of each event. Although there is no consensus in the literature whether or not betting odds are efficient, it seems that any inefficiency that arises is very small. Thus, the probabilities embedded in betting odds can be seen as true probabilities. The European domestic football leagues are generally organized in a double round robin basis, where each team plays against the other teams once at home and once away. Winning teams are awarded three points, a draw earns one point for each team, and the losing team earns no points. The final league ranking is made on the basis of points earned at the end of the double round robin.
Basic probability theory 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, that is, the density function of total points at the end of the season. 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 in order to calculate the density functions for the 2012-13 season for English Premier League. The cumulative density function can be used to calculate the probability of getting more points than the actual result. The inverse of that probability can be viewed as an efficiency index for managers in the sense than the closer the value to one will reflect a better performance and the closer the value to zero will reflect a worse performance. The managers with higher efficiencies would be the teams that have overperformed the results expected from the odds. The overperforming could be due to luck or fortunate referees’ decisions, but the most plausible reason to overperform the expected results from the odds is good coaching, in the same way the underperformance of the teams could be due to injuries or bad luck but the most plausible is bad coaching. Thus, the efficiency index can be understood as a measure of the performance of the managers.
Table 1 shows the managers’ efficiency in the season 2012/2013. It can be seen that Sir Alex Ferguson in his last season was the best manager followed by André Villas-Boas from the Tottenham Hotpsur. On the other hand the worst managers have been Mark Hughes and Harry Redknapp. It is noteworthy to indicate that the weighted average of the efficiency was 0.52, so all managers with a efficiency over 0.52 have been more efficient than the average.
Table 1. Efficiencies of the managers in the season 2012/2013
Manager
Team
Points
Matches
Efficiency
Alex Ferguson
Manchester United
89
38
0.96
André Villas-Boas
Tottenham Hotpsur
72
38
0.90
Chris Hughton
Norwich City
44
38
0.74
Arsène Wenger
Arsenal
73
38
0.71
Steve Clarke
West Bromwich Albion
49
38
0.70
Roberto Di Matteo
Chelsea
24
12
0.69
Paolo Di Canio
Sunderland
8
7
0.68
Rafa Benítez
Chelsea
51
26
0.66
Roberto Mancini
Manchester City
75
36
0.66
David Moyes
Everton
63
38
0.65
Sam Allardyce
West Ham United
46
38
0.63
Paul Lambert
Aston Villa
41
38
0.57
Michael Laudrup
Swansea City
46
38
0.55
Mauricio Pochettino
Southampton
19
16
0.50
Nigel Adkins
Southampton
22
22
0.46
Tony Pulis
Stoke City
42
38
0.40
Nigel Adkins
Reading
5
8
0.37
Martin Jol
Fulham
43
38
0.35
Martin O’Neill
Sunderland
31
31
0.33
Brian Kidd
Manchester City
3
2
0.29
Brendan Rogers
Liverpool
61
38
0.24
Brian McDermott
Reading
23
29
0.21
Roberto Martínez
Wigan Athletic
36
38
0.21
Alan Pardew
Newcastle United
41
38
0.21
Harry Redknapp
Queens Park Rangers
21
26
0.16
Mark Hughes
Queens Park Rangers
4
12
0.03
Eamonn Dolan
Reading
0
1
0.00
Notes: elaborated using the betting odds from CODERE APUESTAS
Managers are ordered by the efficiency level.
In italics managers that were fired
Julio del Corral Cuervo is Associate professor in Economics at the University of Castilla-La Mancha (Spain)
* These results will be presented at the Fourth Spanish Conference in Sports Economics (Madrid, June 2013).

Balance competitivo en el fútbol europeo en la temporada 2011-2012: La liga española fue más escocesa que la liga escocesa

El análisis del balance competitivo en la Economía del Deporte comenzó cuando Simon Rottenberg escribió en 1956 “Es necesario que la distribución del talento sea más o menos equitativa para que exista incertidumbre sobre el resultado y la existencia de incertidumbre en el resultado es necesaria para que los consumidores estén dispuestos a pagar por ver un partido”. Desde entonces muchos autores han analizado la relación entre la igualdad de los competidores con el espectáculo que se genera. Aunque la evidencia empírica no ha resultado siempre favorable a la hipótesis de Rottenberg si se ha corroborado dicha hipótesis en muchos trabajos.
La mayoría de las medidas del balance competitivo usadas en la literatura usan los resultados en la liga para calcular diferentes medidas de desigualdad en los puntos conseguidos por los equipos (desviación típica de los puntos, ratio entre puntos conseguidos por los 4 primeros clasificados y los 4 últimos clasificados). Es decir, son medidas ex posten el sentido que valoran los resultados pero no valoran la percepción de los aficionados de si los partidos ex antevan a ser partidos disputados. Obviamente la percepción de que el partido va a ser disputado es el dato relevante para los aficionados a la hora de tomar la decisión de asistir al partido o no asistir, dado que lo importante no es que el partido resulte disputado sino que los aficionados prevean que el partido va a ser disputado.  
Las casas de apuestas deportivas establecen cuotas para los diferentes resultados de un partido. Las cuotas reflejan el inverso de las probabilidades establecidas por la casa de apuestas ajustadas ligeramente para que la casa tenga un pequeño margen (este suele rondar el 10%). Una vez conocida la probabilidad de cada resultado para todos los equipos  en todos los partidos de una liga es posible determinar cuál es la probabilidad de cada equipo de obtener un determinado número de puntos. Lo único que se necesita conocer es que la probabilidad de dos sucesos independientes es el producto de las probabilidades de cada uno de ellos. Por ejemplo si la probabilidad de que el Real Madrid gane al Sporting de Gijón es del 90% y la probabilidad de que el Real Madrid gane al Barcelona es del 40%, la probabilidad de que el Real Madrid gane esos dos encuentros es del 36% (0,9 x 0,4=0,36). [1]
Replicando esto para todos los partidos se puede construir lo que se conoce como una distribución de la probabilidad del número de puntos para cada equipo. Así, si un equipo tiene probabilidades altas de victoria en todos sus partidos, la probabilidad de obtener una gran cantidad de los puntos en disputa al final de la liga (por ej., 90 puntos si en la liga se disputan 38 partidos) será “relativamente” alta, mientras que la probabilidad de obtener pocos puntos al final de la liga (por ej., 40 puntos) será baja, esto lo que ocurre a los mejores equipos de la liga como El Real Madrid, F.C. Barcelona o Bayern Munich. Por el contrario, los equipos que tienen probabilidades bajas de victoria en la mayoría de partidos de la liga, tendrán una probabilidad muy baja de obtener una gran cantidad de puntos y una probabilidad más alta de obtener 40 puntos.
El siguiente gráfico muestra la distribución de la probabilidad del número de puntos para cada equipo en las 5 principales ligas europeas en la temporada 2011-2012: Liga BBVA (España), Bundesliga (Alemania), Serie A (Italia), Premier League (Inglaterra) y Ligue 1 (Francia). Adicionalmente se incorpora la Scottish Premier League por ser el ejemplo típico de una liga desigual con dos equipos con una diferencia de calidad notoria respecto al resto.
En el gráfico se observa claramente como la liga española es la más desigual, donde hay dos equipos (Real Madrid y F.C. Barcelona), muy por encima del resto. Esta tendencia es similar en la liga escocesa pero de forma más atenuada La liga donde existía un mayor balance competitivo era la francesa seguida de la italiana. En la liga inglesa hay dos grupos relativamente diferenciados. En uno primero estarían Manchester United, Manchester City, Chelsea, Arsenal, Liverpool y Tottenham y en el otro el resto.
En la siguiente tabla se muestra las probabilidades de cada uno de los equipos de la liga española de haber conseguido como máximo los puntos con los que descendió el equipo con mejor puntuación que descendió (Villarreal), y como mínimo los puntos que dieron acceso a la Europa League (Levante), a la Champions League (Málaga), a la Champions League de forma directa (F.C. Barcelona) y a ganar la liga (Real Madrid).[1]
Equipo
Descenso
UEFA
Champions
Champions directa
Liga
Athletic Bilbao
4,440%
47,932%
32,590%
0,000%
0,000%
Atlético de Madrid
0,611%
77,843%
64,425%
0,002%
0,000%
F.C. Barcelona
0,000%
100,000%
100,000%
67,692%
14,764%
Real Betis
19,794%
18,734%
9,992%
0,000%
0,000%
R.C.D. Español
21,449%
16,960%
8,814%
0,000%
0,000%
Getafe C.F.
28,540%
11,900%
5,747%
0,000%
0,000%
Granada C.F.
61,695%
2,016%
0,720%
0,000%
0,000%
Levante U.D.
34,124%
9,011%
4,121%
0,000%
0,000%
Málaga C.F.
1,688%
64,548%
48,964%
0,000%
0,000%
R.C.D. Mallorca
34,008%
9,172%
4,224%
0,000%
0,000%
C.A. Osasuna
28,265%
12,126%
5,888%
0,000%
0,000%
Real Madrid
0,000%
100,000%
100,000%
46,532%
5,574%
Racing de Santander
69,965%
1,130%
0,371%
0,000%
0,000%
Sevilla F.C.
1,912%
62,996%
47,413%
0,000%
0,000%
Real Sociedad
41,095%
6,314%
2,701%
0,000%
0,000%
Sporting de Gijón
58,824%
2,428%
0,893%
0,000%
0,000%
Valencia C.F.
0,344%
83,045%
71,120%
0,004%
0,000%
Rayo Vallecano
36,838%
8,188%
3,718%
0,000%
0,000%
Villarreal C.F.
11,633%
29,265%
17,335%
0,000%
0,000%
Real Zaragoza
52,614%
3,357%
1,287%
0,000%
0,000%
Nota: elaboración propia con datos provenientes de CODERE. Los datos están redondeados al tercer decima.
La cifra más representativa que muestra la desigualdad en la liga entre los dos primeros equipos y el resto, es la probabilidad de haber conseguido al menos los puntos que dieron acceso a la Champions directa. Esta probabilidad fue del 0,004% para  el Valencia y un 0,002% en el caso del Atlético de Madrid, mientras que para el resto de equipos fue menor que el 0,001%. También es llamativa que la probabilidad del Real Madrid y el F.C. Barcelona de conseguir al menos los puntos necesarios para clasificarse para Champions es del 100%.
Visto esto y después de hablar con muchos aficionados está claro que la Liga BBVA está perdiendo mucho atractivo, y si bien juegan en ella dos de los mejores equipos del mundo el resto de equipos son cada vez más flojos y lo que es peor con una tendencia decreciente. Como muestra el Málaga y el Valencia tuvieron que vender el verano pasado  a varios de sus mejores jugadores y son dos de los equipos que podrían optar a estar cerca de los dos grandes. El 27 de Abril habrá elecciones en la Liga de Fútbol Profesional, así que por el bien del fútbol español esperemos que el nuevo presidente sea capaz de reconducir esta situación.
Julio del Corral Cuervo es profesor contratado doctor en Economía en la Universidad de Castilla-La Mancha.
* Agradezco la ayuda en el tratamiento de la información de Jesús Gómez-Roso, así como los comentarios recibidos de Fernando del Corral.
** Una versión de este trabajo se ha enviado para su posible publicación a una revista científica, la versión completa puede descargarse en este enlace:



[1]Bueno y un poquito de programación dado que hacer esto a mano es totalmente inviable.

A comparison between efficiency from traditional techniques and those derived from odds

The traditional methodology to calculate the efficiency of decision making units is to estimate a frontier either using non-parametric 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/ranking-football-managers-big-5-leagues-2011-12-season) 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 2011-2012 (http://footballperspectives.org/efficiency-managers-spanish-football-league-2011-12-season) 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 frontier-squad
0.34
Corr TE odds-squad
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.

A comparison between efficiency from traditional techniques and those derived from odds

The traditional methodology to calculate the efficiency of decision making units is to estimate a frontier either using non-parametric 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/ranking-football-managers-big-5-leagues-2011-12-season) 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 2011-2012 (http://footballperspectives.org/efficiency-managers-spanish-football-league-2011-12-season) 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 frontier-squad
0.34
Corr TE odds-squad
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 over-performing 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 over-performing 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 well-established production function approach.
* I acknowledge the valuable assistance in recording the data from Fernando del Corral, Raúl Laguna and Jesús Gómez-Roso.

[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.