Journal of Sports Economics 2013 Ruiz 276 302

8/11/2019 Journal of Sports Economics 2013 Ruiz 276 302

1/28

http://jse.sagepub.com/Journal of Sports Economics

http://jse.sagepub.com/content/14/3/276Theonline version of this article can be found at:

DOI: 10.1177/1527002511421952

20112013 14: 276 originally published online 27 SeptemberJournal of Sports Economics

Jos L. Ruiz, Diego Pastor and Jess T. Pastor(DEA)

Assessing Professional Tennis Players Using Data Envelopment Analysis

Published by:

http://www.sagepublications.com

On behalf of:

The North American Association of Sports Economists

can be found at:Journal of Sports EconomicsAdditional services and information for

http://jse.sagepub.com/cgi/alertsEmail Alerts:

http://jse.sagepub.com/subscriptionsSubscriptions:

http://www.sagepub.com/journalsReprints.navReprints:

http://www.sagepub.com/journalsPermissions.navPermissions:

http://jse.sagepub.com/content/14/3/276.refs.htmlCitations:

What is This? - Sep 27, 2011OnlineFirst Version of Record

- May 7, 2013Version of Record>>

at University of Birmingham on May 22, 2014jse.sagepub.comDownloaded from at University of Birmingham on May 22, 2014jse.sagepub.comDownloaded from
http://jse.sagepub.com/http://jse.sagepub.com/http://jse.sagepub.com/http://jse.sagepub.com/content/14/3/276http://jse.sagepub.com/content/14/3/276http://www.sagepublications.com/http://www.kennesaw.edu/naase/http://jse.sagepub.com/cgi/alertshttp://jse.sagepub.com/subscriptionshttp://jse.sagepub.com/subscriptionshttp://www.sagepub.com/journalsReprints.navhttp://www.sagepub.com/journalsReprints.navhttp://www.sagepub.com/journalsPermissions.navhttp://jse.sagepub.com/content/14/3/276.refs.htmlhttp://jse.sagepub.com/content/14/3/276.refs.htmlhttp://online.sagepub.com/site/sphelp/vorhelp.xhtmlhttp://online.sagepub.com/site/sphelp/vorhelp.xhtmlhttp://jse.sagepub.com/content/early/2011/09/23/1527002511421952.full.pdfhttp://jse.sagepub.com/content/early/2011/09/23/1527002511421952.full.pdfhttp://jse.sagepub.com/content/14/3/276.full.pdfhttp://jse.sagepub.com/content/14/3/276.full.pdfhttp://jse.sagepub.com/http://jse.sagepub.com/http://jse.sagepub.com/http://jse.sagepub.com/http://jse.sagepub.com/http://jse.sagepub.com/http://online.sagepub.com/site/sphelp/vorhelp.xhtmlhttp://jse.sagepub.com/content/early/2011/09/23/1527002511421952.full.pdfhttp://jse.sagepub.com/content/14/3/276.full.pdfhttp://jse.sagepub.com/content/14/3/276.refs.htmlhttp://www.sagepub.com/journalsPermissions.navhttp://www.sagepub.com/journalsReprints.navhttp://jse.sagepub.com/subscriptionshttp://jse.sagepub.com/cgi/alertshttp://www.kennesaw.edu/naase/http://www.sagepublications.com/http://jse.sagepub.com/content/14/3/276http://jse.sagepub.com/


2/28

Article

Assessing ProfessionalTennis Players UsingData EnvelopmentAnalysis (DEA)

JoseL. Ruiz1, Diego Pastor2, and Jesus T. Pastor1

Abstract

The authors assess the performance of professional tennis players from theperspective of the efficiency of their game using data envelopment analysis (DEA).

This research can complement the information provided by the Association of

Tennis Professionals (ATP) ranking, which is concerned with their competitive

performance. The DEA provides an index of the overall performance of players by

aggregating the ATP statistics regarding the different aspects of the game. The DEAbenchmarking analysis also allows identifying strengths and weaknesses of the game

of the players. To the ranking of players, the authors use the cross-efficiency evalua-

tion, which assesses the players in a peer evaluation with different patterns of game.

Keywords

tennis, data envelopment analysis, cross-efficiency evaluation, assessment of players

Introduction

Tennis is a sport that moves many players and attracts millions of spectators all

around the world. A large number of tournaments and related events take place in

the five continents, both for professional and for amateur and senior players.

1 Centro de Investigacion Operativa, Universidad Miguel Hernandez, Alicante, Spain2 Division Educacion Fsica y Deportiva, C.I.D., Universidad Miguel Hernandez, Alicante, Spain

Corresponding Author:

JoseL. Ruiz, Centro de Investigacion Operativa, Universidad Miguel Hernandez, Avd. de la Universidad,

s/n, 03202-Elche, Alicante, Spain.

Email: [email protected]

Journal of Sports Economics

14(3) 276-302

The Author(s) 2013

Reprints and permission:sagepub.com/journalsPermissions.nav

DOI: 10.1177/1527002511421952

jse.sagepub.com

at University of Birmingham on May 22, 2014jse.sagepub.comDownloaded from
http://www.sagepub.com/journalsPermissions.navhttp://jse.sagepub.com/http://jse.sagepub.com/http://jse.sagepub.com/http://jse.sagepub.com/http://jse.sagepub.com/http://jse.sagepub.com/http://www.sagepub.com/journalsPermissions.nav


3/28

In competition, each player uses different game strategies trying to exploit their

strengths and hide their weaknesses as well as trying to highlight the weaknesses of

his partner. Each player usually follows a specific pattern of game, which is designedto take advantage of his own strengths. For this reason, identifying the strengths and

weaknesses of tennis players is crucial, particularly for top tennis players.

The Association of Tennis Professionals (ATP) provides a ranking of players that

is based on the points they get during the season. The players are given a specific

amount of points that depends on both the relevance of the tournament and the

matches they win in each of them. To give but one example, the smallest amount

of points obtained for reaching a semifinal corresponds to a Series 250 tournament

while the highest one corresponds to a Grand Slam. Thus, the ATP ranking is con-

cerned with the competitive performance of players. However, the ATP also pro-vides statistics regarding their game performance. For instance, its official

webpage reports data regarding the percentage of first serve points won or the per-

centage of return games won, which determine a ranking of players regarding their

performance in each of these aspects of the game separately. Unfortunately, we do

not have an index of performance of players that allows us to derive a ranking based

on the overall performance of their game. To have one such index, we should be able

to aggregate into a single scalar the values of the statistics corresponding to the dif-

ferent factors of the game, and this aggregation should be made by incorporating

information regarding the relative importance of these factors.In this article, we propose the use of Data Envelopment Analysis (DEA) to carry out a

research on the relative efficiency of the game of professional tennis players. DEA, as

introduced in Charnes, Cooper, and Rhodes (1978), is a methodology for the analysis

of the relative efficiency of decision-making units (DMUs) involved in a production

process. It provides for each DMU an efficiency score that assessesthe relative efficiency

of its performance in the use of several inputs to produce several outputs. The DEA has

been successfully used in many real applications to the analysis of efficiency of hos-

pitals, airlines, universities, financial institutions, municipalities, countries, and so on.

In the analysis in the present article, we assess the performance of the game oftennis players, which play the role of DMUs. As for the variables to be used, the ATP

statistics regarding the different aspects of the game are considered as outputs and

we do not consider explicitly any inputs, so we make a performance evaluation of

pure output data. To measure relative efficiency, DEA uses an empirical technology

of reference, which is constructed from the data by assuming some conditions (con-

vexity, . . . ). This technology is actually a set of game possibilities that is used as

reference in the assessment of the performance of each player. Thus, the assessment

of efficiency in DEA is based on a benchmarking analysis: The different players are

classified into efficient and inefficient, so the latter are assessed with reference to abest practice frontier determined by the former, which is actually the frontier of the

technology. In particular, for each inefficient player DEA determines an efficient

referent player, real or virtual (in the latter case, it would be a combination of real

players), which can be used for benchmarking. To be specific, the coordinates of this

Ruiz et al. 277

http://jse.sagepub.com/http://jse.sagepub.com/http://jse.sagepub.com/http://jse.sagepub.com/


4/28

benchmark represent efficient targets for the player being evaluated, that is, levels of

performance in each aspect of the game that would make his game perform effi-

ciently. Therefore, comparing actual data and targets allows the analyst to identifythe sources of inefficiency in the game of the players and quantify such inefficiency,

and this information can be used to suggest possible directions of improvement. We

note that with the raw ATP statistics a given player cannot know either how or how

much to improve his game in order to perform efficiently. He could compare himself

with Federer regarding the percentage of service games won, since that is the best

player in this aspect of the game (90%), with Nadal regarding the percentage of

return games won (34%), with Roddick regarding the percentage of first serve

(70%), with Tsonga regarding the percentage of first points won (80%), . . . but this

would mean to use as benchmark a player (virtual) that in each dimension of thegame plays like the best, which is very unrealistic. Perhaps, the players can achieve

the efficiency without the need to play at these maximum levels of performance. In

other words, with the raw statistics the players cannot set achievable levels of per-

formance (lower than the maximum in each factor game) that ensure the relative

efficiency of their game. Besides, each player may have a different way to achieve

the efficiency of performance, which will depend obviously on the own characteris-

tics of his game. This suggests the need of a methodology that, based on such statis-

tics, allows the analyst to set more realistic targets, which are specific of the player

under assessment and show him the way to an efficient performance of his game.The comparison between the actual data of a given player and the targets provided

by DEA also yields a measure of the overall performance of his game. To be specific,

the DEA efficiency scores reflect the distance between the actual data and the corre-

sponding benchmark, so the closer the data to the benchmark, the higher the efficiency

of his game. We note again that the ATP statistics provide information on the perfor-

mance of the players in each aspect of the game separately, but we do not have an

index of the overall performance of their game. The DEA efficiency score of each

player has the form of a weighted sum of the values in the statistics considered for the

analysis. One of the most appealing features of the DEA methodology is that we do notneed to a priori know the value of these weights, which represent the relative impor-

tance of the different aspects of the game. Hence, in absence of this information, DEA

is a useful methodology since it provides weights that represent a relative value system

of the game factors. To be specific, DEA provides such weights trying to show the

player under assessment in his best possible light. We note that this gives total freedom

to each player in the choice of weights that he makes and this allows him to exploit the

strengths of his game in the assessments.

We should finally stress that the use of weights that are player-specific makes

impossible to derive a ranking of players based on the resulting efficiency scores,since each player is assessed with a set of weights that is usually different from those

of the others. This is why we also use here the cross-efficiency evaluation introduced

in Sexton, Silkman, and Hogan (1986) and Doyle and Green (1994), which is an

extension of DEA aimed at ranking the DMUs. The idea behind cross-efficiency

278 Journal of Sports Economics 14(3)



5/28

evaluation here is to assess each player not only with its own weights but also with

those of other players. This provides a peer evaluation of the players, as opposed to

the DEA self-evaluation, which makes it possible to derive an ordering.DEA has been successfully used in many real-world applications both in public and in

private sectors and, in particular, it has been used in the context of sports. For instance,

Cooper, Ruiz, and Sirvent (2009) deal with the assessment of basketball players in the

context of the Spanish premier basketball league, the Asociacion de Clubes de

Baloncesto (ACB) league. In Cooper, Ramon, Ruiz, & Sirvent (2011) the authors use

the cross-efficiency evaluation in order to provide a ranking of basketball players. The

DEA models have also been used for evaluating baseball players (Anderson & Sharp,

1997; Chen & Johnson, 2010; Sexton & Lewis, 2003; Sueyoshi, Ohnishi, & Kinase,

1999), golf players (Fried, Lambrinos, & Tyner, 2004; Fried & Tauer, 2011; Ueda &Amatatsu, 2009), and football (Alp, 2006). In football, this methodology has been applied

from the point of view of the soccer teams (see Bosca, Liern, Martnez, & Sala, 2009;

Espitia-Escuer & Garca-Cebrian, 2004; Gonzalez-Gomez & Picazo-Tadeo, 2010; Haas,

2003; Haas, Kocher, & Sutter, 2004), the coaches (Dawson, Dobson, & Gerrard, 2000),

and that of the clubs (Barros, Assaf, & Sa-Earp, 2010). Relative efficiency in sports at the

level of countries has also been measured with DEA models, in particular for measuring

the performance of the participating nations at the Summer Olympics Games (Lozano,

Villa, Guerrero, & Cortes, 2002; Soares, Angulo-Meza, & Branco Da Silva, 2009;

Wu, Zhou, & Liang, 2010; Zhang, Li, Meng, & Liu, 2009). Finally, we can find applica-tions of DEA analyzing the efficiency in sports from other perspectives: Fizel and

DItri (1999) study the impact on organizational performance of practices like firing and

hiring managers, Volz (2009) provides efficiency scores not only of team performance

but also of player salaries in Major League Baseball, and Einolf (2004) measures

franchise payroll efficiency in National Football League and Major League Baseball.

As far as we know, DEA has not yet been used in the context of tennis. Statistical

data have been used without any connection to DEA in ODonoghue and Ingram

(2001). By using the data of Wimbledon, in Klaassen and Magnus (2009) the authors

develop a model that maximizes the probability of winning a point on service. Thereare other papers that deal with either physiological (Fernandez, Mendez-Villanueva,

& Pluim, 2006) or psychological aspects (Santos-Rosa, Garca, Jimenez, Moya, &

Cervello, 2007). The purpose of the present article is to show how DEA can be used

to assess the relative efficiency of professional tennis players with the aim of inves-

tigating the keys of the performance of their game. We have used the basic DEA

models, which have provided useful information, in particular for improvement of

performance and coaching. Nevertheless, in this study we also raise a number of

questions for future research, which open the door to the use of some of the exten-

sions and enhancements of the basic DEA methodology to address them.The article unfolds as follows: In the section on Material and Method, we

described both the data set and the models used in the analysis. The succeeding sec-

tion reports the results of the analyses and the conclusions that can be drawn are in

the section Discussion. The last section concludes.

Ruiz et al. 279



6/28

Material and Method

The purpose in this application is to assess professional tennis players from the pointof view of the efficiency of their game by means of a DEA benchmarking analysis

and to rank them using a cross-efficiency evaluation. The data have been taken from

the official website of the ATP, http://www.atpworldtour.com/, on December 28,

2009, that is, at the end of the 2009 season. For each player, we consider for the anal-

ysis here all the variables that are recorded in this webpage, which are identified as

the RICOH ATP MATCHFACTS (individual statistics) and are devoted to differ-

ent aspects of both service and return. These are the following:

y11serve% percentage of 1st serve;

y2ptos1servw% percentage of 1st serve points won;

y3ptos2servw% percentage of 2nd serve points won;

y4gamesservw% percentage of service games won;

y5ptosbreaksav% percentage of break points saved;

y6ptosret1servw% percentage of points won returning 1st serve;y7ptosret2servw% percentage of points won returning 2nd serve;

y8ptosbreakconv% percentage of break points converted;

y9gamesretw% percentage of return games won:

The ATP provides these individual statistics for the top 100 players but, in order

to have reliable data, we have selected a sample of 53 players, which are those that

have played more than 40 matches in Grand Slam and World Tour. Since the com-petition in professional tennis does not consist of a number of games that each player

has to play during the season (e.g., as it happens in a basketball league), we have

taken here as reference the player that has played more matches in 2009, Djokovic

(97 games), and 40 matches is approximately 40% of that maximum number of

games played in a season (in the European basketball leagues the players are

required to play at least two third of the games of the competition to appear in the

statistics). With these 40 matches, we seek not only the reliability of the data but also

to have a sample size large enough so as to avoid problems with the dimensionality

of the models used, as we have many variables (9). We note that this exclusion ofplayers should not affect the conclusions we have drawn in this article, since the

excluded players are those with a worse performance, which have failed to progress

far in the tournaments. Therefore, they are not expected to be efficient players, so

they do not play a role in the assessments. The data are recorded in Table 1.




7/28

Table1.DataSeason2009.

Player

1serve

ptos1servwp

tos2servw

gamesservwptosbreaksav

ptosret1servw

ptosret2servw

ptosbreakconv

gamesretw

Federer

62

79

57

90

69

31

51

41

24

Nadal

68

71

57

84

65

33

57

47

34

Djokovic

63

79

54

85

66

33

54

42

31

Murray

58

76

54

85

65

35

56

46

33

delPotro

62

74

53

84

65

31

53

42

27

Davydenko

67

71

55

83

64

34

54

41

31

Roddick

70

79

57

91

64

26

49

37

19

Soderling

60

78

54

86

65

31

51

44

25

Verdasco

69

72

54

85

66

31

53

45

28

Tsonga

63

80

54

89

67

28

47

38

19

Gonzalez

63

77

53

88

71

29

49

38

22

Stepanek

61

73

50

80

62

31

53

40

25

Monfils

62

76

50

84

63

30

50

42

25

Cillic

56

75

54

84

65

33

51

38

27

Simon

55

74

54

82

67

30

52

43

25

Robredo

63

72

54

81

61

31

51

44

25

Ferrer

61

69

52

77

60

32

55

43

32

Haas

60

77

53

85

66

28

48

39

21

Youzhny

62

71

52

80

60

31

51

40

26

Berdych

59

74

53

81

61

30

50

37

22

Wawrinka

58

73

52

81

66

32

50

38

25

Hewitt

53

76

53

81

62

31

53

39

28

Ferrero

67

68

54

78

60

29

53

43

26

Ljubicic

59

78

51

85

67

27

48

35

16

Querrey

60

79

52

86

60

27

48

39

19

Almagro

59

75

52

82

60

28

49

40

21

(continued)

281at University of Birmingham on May 22, 2014jse.sagepub.comDownloaded from


8/28

Table1(continued)

Player

1serve

ptos1servwp

tos2servw

gamesservwptosbreaksav

ptosret1servw

ptosret2servw

ptosbreakconv

gamesretw

Kohlschreiber

66

70

56

82

66

30

50

41

24

Melzer

60

74

51

81

59

30

47

40

21

Troicki

59

71

45

73

63

30

50

44

25

Montantes

58

71

50

76

60

30

51

45

24

Chardy

59

75

52

82

62

26

47

36

19

Mathieu

57

71

50

78

61

26

49

42

20

Isner

67

75

56

89

70

22

42

32

11

Andreev

62

71

52

80

61

29

48

37

20

Karlovic

67

85

54

92

69

21

42

32

10

Tipsarevic

56

74

52

80

61

28

50

43

24

Beck

59

72

52

81

66

28

50

33

19

Garcia-Lopez

61

69

49

75

59

30

53

40

27

Blake

57

74

52

82

63

27

48

38

19

Bennetau

66

70

48

77

59

29

50

40

22

Lopez

62

74

52

83

62

28

42

38

14

Hanescu

69

69

52

80

62

28

51

36

21

Seppi

60

69

49

74

57

31

50

35

24

Acasuso

56

73

51

79

62

27

50

39

22

Fognini

59

65

46

66

53

31

48

42

25

Gicquel

56

72

50

79

64

29

49

37

21

Serra

61

68

50

76

63

27

50

37

21

Hernandez

66

66

47

71

60

28

49

39

20

Schuettler

63

65

49

70

57

28

52

43

23

Rochus

60

65

46

66

52

28

51

41

23

Gulbis

63

73

47

79

60

27

47

38

18

Granollers

59

68

48

71

57

32

47

40

23

VasslloArgue

llo

69

66

49

73

59

28

44

39

18

Source.http://w

ww.atpworldtour.com/.



9/28

In a DEA efficiency analysis, we have nDMUs which use minputs to produce

s outputs. Each DMUj can be described by means of the vector Xj; Yj

x1j;. . .;xmj;y1j;. . .;ysj; j 1;. . .; m:As said before, the DEA models assess efficiency with reference to an empiricaltechnology or production possibility set (PPS), which is constructed from the

observations by assuming some postulates. For instance, if we assume convexity,

constant returns to scale (CRS), and free disposability (which means that if we can

produce Ywith X, then we can both produce less than Ywith XandYwith more

than X), then it can be shown that the technology is the set T f X; Y 2Rms =

X Pn

j1ljXj; YP

n

j1ljYj; lj0; j 1; :::; ng: The original DEA model by

Charnes, Cooper and Rhodes, the CCR model, provides as measure of the relative

efficiency of a given DMU0the minimum valuey0such that y0X0; Y0 2T:There-fore, this value can obviously be obtained by solving the following linear program-

ming (LP) problem

Min y0

s:t:Pn

j1ljxijy0xi0 i 1; :::; m

Pnj1

ljyrjyr0 r 1; :::;s

lj0 8j

; 1

which is the so-called primal envelopment formulation of the CCR model. Thus,

DMU0is said to be efficient if, and only if, its efficiency score equals 1. Otherwise,

it is inefficient, and the lower the efficiency score, the lesser its efficiency.

The model in Banker, Charnes, & Cooper (1984), the BCC model, is that resulting

from eliminating the CRS postulate and allowing for variable returns to scale (VRS)

in the PPS. Its formulation is the LP problem resulting from adding the constraintPn

j1lj1 to (1).

In the performance evaluation in the present article, the 53 players in the sample are

the DMUs. The nine variables previously listed, which, as said before, are all those the

ATP provides, are incorporated as outputs in the models used. Note that a higher value

in each of these variables corresponds to a better performance. Finally, we do not con-

sider any explicit inputs, since in our analysis there is no reference to resources con-

sumed.1 We only include in the models a single constant input equals 1, which means

that every player is doing the best for playing his game, that is, each player is perform-

ing as good as he can. It should be noted that, in the case of having one constant input,

the optimal solutions of (1) satisfy the conditionPn

j1lj1. Therefore, in these special

Ruiz et al. 283



10/28

circumstances, the specification of returns to scale is not particularly relevant (see

Lovell & Pastor, 1995, 1999 for details and discussions).

The primal envelopment formulation is that used in the benchmarking analysis. Itis usually solved in two stages in order to avoid the problems with the slacks in the

constraints of (1). Note that the existence of nonzero slacks in some of the con-

straints of (1) means that the point provided by this model as benchmark,

y0X0; Y0

; wherey0is the radial efficiency score of DMU0obtained with (1), is not

actually efficient since we can find another point in the PPS that uses less inputs and/

or produces more outputs. In other words, as a result of the radial measurement of the

efficiency with (1), we cannot guarantee an assessment of the efficiency in the sense

of Pareto. In the two-step procedure, Model 1 is first solved and, second, the slacks

associated with the obtained projection of DMU0 are maximized.

MaxPm

i1si0

Ps

r1sr0

s:t:Pn

j1ljxij s

i0 y

0xi0 i 1; :::; m

Pn

j1ljyrjs

r0 yr0 r 1; :::;s

lj;si0;s

r0 0 8i; r;j

: 2

Thus, DMU0is said to be efficient, in the Pareto sense, if, and only if,y01 and the

optimal value of (2) equals 0. In addition, with the optimal solutions lj,j 1, . . . ,n,

of (2) we can set efficient targets for DMU0asPn

j1ljxij; for each inputi1, . . . ,m,

andPn

j1ljyrj; for each outputr1, . . . ,s, so the evaluation with reference to points

Pareto-efficient of the PPS is guaranteed.

The model dual to (1) is the so-called dual multiplier formulation of the CCR

DEA model, whose formulation is the following LP problem

MaxPs

r1ury0

s:t: Pm

i1vixi0 1

Pm

i1vixij

Ps

r1uryrj0 j 1; :::; n

vi;ur0 8i; r

: 3

We can see that (3) provides the weights of the inputs and outputs that show DMU0 in

its best possible light. It should be noted that the DEA total weight flexibility can be a

source of trouble, since the weights provided are sometimes inconsistent with the

expert opinion. In particular, it may happen that a given player is assessed by putting




11/28

the weight only on a few of the game factors, ignoring the aspects of his game with

poor performance by assigning them a zero weight. For this reason, the literature has

widely claimed the need to avoid zero weights in efficiency assessments.The linear problem (3) is actually the result of the conversion of the following

fractional problem using the transformations in Charnes and Cooper (1962)

Max

Psr1

uryr0

Pmi1

vixi0

s:t: :

Psr1

uryrj

Pm

i1 vixij

1 j 1; :::; n

vi; ur0 8i; r

: 4

Model (4) is the CCR model in the ratio form. It has been widely claimed in the lit-

erature that DEA generalized to so-called engineering ratio traditionally used in effi-

ciency analyses in engineering and economics to the case of having multiple inputs

and multiple outputs. Models (3) and (4) are those used in the cross-efficiency

evaluation.

The cross-efficiency evaluation is an extension of DEA that is mainly aimed at

ranking the DMUs. In the standard cross-efficiency evaluation, the optimal solutionsof (3) for each DMUd,vd1 ; :::; v

dm; u

d1 ; :::; u

ds ; provide the profiles of weights that are

used to calculate the corresponding cross-efficiency of a given DMUj,j 1, . . . ,n,as follows:

Edj

Ps

r1udryrj

Pm

i1vdixij

: 5

Edjprovides a measure of the efficiency of DMUjwith the weights of DMUd. The cross-efficiency score of DMUjis defined as the average of the these cross-efficiencies

Ej1

n

Xn

d1

Edj; j 1; :::; n; 6

which measures the average efficiency according to all DMUs. In the assessment of

tennis players, the cross-efficiency score of each player provides an evaluation of his

performance with the different patterns of game that the different players have used

in their DEA self-evaluation.The main difficulty with cross-efficiency evaluation is the possible existence of

alternate optima for the weights when solving (3), which may lead to different cross-

efficiency scores depending on the choice of the profile of weights that each DMU

makes. The use of alternative secondary goals to the choice of weights among the

Ruiz et al. 285



12/28

alternative optimal solutions has been suggested as a potential remedy to the possible

influence of this difficulty which may reduce the usefulness of cross-efficiency

evaluations (see Liang, Wu, Cook, & Zhu, 2008). There are many approaches to dealwith this issue. In the application here, we use the optimal solutions for the weights

provided by the following model in Ramon, Ruiz, and Sirvent (2010b), which makes

a choice of weights among the alternate optima in Model 3 of DMUd trying to

avoid large differences in the relative importance attached to the inputs and to the

outputs as measured by the corresponding virtual weights (vixi0; i 1, . . . , m anduryr0; r 1, . . . , s)

Max jd

s:t:: Pm

i1vdixid1

Ps

r1udryrdy

d

Pm

i1vdixij

Ps

r1udryrj0 j 1; :::; n

zIvdixidhI i 1; :::; m

zO udryrdhO r 1; :::;s

zIhI

jdzOhO

jdzI;zO 0

: 7

The valuejdprovides us with an insight into how much DMUdneeds to unbalance

the relative importance attached to the different inputs and to the different outputs in

the assessment of its efficiency. jd2 0; 1; and the lower the value ofjd, the larger

the differences in the relative importance attached to the variables considered. In the

context of tennis if, for example,j

d0:1; this means that the corresponding playerwould not have his efficiency score with a set of weights in which the game factorwith lowest virtual were higher than 10%of that with highest one. Thus, this player

needs to unbalance very much the importance attached to the different aspects of the

game in order to achieve his DEA efficiency score. On the contrary, ifjd1; thismeans that this player would achieve his efficiency rating even with a profile of

weights with the same virtual for all the game factors, that is, by giving all these fac-

tors the same importance. This latter case might be indicating a good performance of

this player in all the aspects of the game.

We also note that (7) ensures a choice of nonzero weights for the DMUs that haveoptimal solutions without zeros, and this guarantees that none of the inputs or out-

puts are ignored in the assessments (Ramon, Ruiz, & Sirvent, 2010b) actually

extends to its use in cross-efficiency evaluations the multiplier bound approach to

the assessment of efficiency without slacks (Ramon, Ruiz, & Sirvent, 2010a).




13/28

Therefore, in our application, this model provides weights trying to avoid that the

relative importance attached to the different aspects of the game is extremely differ-

ent, and it also guarantees that none of these game factors are ignored in the assess-ments, in particular in the case of the efficient players.

Results

The DEA model (2) revealed that 11 (of the 53 players considered) were rated as effi-

cient. We note that 8 of them are ATP top-10 players (the other 3 are ATP ranked as 11

[Gonzalez], 33 [Isner], and 35 [Karlovic]). In Table 2, we have recorded for them the

ATP points on December 28, 2009. For each of the efficient players, Table 2 also

records the contributions to the efficiency of each of the factors of the game, together

with the corresponding optimal value of (7), jd. These contributions, which are

called virtual weights, are the product of the absolute weights obtained with (7) and

the corresponding actual output values. They are dimensionless and represent the

percentages of contribution of each factor to the total efficiency (100%), so they can

be seen as the relative importance attached to each aspect of the game in the assess-

ment of each player. As explained before, the valuejd, which is eventually the ratio

between the minimum and the maximum virtuals in each row of the table, provides

us with an insight into how much each of these players needs to unbalance the impor-

tance attached to the different game factors in order to be rated as an efficient player.

We also record in this table the number of times these players have acted as referent

in the assessments of the inefficient players, which is determined as the number of times

theljcorresponding to that efficient player in (2) is strictly positive. This shows which

players have played a relevant role as benchmarks in the analysis of relative efficiency.

The benchmarking analysis provided by DEA is one of the key features of this

methodology. This is shown in Tables 3 and 4. Table 3 records, as representative

cases, the actual data (in the first row of each player) and the corresponding efficient

targets provided by (2) of some inefficient players in the top ATP ranking (in the sec-

ond row). We only report these results just for reasons of space. The third row shows

the room for improvement in each dimension, as the difference between the target and

the actual data in relation to the actual data. In this table, we also record the DEA effi-

ciency score of the inefficient players provided by (1). We note that this radial measure

may sometimes give a misleading idea of the efficiency of the game of the player

under assessment, since it does not account for the inefficiency in the slacks. In fact,

this score does not reflect the inefficiency that has been accounted for in the setting of

the targets provided in Table 3, since these targets are the coordinates of a benchmark

provided by (2) that is efficient in the Pareto sense. Table 4 records for each of these

inefficient players those efficient players that have acted as benchmarks in their

assessment, together with the corresponding value lj >0. Obviously, the larger thevalue oflj, the larger the role of the corresponding efficient player as referent for the

inefficient player under assessment (these lj are called intensities).

Ruiz et al. 287



14/28

Table2.EfficientPlayers:ContributionstotheEfficiencyofGameFactorsandNumberofTimesActingasReferent.

Player

1serve

(%)

ptos1servw

(%)

ptos

2servw

(%)

gamesservw

(%)

ptosbreaksav

(%)

ptosret1servw

(%)

ptosret2se

rvw

(%)

ptosbreakconv

(%)

gamesretw(%)

Total

j d

#Ref.

Federer(10,550)

5.56

20.72

10.02

20.72

20.72

5.56

5.56

5.56

5.56

100

0.27

20

Nadal(9,205)

11.11

11.11

11.11

11.11

11.11

11.11

11.11

11.11

11.11

100

1

29

Djokovic(8,310)

6.91

20.71

6.41

19.64

20.71

6.41

6.41

6.41

6.41

100

0.31

22

Murray(7,030)

7.92

17.50

7.92

17.50

7.92

17.50

7.92

7.92

7.92

100

0.45

15

Davydenko(4,930)

20.36

3.58

3.58

3.58

3.58

54.54

3.58

3.58

3.58

100

0.06

4

Roddick(4,410

)

26.16

26.16

3.59

26.16

3.59

3.59

3.59

3.59

3.59

100

0.14

12

Soderling(3,410)

1.13

55.73

1.13

14.98

1.13

1.13

1.13

22.52

1.13

100

0.02

8

Verdasco(3,30

0)

28.05

2.66

2.64

28.05

28.05

2.64

2.64

2.64

2.64

100

0.09

7

Gonzalez(2,870)

6.31

4.56

4.56

4.56

61.78

4.56

4.56

4.56

4.56

100

0.07

6

Isner(1,067)

27.21

1.01

6.75

23.86

37.10

1.01

1.01

1.01

1.01

100

0.03

1

Karlovic(1,015

)

23.02

23.02

1.58

23.02

23.02

1.58

1.58

1.58

1.58

100

0.07

8

Note.Valuesin

parenthesesdenoteATPpointsonDecember28,2009.



15/28

Table3.Ben

chmarkingAnalysis:ActualDataandEfficientTargets(Som

eInefficientPlayers).

Player

DEAScore

1serve

ptos1servw

ptos2servw

gamesservw

ptosbreaksav

ptosret1servw

pto

sret2servw

ptosbreakconvga

mesretw

delPotro

0.9765

62

74

53

84

65

31

53

42

27

63.50

75.78

56.09

86.45

66.57

32.62

54.28

44.01

29.85

2.41%

2.41%

5.83%

2.92%

2.41%

5.23%

2.41%

4.78%

10.54%

Tsonga

0.9901

63

80

54

89

67

28

47

38

19

63.64

80.80

55.67

89.89

68.57

28.28

48.72

38.44

20.79

1.02%

1.00%

3.10%

1.00%

2.35%

1.00%

3.67%

1.16%

9.41%

Stepanek

0.9593

61

73

50

80

62

31

53

40

25

63.59

76.09

54.89

84.70

65.52

33.36

55.25

44.20

32.25

4.24%

4.24%

9.78%

5.88%

5.69%

7.61%

4.24%

10.49%

28.99%

Monfils

0.9795

62

76

50

84

63

30

50

42

25

63.30

77.59

54.68

85.76

65.34

31.67

52.92

42.88

28.26

2.10%

2.10%

9.36%

2.10%

3.72%

5.58%

5.84%

2.10%

13.02%

Cillic

0.9807

56

75

54

84

65

33

51

38

27

59.63

76.79

55.06

86.57

66.28

33.65

54.43

44.43

30.15

6.47%

2.38%

1.97%

3.06%

1.97%

1.97%

6.73%

16.93%

11.68%

Simon

0.9931

55

74

54

82

67

30

52

43

25

64.53

75.58

56.77

87.41

67.47

31.71

53.36

43.30

28.01

17.33%

2.14%

5.12%

6.60%

0.70%

5.69%

2.61%

0.70%

12.03%

Robredo

0.9699

63

72

54

81

61

31

51

44

25

64.96

74.24

55.68

84.72

65.16

32.44

54.84

45.37

30.99

3.10%

3.10%

3.11%

4.59%

6.82%

4.63%

7.52%

3.10%

23.96%

Ferrer

0.9663

61

69

52

77

60

32

55

43

32

67.19

71.41

56.76

84.08

65.00

33.16

56.92

46.92

33.92

10.14%

3.49%

9.15%

9.20%

8.33%

3.63%

3.49%

9.11%

6.00%

(c

ontinued)



16/28

Table3(continued)

Player

DEAScore

1serve

ptos1servw

ptos2servw

gamesservw

ptosbreaksav

ptosret1servw

pto

sret2servw

ptosbreakconvga

mesretw

Haas

0.9670

60

77

53

85

66

28

48

39

21

62.31

79.63

56.03

89.53

68.26

29.64

49.77

40.33

22.28

3.85%

3.42%

5.72%

5.33%

3.42%

5.85%

3.70%

3.42%

6.09%

Youzhny

0.9450

62

71

52

80

60

31

51

40

26

65.61

75.13

55.45

84.68

65.40

32.81

55.12

43.94

31.88

5.82%

5.82%

6.64%

5.85%

8.99%

5.82%

8.09%

9.85%

22.61%

Berdych

0.9477

59

74

53

81

61

30

50

37

22

63.04

78.09

55.93

87.52

67.47

31.94

52.76

42.04

27.65

6.85%

5.52%

5.52%

8.05%

10.60%

6.48%

5.52%

13.63%

25.68%

Wawrinka

0.9800

58

73

52

81

66

32

50

38

25

60.20

77.25

54.96

87.28

67.35

32.65

53.14

42.99

28.08

3.80%

5.82%

5.69%

7.75%

2.04%

2.04%

6.28%

13.14%

12.34%

Hewitt

0.9746

53

76

53

81

62

31

53

39

28

63.64

77.98

54.38

84.87

65.87

33.00

54.38

42.64

31.38

20.07%

2.61%

2.61%

4.78%

6.25%

6.45%

2.61%

9.32%

12.08%

Ferrero

0.9757

67

68

54

78

60

29

53

43

26

68.67

72.77

56.09

85.59

65.12

31.11

54.32

44.56

29.43

2.49%

7.01%

3.87%

9.72%

8.53%

7.28%

2.49%

3.63%

13.20%

Ljubicic

0.9727

59

78

51

85

67

27

48

35

16

63.03

80.19

56.28

90.19

68.88

29.11

49.34

39.26

21.52

6.83%

2.80%

10.36%

6.11%

2.80%

7.80%

2.80%

12.18%

34.51%



17/28

Table 5 records the cross-efficiencies and the cross-efficiency scores. The rows of

this table correspond to each of the player in the sample and in each of them we have

the evaluations of their game with the weights of each of the efficient players (underthe corresponding column). To be specific, for the calculation of the cross-

efficiencies with (5) we use the absolute weights provided by (7) for each of the effi-

cient players, and the cross-efficiency score of each player is the average of such

cross-efficiencies in the corresponding row (like in Equation 6), which are recorded

in the last column of this table. These latter determine the ranking of players. We can

see, for instance, that Nadal ranks first followed by Djokovic and Federer, in this

order. It should be noted that in this cross-efficiency evaluation we have assessed

the different players only with the weights of the efficient players (one such

approach is called peer restricted cross-efficiency evaluation in Ramon, Ruiz, &Sirvent, 2011). This is because the optimal solutions for the weights provided by

(3) for the inefficient players have all zeros, so their use would mean to assess the

players with patterns that ignore some of the factors of the game.

Finally, Table 6 records the ranking of the full sample of 53 players, ordered by

their cross-efficiency scores. The last column shows the same sample of players

ordered by their ATP ranking.

Discussion

The assessment of professional tennis players from the point of view of the factors

that describe their game provides a useful insight into their performance. In

Table 4. Inefficient Players and Benchmarks (lj Intensities).

Inefficient Player Benchmarks

del Potro Federer (0.36), Nadal (0.34), Djokovic (0.13), Murray (0.17)Tsonga Federer (0.56), Djokovic (0.14), Karlovic (0.30)Stepanek Nadal (0.30), Djokovic (0.52), Murray (0.18)Monfils Nadal (0.13), Djokovic (0.44), Roddick (0.09), Soderling (0.34)Cillic Federer (0.32), Nadal (0.03), Murray (0.65)Simon Federer (0.53), Nadal (0.41), Gonzalez (0.06)Robredo Nadal (0.56), Djokovic (0.16), Soderling (0.28)Ferrer Nadal (0.92), Murray (0.08)Haas Federer (0.68), Soderling (0.19), Karlovic (0.13)

Youzhny Nadal (0.43), Djokovic (0.48), Davydenko (0.05), Roddick (0.04)Berdych Federer (0.53), Nadal (0.11), Djokovic (0.36)Wawrinka Federer (0.37), Murray (0.47), Gonzalez (0.16)Hewitt Nadal (0.13), Djokovic (0.87)Ferrero Nadal (0.52), Roddick (0.18), Verdasco (0.30)Ljubicic Federer (0.76), Djokovic (0.04), Karlovic (0.20)

Ruiz et al. 291



18/28

Table5.Cro

ss-EfficiencyEvaluation.

Player

EfficientPlayerProvidingWeights

Score

Federer

Nadal

Djoko

vic

Murray

Davydenko

Roddick

Soderling

Verdasco

Gonzalez

Isner

Karlovic

Federer

1

0.952

1

0.975

0.938

0.992

1

0.998

1

1

1

0.987

Nadal

1

1

1

1

1

1

0.970

1

1

1

0.993

0.997

Djokovic

1

0.974

1

0.990

0.978

0.997

1

0.981

0.991

0.980

0.994

0.990

Murray

1

0.986

1

1

1

0.979

1

0.963

0.990

0.958

0.973

0.986

delPotro

0.965

0.938

0.966

0.952

0.933

0.961

0.959

0.962

0.965

0.961

0.963

0.957

Davydenko

0.974

0.965

0.974

0.976

1

0.979

0.935

0.980

0.973

0.980

0.975

0.974

Roddick

0.966

0.912

0.967

0.935

0.869

1

0.977

1.000

0.940

1.000

0.999

0.960

Soderling

0.975

0.940

0.976

0.960

0.929

0.970

1

0.960

0.964

0.958

0.969

0.964

Verdasco

0.979

0.962

0.982

0.969

0.959

0.990

0.964

1

0.987

1

0.991

0.980

Tsonga

0.965

0.903

0.966

0.932

0.879

0.973

0.985

0.978

0.959

0.979

0.983

0.955

Gonzalez

0.976

0.919

0.978

0.941

0.899

0.971

0.964

0.995

1

1

0.992

0.967

Stepanek

0.930

0.907

0.933

0.924

0.920

0.933

0.933

0.927

0.926

0.924

0.932

0.926

Monfils

0.949

0.916

0.952

0.936

0.911

0.958

0.971

0.948

0.938

0.944

0.956

0.943

Cillic

0.959

0.925

0.958

0.948

0.944

0.940

0.945

0.936

0.957

0.937

0.944

0.945

Simon

0.955

0.920

0.955

0.933

0.895

0.926

0.959

0.933

0.970

0.937

0.937

0.938

Robredo

0.939

0.922

0.940

0.935

0.931

0.945

0.948

0.937

0.926

0.935

0.937

0.936

Ferrer

0.935

0.935

0.935

0.939

0.945

0.931

0.918

0.918

0.928

0.915

0.921

0.929

Haas

0.947

0.896

0.948

0.918

0.869

0.945

0.962

0.950

0.949

0.952

0.955

0.935

Youzhny

0.923

0.905

0.924

0.920

0.923

0.931

0.919

0.922

0.910

0.920

0.924

0.920

Berdych

0.919

0.885

0.920

0.907

0.893

0.921

0.924

0.912

0.905

0.912

0.919

0.911

Wawrinka

0.941

0.907

0.942

0.927

0.928

0.926

0.924

0.934

0.956

0.937

0.936

0.933

Hewitt

0.944

0.913

0.942

0.932

0.903

0.923

0.951

0.902

0.928

0.901

0.920

0.924

Ferrero

0.922

0.914

0.923

0.917

0.909

0.937

0.910

0.937

0.917

0.937

0.930

0.923

Ljubicic

0.929

0.864

0.932

0.893

0.840

0.929

0.946

0.941

0.938

0.945

0.946

0.919

Querrey

0.928

0.878

0.930

0.906

0.848

0.945

0.975

0.925

0.891

0.920

0.939

0.917

Almagro

0.917

0.879

0.918

0.900

0.861

0.924

0.946

0.910

0.894

0.907

0.919

0.907

Kohlschreiber

0.948

0.922

0.949

0.931

0.923

0.949

0.921

0.970

0.965

0.976

0.958

0.946

(continued)



19/28

Table5(continued)

Player

EfficientPlayerProvidingWeights

Score

Federer

Nadal

Djoko

vic

Murray

Davydenko

Roddick

Soderling

Verdasco

Gonzalez

Isner

Karlovic

Melzer

0.909

0.877

0.911

0.901

0.893

0.921

0.938

0.906

0.885

0.902

0.914

0.905

Troicki

0.902

0.884

0.907

0.893

0.893

0.897

0.924

0.896

0.924

0.896

0.904

0.902

Montantes

0.907

0.892

0.910

0.903

0.893

0.903

0.935

0.892

0.903

0.890

0.899

0.902

Chardy

0.907

0.856

0.907

0.878

0.823

0.913

0.924

0.912

0.897

0.913

0.918

0.895

Mathieu

0.891

0.857

0.894

0.869

0.819

0.888

0.919

0.889

0.892

0.889

0.892

0.882

Isner

0.925

0.841

0.926

0.866

0.777

0.942

0.913

0.987

0.950

1.000

0.972

0.918

Andreev

0.902

0.868

0.903

0.888

0.879

0.912

0.899

0.916

0.897

0.917

0.914

0.900

Karlovic

0.947

0.848

0.950

0.883

0.764

0.979

0.989

0.994

0.943

1.000

1.000

0.936

Tipsarevic

0.922

0.891

0.922

0.904

0.858

0.913

0.952

0.900

0.909

0.899

0.911

0.907

Beck

0.912

0.862

0.913

0.883

0.855

0.905

0.888

0.926

0.932

0.932

0.922

0.903

Garcia-Lopez

0.900

0.890

0.902

0.899

0.900

0.906

0.895

0.896

0.897

0.894

0.900

0.898

Blake

0.911

0.863

0.912

0.884

0.835

0.907

0.928

0.910

0.908

0.912

0.914

0.899

Bennetau

0.894

0.875

0.898

0.889

0.892

0.920

0.903

0.916

0.888

0.913

0.915

0.900

Lopez

0.899

0.846

0.901

0.876

0.857

0.915

0.927

0.922

0.891

0.924

0.921

0.898

Hanescu

0.908

0.881

0.911

0.894

0.887

0.934

0.882

0.949

0.914

0.951

0.938

0.914

Seppi

0.876

0.860

0.877

0.878

0.902

0.887

0.866

0.874

0.864

0.872

0.880

0.876

Acasuso

0.906

0.868

0.907

0.883

0.835

0.898

0.922

0.895

0.905

0.896

0.903

0.893

Fognini

0.838

0.845

0.841

0.853

0.895

0.850

0.857

0.828

0.828

0.824

0.836

0.845

Gicquel

0.905

0.865

0.907

0.884

0.863

0.894

0.904

0.902

0.919

0.905

0.905

0.896

Serra

0.887

0.856

0.889

0.866

0.843

0.888

0.871

0.905

0.909

0.911

0.900

0.884

Hernandez

0.862

0.847

0.867

0.855

0.868

0.883

0.858

0.895

0.882

0.898

0.888

0.873

Schuettler

0.864

0.863

0.867

0.864

0.866

0.876

0.870

0.874

0.868

0.874

0.871

0.869

Rochus

0.828

0.831

0.831

0.836

0.848

0.846

0.851

0.824

0.813

0.820

0.831

0.833

Gulbis

0.886

0.847

0.890

0.869

0.845

0.910

0.914

0.907

0.879

0.905

0.911

0.888

Granollers

0.867

0.857

0.869

0.874

0.914

0.873

0.878

0.861

0.863

0.859

0.867

0.871

VasslloArguello

0.859

0.840

0.864

0.853

0.873

0.893

0.860

0.906

0.871

0.909

0.895

0.875



20/28

Table 6. Full Rankings of Players.

Ranking Cross-Efficiency Score ATP Ranking

Nadal 0.997 FedererDjokovic 0.990 NadalFederer 0.987 DjokovicMurray 0.986 MurrayVerdasco 0.980 del PotroDavydenko 0.974 DavydenkoGonzalez 0.967 Roddick Soderling 0.964 SoderlingRoddick 0.960 Verdasco

del Potro 0.957 TsongaTsonga 0.955 GonzalezKohlschreiber 0.946 Stepanek Cillic 0.945 MonfilsMonfils 0.943 CillicSimon 0.938 SimonKarlovic 0.936 RobredoRobredo 0.936 FerrerHaas 0.935 HaasWawrinka 0.933 YouzhnyFerrer 0.929 BerdychStepanek 0.926 WawrinkaHewitt 0.924 HewittFerrero 0.923 FerreroYouzhny 0.920 LjubicicLjubicic 0.919 QuerreyIsner 0.918 AlmagroQuerrey 0.917 KohlschreiberHanescu 0.914 MelzerBerdych 0.911 TroickiTipsarevic 0.907 Montantes

Almagro 0.907 ChardyMelzer 0.905 MathieuBeck 0.903 IsnerMontantes 0.902 AndreevTroicki 0.902 KarlovicBennetau 0.900 TipsarevicAndreev 0.900 Beck Blake 0.899 Garcia-LopezGarcia-Lopez 0.898 BlakeLopez 0.898 Bennetau

Gicquel 0.896 LopezChardy 0.895 HanescuAcasuso 0.893 Seppi

(continued)




21/28

particular, the DEA analysis of relative efficiency provides very useful information

regarding the strengths and weaknesses of each player, which may help them to

improve their game toward the level of the best players and it could also be used

to analyze the game of the partners in the preparation of the matches.

As said before, the DEA analysis of efficiency has revealed 11 efficient players

out of the sample of 53, which shows that DEA has made a selective classification of

players. The different profiles of weights recorded in Table 2 show that these effi-cient players achieve the efficiency with different patterns of game.

On one hand, we have players like Nadal, Murray, Djokovic, or Federer who have

been assessed with profiles of weights without an extreme disequilibrium, and this

shows a good performance of these players in the different aspects of the game. It is

particularly noticeable the case of Nadal, where the valuejdequals 1 indicates that

in his assessment the contributions to the efficiency of the different game factors

coincide, and this means that his game can be rated as efficient even with a profile

of weights that attaches the same importance to all these variables. It can be con-

cluded, therefore, that Nadal has a very good performance in all the aspects of thegame. In contrast, the valuejd0:45 of Murray reflects that he needs to put somemore weight in some of the game factors in order to be rated as efficient. What Table

2 shows is that Murray has exploited to some extent his relative strength in

ptos1servw%, gamesservw%, and ptosret1servw% in the achievement of the effi-

ciency. We have a similar situation with Djokovic and Federer. We can also see

in this table that these players have played an important role as benchmarks for the

remaining players. To be specific, we point out again the case of Nadal, who has

been a referent in the assessment of 29 of the inefficient players. Djokovic and Fed-

erer, and also Murray, have frequently acted as referents too, 22, 20, and 15 times,respectively. On the other hand, we have players who have needed to unbalance very

much the importance attached to the different aspects of the game in order to be rated

as efficient. In some cases, like those of Isner (jd0:03) and Karlovic (jd0:07),

the need for such differences in the weights used can be explained by a higher degree

Table 6 (continued)

Ranking Cross-Efficiency Score ATP Ranking

Gulbis 0.888 AcasusoSerra 0.884 FogniniMathieu 0.882 GicquelSeppi 0.876 SerraVassllo Arguello 0.875 HernandezHernandez 0.873 SchuettlerGranollers 0.871 RochusSchuettler 0.869 GulbisFognini 0.845 GranollersRochus 0.833 Vassllo Arguello

Ruiz et al. 295



22/28

of specialization of their game. These two players have a poor performance in the

game factors concerned with return, and this is why they put the weight on the vari-

ables regarding service. But that is not the case of other players like Soderling,Gonzalez, or Davydenko, who also use a very unbalanced profile of weights in

their assessments (the values of jd for them are, respectively, 0.02, 0.07, and

0.06). These players have, in general, a relatively good performance in all the

aspects of the game but, as a consequence of the relative nature of the DEA, with

a more balanced profile of weights than that used they would be rated worse than

other players like Nadal or Federer, that is, they would not be rated as efficient.

Therefore, they manage to achieve the efficiency in their game by exploiting to a

large extent their relative strengths in some aspects of the game: in ptos1serw%in

the case of Soderling, whose virtual 55.73% is very large virtual as compared tothose of the other players (this player is also outstanding in ptosbreakconv%, with

a virtual of 22.52%); in ptosbreaksav% in Gonzalez, with a large virtual of

61.78%; and in ptosret1serw%in Davydenko, whose virtual is 54.54%. This shows

how the DEA methodology allows us to identify the strengths of the game of the

players that make them perform efficiently.

As for the inefficient players, Tables 3 and 4 provide useful information for

benchmarking purposes. To be specific, for each of them, we have efficient targets

in each dimension of the game and also the efficient players that have acted as refer-

ents in their assessments. The comparison between actual data and efficient targetsallows us to identify and quantify the sources of inefficiency of the game of the inef-

ficient players and, therefore, to suggest potential directions of improvement. For

example, we can see that Tsonga is very similar to his benchmark, which is a virtual

player resulting from a combination of Federer (with weightlj 0:56), Djokovic(lj 0:14), and Karlovic (l

j 0:30). His actual data are very close to the targets

provided, except perhaps in gamesretw%. Therefore, since his percentage of

improvement in this latter variable is 9.41%, we can say that with practically a raise

in gamesretw%from the actual 19%to the efficient target 20.78%the performance

of his game would be at the level of the efficient players. A similar conclusion can bedrawn for Del Potro. Haas is also a player that seems not to have important weak-

nesses in his game.

In contrast, other players have to improve in different aspects of their game and/or

have a deep weakness in one or several game factors. For example, the percentage of

improvement in gamesretw%for Ljubicic is quite large (34.51%) and he should also

improve ptosbreakconv% (12.18%) and ptos2serw% (10.36%). These percentages

of improvement are the result of comparing Ljubicic with a virtual player in which

Federer plays a very important role (theljcorresponding to Federer in that bench-

mark is 0.76). Table 3 provides a similar portrayal of Stepanek. The comparison ofHewitt with a virtual player where Djokovic has a very large weight (his lj is 0.87)

reveals that he should improve mainly in 1serve% (20.07%), and also in

ptosbreakconv% (9.32%) and gamesretw% (12.08%). Simon should also improve




23/28

in 1serve% (17.33%). Finally, Ferrer, who compares himself with a virtual player

mainly determined by Nadal (his lj in the benchmark is 0.92), should make some

improvement in some game factors like 1serve%(10.14%), ptos2serw%(9.15%),gamesservw% (9.20%), or ptosbreakconv% (9.11%) in order to become an effi-

cient player.

The benchmarking analysis has detected in gamesretw% the main weakness of

the game of many inefficient players, as shown by their percentage of improvement

in this aspect of the game: see Ljubicic (34.51%), Stepanek (28.99%), Berdych

(25.68%), Robredo (23.96%), and Youznhy (22.61%). For some of them, like

Ljubicic, Stepanek, and Berdych, the DEA might be providing an explanation of this

(at least partly) in their poor performance in ptobreakconv%, as it reveals the corre-

sponding percentage of improvement, 12.18%, 10.49%, and 13.63%, respectively. Butthis is not the case of, for example, Robredo, who is good in break points converted.

Since Robredo does not have any remarkable weakness in variables concerned with

return, this player should investigate the way to improve this aspect of the game.

We finally note that in Table 3 we also have players, like Cillic and Wawrinka, with

a relative poor performance in ptobreakconv%, but in those cases that seems not to

affect gamesretw%so seriously as with the players previously mentioned.

The DEA benchmarking analysis has been complemented with a cross-efficiency

evaluation, which provides a peer evaluation of the players that makes it possible to

rank them. In the row of Nadal in Table 5 we can see that all the cross-efficienciesequal 1, except those in the columns associated with Soderling and Karlovic (where

we also find a large efficiency evaluation). This means that Nadal is rated as efficient

with the profiles of weights of almost all the efficient players, or in other words, that

the game of this player is assessed with the maximum efficiency (1) with a wide vari-

ety of patterns of game. This is why he eventually ranks first: his cross-efficiency

score 0.997 is the largest in the last column of this table. Djokovic, Federer, and

Murray, who rank second, third, and fourth, respectively, according to their cross-

efficiency scores, have large cross-efficiencies and, in particular, are rated as effi-

cient with the weights of some of the other efficient players, aside from with theirown self-evaluation. To be specific, Federer is assessed as efficient with the

weights of Djokovic, Soderling, Gonzalez, Isner, and Karlovic, aside from with his

own weights. Eventually, his average of cross-efficiencies is 0.987, which makes

him rank second; Djokovic is rated as efficient with his weights and with those of

Federer and Soderling; and Murray with his and with those of Federer, Djokovic,

Davydenko, and Soderling. These players are followed in the ranking by Verdasco,

Davydenko, Gonzalez, Soderling, and Roddick. After these efficient players, the

ranking follows with Del Potro, Tsonga, Kohlschreiber, Cilic, Monfils, Simon,

Karlovic, Robredo, Haas, Wawrinka . . . and this ranking could be seen as consis-tent with expert professional tennis opinion.

The full ranking of players provided by the cross-efficiency evaluation is

recorded in Table 6. In particular, the cross-efficiency evaluation has made it

Ruiz et al. 297



24/28

possible to discriminate between the players that have been initially rated as efficient

in the DEA self-evaluation. We can see that the efficient players are all in the top

positions of the ranking, except Isner and Karlovic. If we have a look at their rowsin Table 5, we realize that in the peer evaluation these two players are rated as

efficient only when they are assessed with their own profile of weights and that the

efficient players give them in general lower ratings, especially Nadal and Murray

(and also Davydenko). Eventually, their cross-efficiency scores are 0.918 and

0.936, respectively, which are lower than those of some DEA inefficient players like

Del Potro, Tsonga, Kohlschreiber, Cillic, Monfils, and Simon. A possible explana-

tion for this can be the previously mentioned higher degree of specialization of the

game of these players, which are penalized when they are assessed with more

balanced profiles of weights like those of Nadal and Murray (Nadal gives a cross-efficiency of 0.841 to Isner and 0.848 to Karlovic and the cross-efficiencies provided

by Murray for these two players are, 0.866 and 0.883, respectively).

As could be expected, we can see differences between the ranking provided by

the cross-efficiency evaluation and that of the ATP. This is mainly because the anal-

ysis in the present article is based on the variables that describe the game of the play-

ers while the ranking ATP is based on the points they get in the tournaments in which

they participate during the season. That is, the ATP ranking is concerned with the

competitive performance of the players while that provided in this article is con-

cerned with the efficiency performance of their game. Nevertheless, we do not thinkthe pictures that both ranking provide are too different (in fact, the Spearman rank

correlation is .933, with a bilateral significance equals 0). We rather believe that both

analyses can complement each other with some information of interest. Among the

differences detected, we notice the fact that Nadal and Djokovic would outperform

Federer with the data of their game during the 2009 season. Verdasco and Gonzalez

would gain both four positions with respect to the ATP ranking (perhaps, they do not

exploit sufficiently in competition the good performance of their game), while Del

Potro would lose five (which may be showing that he is a strong competitor).

Conclusions

The purpose of this article has been to show the possibilities of the basic DEA as

methodology for the assessment of professional tennis players from the perspective

of the efficiency of their game. DEA provides an analysis of relative efficiency

where the performance of the game of the players is evaluated with respect to that

of the others. This allows to identifying the strengths and weaknesses of the game

of the players and, in particular, suggesting possible directions of improvement.We have also used the cross-efficiency evaluation, which is an extension of DEA

that has made it possible to rank the players. The analysis with the classical DEA

models has provided useful information, but this study also raises a number of ques-

tions for interesting future research. As we explain next, some of these issues we




25/28

raise here can in fact be addressed with some of the extensions and enhancements of

the basic DEA methodology that have been proposed in the literature. For example,

concerning the model we have used in this article, it would be of interest to analyzethe role that the different game factors considered have played in the evaluation of

players. To do it, we can use the approach in Pastor, Ruiz, and Sirvent (2002), and in

particular the backward and forward procedures proposed there in order to eliminate

irrelevant variables (if any) and incorporate others (if data for them available) whose

consideration may help to have a better description of the game. Another issue of inter-

est is that regarding the relative importance of the factors of the game. In the analysis

here the game factors considered have been given the same treatment, without any pre-

ferences of some of them over the others. However, the experts may think, for instance,

that the percentage of service games won is a factor that should be considered as havingmore importance than the percentage 1st serve. If this type of information is available

from experts, it can be incorporated into the analysis using the so-called assurance

region AR models (Thompson, Singleton, Thrall, & Smith, 1986), which are models

that result from adding to (3) weights restrictions that reflect preferences regarding the

relative importance of the aspects of the game involved. Thus, the conclusions that can

be drawn from the analysis would be more consistent with the accepted views and

beliefs of experts regarding tennis game. Besides, we can also carry out a similar anal-

ysis to that we have made here by only using the data of some specific tournaments, in

particular those played in a specific surface, or only using the variables regarding eitherservice or return. All done with the purpose of having a precise view of the game of

tennis players, which allows identifying the keys for improvement and also having

a thorough knowledge of the partners in the preparation of the matches.

Declaration of Conflicting InterestsThe authors declared no potential conflicts of interest with respect to the research,

authorship, and/or publication of this article.

FundingThe authors disclosed receipt of the financial support for the research in this articleby Ministerio de Ciencia e Innovacion (MTM2009-10479) and Generalitat Valenciana

(ACOMP/2011/115).

Note

1. As a consequence, although we use the term efficiency throughout the article, we are

actually concerned with effectiveness (see Prieto & Zofio, 2001 for discussions).

ReferencesAlp, I. (2006). Performance evaluation of goalkeepers of the world cup. G.U. Journal of

Science, 19, 119-125.

Anderson, T. R., & Sharp, G. P. (1997). A new measure of football batters using DEA.Annals

of Operations Research, 73, 141-155.

Ruiz et al. 299



26/28

Banker, R. D., Charnes, A., & Cooper, W. W. (1984). Some models for estimating technical and

scale inefficiencies in data envelopment analysis.Management Science, 30, 1078-1092.

Barros, C. P., Assaf, A., & Sa-Earp, F. (2010). Brazilian football league technical efficiency:A Simar and Wilson approach. Journal of Sports Economics,11, 641-651.

Bosca, J. E., Liern, V., Martnez, A., & Sala, R. (2009). Increasing offensive or defensive

efficiency? An analysis of Italian and Spanish football.Omega, 37, 63-78.

Charnes, A., & Cooper, W. W. (1962). Programming with linear fractional functionals. Naval

Research Logistics Quarterly, 9, 181-186.

Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision

making units. European Journal of Operations Research, 2, 429-444.

Chen, W.-C., & Johnson, A. L. (2010). The dynamics of performance space of major league

baseball pitchers 1871-2006.Annals of Operations Research, 181, 287-302.Cooper, W.W., Ramon, N., Ruiz, J.L., & Sirvent, I. (2011). Avoiding Large Differences in

Weights in Cross-Efficiency Evaluations: Application to the Ranking of Basketball Play-

ers. Journal of Centrum Cathedra, 4, 197-215.

Cooper, W. W., Ruiz, J. L., & Sirvent, I. (2009). Selecting non-zero weights to evaluate

effectiveness of basketball players with DEA.European Journal of Operational Research,

195, 563-574.

Dawson, P., Dobson, S., & Gerrard, B. (2000). Estimating coaching efficiency in professional

team sports: Evidence from English association football. Scottish Journal of Political

Economy, 47, 399-419.Doyle, J. R., & Green, R. H. (1994). Efficiency and cross-efficiency in DEA: Derivations,

meanings and uses. Journal of the Operational Research Society,45, 567-578.

Einolf, K. W. (2004). Is winning everything? A data envelopment analysis of major league

baseball and the national football league.Journal of Sports Economics, 5, 127-151.

Espitia-Escuer, M., & Garca-Cebrian, L. I. (2004). Measuring the efficiency of Spanish first-

division soccer teams. Journal of Sports Economics,5, 329-346.

Fernandez, J., Mendez-Villanueva, A., & Pluim, B. M. (2006). Intensity of tennis match play.

British Journal of Sports Medicine, 40, 387-391.

Fizel, J. L., & DItri, M. P. (1999). Firing and hiring of managers: Does efficiency matter?Journal of Management, 25, 567-585.

Fried, H. O., Lambrinos, J., & Tyner, J. (2004). Evaluating the performance of professional

golfers on the PGA, LPGA and SPGA tours. European Journal of Operational Research,

154, 548-561.

Fried, H. O., & Tauer, L. W. (2011). The impact of age on the ability to perform under

pressure: Golfers on the PGA tour.Journal of Productivity Analysis, 35, 75-84.

Gonzalez-Gomez, F., & Picazo-Tadeo, A. J. (2010). Canwe be satisfied with our football team?

Evidence from Spanish professional football.Journal of Sports Economics,11, 418-442.

Haas, D. J. (2003). Technical efficiency in the major league soccer. Journal of Sports

Economics,4, 203-215.

Haas, D. J., Kocher, M. G., & Sutter, M. (2004). Measuring efficiency of German football

teams by data envelopment analysis. Central European Journal of Operations Research,

12, 251-268.




27/28

Klaassen, F. J. G. M., & Magnus, J. R. (2009). The efficiency of top agents: An analysis

through service strategy in tennis. Journal of Econometrics, 148, 72-85.

Liang, L., Wu, J., Cook, W. D., & Zhu, J. (2008). Alternative secondary goals in DEAcross-efficiency evaluation.International Journal of Production Economics, 113, 1025-1030.

Lovell, C. A. K., & Pastor, J. T. (1995). Units invariant and translation invariant DEA models.

Operations Research Letters, 18, 147-151.

Lovell, C. A. K., & Pastor, J. T. (1999). Radial DEA models without inputs or without outputs.

European Journal of Operational Research, 118, 46-51.

Lozano, S., Villa, G., Guerrero, F., & Cortes, P. (2002). Measuring the performance of nations

at the summer Olympics using data envelopment analysis. Journal of the Operational

Research Society, 53, 501-511.

ODonoghue, P., & Ingram, B. (2001). A national analysis of elite tennis strategy. Journal ofSport Science, 19, 107-115.

Pastor, J. T., Ruiz, J. L., & Sirvent, I. (2002). A statistical test for nested radial DEA models.

Operations Research,50, 728-735.

Prieto, A. M., & Zofio, J. L. (2001). Evaluating effectiveness in public provision of infrastruc-

ture and equipment: The case of Spanish municipalities.Journal of Productivity Analysis,

15, 41-58.

Ramon, N., Ruiz, J. L., & Sirvent, I. (2010a). A multiplier bound approach to assess rela-

tive efficiency in DEA without slacks.European Journal of Operational Research,203,

261-269.Ramon, N., Ruiz, J. L., & Sirvent, I. (2010b). On the choice of weights profiles in cross-

efficiency evaluations. European Journal of Operational Research, 207, 1564-1572.

Ramon, N., Ruiz, J. L., & Sirvent, I. (2011). Reducing differences between profiles of

weights: A peer-restricted cross-efficiency evaluation. Omega, 39, 634-641.

Santos-Rosa, F. J., Garca, T., Jimenez, R., Moya, M., & Cervello, M (2007). Prediccion de la

satisfaccion con el rendimiento deportivo en jugadores de tenis: Efecto de las claves situa-

cionales.Motricidad: European Journal of Human Movement, 18, 41-60.

Sexton, T. R., & Lewis, H. F. (2003). Two-stage DEA: An application to major league base-

ball.Journal of Productivity Analysis, 19, 227-249.Sexton, T. R., Silkman, R. H., & Hogan, A. J. (1986). Data envelopment analysis: Critique

and extensions. In R. H. Silkman (Ed.),Measuring efficiency: An assessment of data envel-

opment analysis (pp. 73-105). San Francisco, CA: Jossey-Bass.

Soares, J. C., Angulo-Meza, L., & Branco Da Silva, B. P. (2009). A ranking for the Olympic

games with unitary input DEA models. IMA Journal Management Mathematics, 20,

201-211.

Sueyoshi, T., Ohnishi, K., & Kinase, Y. (1999). A benchmark approach for baseball evalua-

tion.European Journal of Operational Research,115, 429-448.

Thompson, R. G., Singleton, F. D., Thrall, R. M., & Smith, B. A. (1986). Comparative site

evaluations for locating a high-energy physics lab in Texas. Interfaces, 16, 35-49.

Ueda, T., & Amatatsu, H. (2009). Determination of bounds in DEA assurance region

methodIts application to evaluation of baseball players and chemical companies. Jour-

nal of the Operations Research Society of Japan, 4, 453-467.

Ruiz et al. 301



28/28

Volz, B. (2009). Minority status and managerial survival in major league baseball. Journal of

Sports Economics, 10, 522-542.

Wu, J., Zhou, Z., & Liang, L. (2010). Measuring the performance of nations at BeijingSummer Olympics using integer-valued DEA model. Journal of Sports Economics, 11,

549-566.

Zhang, D., Li, X., Meng, W., & Liu, W. (2009). Measuring the performance of nations at the

Olympic games using DEA models with different preferences. Journal of the Operational

Research Society, 60, 983-990.

Author Biographies

JoseL. Ruizis an Associate Professor of Statistics and Operations Research at the University

Miguel Hernandez in Spain. He received an MBA in Mathematical Sciences from the Univer-sity of Granada and a Ph.D. from the University Miguel Hernendez. His primary research

interest is the analysis of efficiency and productivity. He has published his research in differ-

ent journals of Operations Research and Management Science.

Diego Pastoris an Assistant Professor of Sports Sciences at the University Miguel Hernandez

in Spain, where he earned his Ph.D. He deve

Journal of Sports Economics 2013 Ruiz 276 302

Documents

Transcript of Journal of Sports Economics 2013 Ruiz 276 302