Indices Development for Player’s Performance Evaluation through the Super-SBM Approach in Each Department for All Three Formats of Cricket

Abstract

Player performance evaluations in all three formats of cricket have been a topic of great concern for sports analysts and research experts. This study proposed a comprehensive performance estimation tool that incorporates all the essential inputs–outputs and evaluates a cricketer’s overall performance. This research introduced three different estimation indices for player efficiency in all three formats of cricket for batting, bowling, and fielding. Further, this research employed the DEA Super-SBM model to evaluate the player’s efficiency in batting, bowling, and fielding departments of all three formats. The study estimates the most efficient batsman, bowler, and fielder in cricketing history by using the data of international cricketers (1877–2019). The results indicate that, compared to the traditional parameters, the proposed study indices are more accurate and comprehensive in nature. The most efficient batsman, bowler, and fielder in all three formats are given, respectively: (i) Sir Bradman, Sachin Tendulkar, and Virat Kohli; (ii) Muralitharan, Mitchell Starc, and Umar Gul; and (iii) Saleem Yousuf, Luke Ronchi, and Scott Edwards. For teams, England, Australia, and India were determined to be the most efficient in batting for all three formats; the West Indies, Australia, and Pakistan are the most efficient in bowling; and the Australian (Test & ODIs) and South African teams are efficient in the fielding department.

1. Introduction

Cricket has been a worldwide phenomenon since the late 19th century. Cricket with a bat and ball was first played internationally in 1844, but it wasn’t until 1877 that the format known as “Test cricket” was established. However, shorter versions of the game were developed much later. The rise of digitalization in the late 20th and early 21st centuries has been a boon to cricket’s market value from a revenue and profitability perspective. Cricket as a game has three formats. Australia and England played in the first official Test match in March 1877. Australia reportedly hosted the first 50-over One Day International (ODI) against England in 1971. T20s, on the other hand, were founded in June of 2003, not long after the millennium change. Rediff reports that T20s represent 92% of the global fans’ interest, while 88% of fans follow ODIs. Conversely, only 70% of spectators were interested in watching a Test match [1].

In terms of economic importance, the sport’s growth is a driving force behind the current sports economy. The International Cricket Council (ICC) counts over a hundred member countries with national cricket teams with great potential to boost the sports economy. Many international leagues have gained popularity since the introduction of competitive 20-over cricket. There was a total estimated value of GBP 4.6 billion for the Indian Premier League in 2018 [2]. According to The Cricket Monthly, the Big Bash made about GBP 1.262 billion thanks to a GBP 732.5-million television deal and GBP 530 million earned by the teams. In addition, the 2019 Caribbean Premier League injected GBP 93 million into the local economy, and the England and Wales Cricket Board’s broadcasting revenues peaked at GBP 7 million. According to Pro Pakistani, the Pakistan Super League, the last of the major leagues, brought in GBP 21.7 million in revenue last year. Adding up everything mentioned above yields a total of GBP 5.983 billion. However, that is only true for the top T20 leagues [3].

The performance evaluation of teams and players is vital for cricket development in the earning and game enhancement perspectives. The performance of any cricket team is usually measured through the percentage of winning against counterparts in international games in all three formats of cricket [4]. Similarly, the performance of any player is gauged through batting averages, as well as the 100s and 50s he scored. In the bowling department, the wickets a bowler took or his bowling average throughout his career gauges his performance. In the fielding department, the more catches, stumps, and run-outs he takes will summarize his performance [5]. Usually, the ICC issues a ranking of players on a monthly, annual, and career basis. This is gauged through their averages and wicket, or the scores they scored in a specific period or during their whole career [6]. However, studies have proved that a single indicator for performance evaluation is not trustworthy and could create bias in the results. To tackle this issue, few studies applied different techniques to estimate the overall efficiency of players in different games [7,8,9,10,11,12,13,14]. DEA is a well-known linear programming technique used to measure efficiency globally in different sectors and industries. DEA incorporates multiple input–outputs to measure the relative efficiency of DMUs (decision-making units) [15].

The literature suggests that very few studies applied DEA to measure the efficiency of players in different games [16,17,18,19]. Some researchers tried to apply the DEA to measure the cricket player’s efficiency for a specific period, or a sports event like the World Cup or the bilateral series [20,21,22,23,24,25,26,27,28]. However, a comprehensive study that could gauge the player’s career efficiency in all three formats of cricket is missing. To this end, this study uses the multiple inputs–outputs and employs DEA Super-SBM to measure the player’s efficiency in all three formats of cricket. As in the conventional DEA model, the efficient DMUs (in our case, a cricket player) cannot be ranked as they all score unity, which makes it difficult for decision-makers to choose the most efficient DMUs. In Super-SBM, the efficiency score could cross 1. Therefore, it ranks the efficient players in all three formats of cricket.

Moreover, this research presents a comprehensive index for player performance evaluation in cricket, which is beneficial for the ICC and other cricketing leagues to estimate player efficiency. This study proposed a comprehensive performance estimation tool that incorporates all the essential inputs–outputs and evaluates a cricketer’s overall performance. This research introduced three different estimation indices for player efficiency in all three formats of cricket for batting, bowling, and fielding. The study compares the performance of all the international players in three formats of cricket from 1878–2019 and ranks the top performers in all formats. It also ranks the countries on the player’s performance and differentiates the cricket-playing nations on the efficiency scores basis. The rest of the study is organized as follows: Section 2 describes the detailed methodology used in the research. Section 3 presents the input–output selection and data sources. The results and discussion and the conclusion are presented in Section 4 and Section 5, respectively.

2. Materials and Methods

The radial DEA model based on CCR cannot fully account for the effect of laziness on productivity. Tone [29] proposed the SBM and super-efficiency SBM models [30,31]. SBM is a non-radial method for evaluating efficiency when input and output vary in a non-proportional manner [32]. Combining the super-efficiency and SBM models, the super-efficiency SBM model is a modelling approach. The fundamental concept of the super-efficiency evaluation method is to remove the effective evaluation unit from the set and evaluate it. Consequently, the original non-effective value evaluation remains unchanged, and the original effective value evaluation can be greater than 1, after which they can be compared [33,34,35,36].

The SBM model can solve input excess and output deficiency promptly. In the SBM model, the data unit is constant, and each input and output slack variable may be increased evenly to compensate for the deficiencies of other models. The key advantage of the SBM model over other models is that it evaluates the efficiency of the less efficient DMU with precision.

Assume that the η is the number of DMUs (decision-making units). Decision-making units are made up of input and expected output. Three vectors $x\in R^M,y^g\in R^{S1},y^b\in R^{S1}$ indicate the expected output of $S_1$ with $m$ units of input. Assuming $X>\mathbf{0},Y^g>$ $0,Y^b>0$, the production possibility set is defined as: $X=\left[{\mathit{x}}_{\mathbf{1}},{\mathit{x}}_{\mathbf{2}},\dots ,{\mathit{x}}_{\mathbf{N}}\right]\mathit{\in }{\mathbf{R}}^{\mathit{N}\mathit{\times }\mathit{M}}$, and the output matrix is expressed as ${\mathbf{Y}}^g=\left[{\mathit{y}}^{g1},{\mathit{y}}^{g2},\dots ,{\mathit{y}}^{g\mathit{N}}\right]\in R^{S1\times N}$.

$$\mathit{P}\mathit{=}\left\{\left(\mathit{x},{\mathit{y}}^g,{\mathit{y}}^b\right)\left|,\mathit{x}\mathit{\ge }\mathit{X}\mathit{\eta }\right|,{\mathit{y}}^g\mathit{\le }\mathit{Y}\mathit{\eta }\left|,{\mathit{y}}^{\mathit{b}}\mathit{\ge }\mathit{Y}\mathit{\eta }\right|\mathit{,}\mathit{\eta }\mathit{\ge }\mathit{0}\right\}$$

(1)

The actual expected output in Equation (1) is less than the optimal expected output on the frontier. Tone’s SBM model takes into account slack in the assessment DMU DMU ($x_0{y}^g{ }_0,y^b{ }_{\mathbf{0}})$ by considering the production possibility set.

$$\gamma \mathit{=}min\left(\frac{1\mathit{-}\frac{\mathit{1}}{M}{{{\displaystyle \sum }}^{\mathit{\text{}}}}_{\mathit{i}\mathit{=}\mathbf{1}}^{\mathit{M}}\frac{{\mathit{S}}_{\mathit{i}}^{\mathit{-}}}{{\mathit{x}}_{\mathit{i}\mathit{o}}}}{\mathbf{1}\mathit{+}\frac{\mathit{1}}{\mathit{S}\mathit{1}\mathit{+}\mathit{S}\mathit{2}}({{{\displaystyle \mathit{\sum }}}^{\mathit{\text{}}}}_{\mathit{r}\mathit{=}\mathit{1}}^{\mathit{S}\mathbf{1}}\frac{{\mathit{S}}_{\mathit{r}}^{\mathit{g}}}{{\mathit{y}}_{\mathit{r}\mathbf{0}}^{\mathit{g}}}+{{{\displaystyle \sum }}^{\mathit{\text{}}}}_{\mathit{r}\mathit{=}\mathit{1}}^{\mathit{S}\mathit{2}}\frac{{\mathit{S}}_{\mathit{r}}^{\mathit{b}}}{{\mathit{y}}_{\mathit{r}\mathit{0}}^{\mathit{b}}}}\right))$$

(2)

$$\mathrm{s}.\mathrm{t}.\{\begin{array}{c}{\mathit{x}}_{\mathit{0}}\mathit{=}\mathit{X}\eta +{\mathit{S}}^{\mathit{-}}\\ {\mathit{y}}_{\mathit{0}}^{\mathit{g}}\mathit{=}{\mathit{Y}}^{\mathit{g}}\mathit{\eta }\mathit{-}{\mathit{S}}^{\mathit{g}}\\ {\mathit{y}}_{\mathit{0}}^{\mathit{b}}\mathit{=}{\mathit{Y}}^{\mathit{b}}\mathit{\eta }+{\mathit{S}}^{\mathit{b}}\\ {\mathit{S}}^{\mathit{-}}\mathit{\ge }\mathit{0}\mathit{,}{S}^g\mathit{\ge }\mathit{0},{\mathit{S}}^{\mathit{b}}\mathit{\ge }\mathit{0},\mathit{\eta }\mathit{\ge }\mathit{0}\end{array}$$

In Equation (2), it denotes the DMU’s efficiency, and the change value ranges from 0 to 1. The symbols $(S^{-}{S}^{-}\mathbf{,}{S}^g,S^b$) denote input, output, and slack, respectively. The DMU is at the vanguard of production only when the technical efficiency $\mathit{\gamma }$ is 1 and $S^{\mathit{-}},S^g,S^b$ are all 0; if $\mathit{\gamma }<1$, the DMU efficiency is inefficient. As shown below, the nonlinear Equation (2) can be transformed into a linear model using the Charnes–Cooper transformation.

$$\mathit{\kappa }\mathit{=}\mathbf{m}\mathit{ }\left(\mathit{T}\mathit{-}\frac{\mathit{1}}{\mathit{M}}{{\displaystyle \mathbf{\sum }}}_{\mathit{i}\mathit{=}\mathit{1}}^{\mathit{M}}\frac{{\mathit{S}}_{\mathit{i}}^{\mathit{-}}}{{\mathit{x}}_{\mathit{i}\mathit{o}}}\right)$$

(3)

$$\{\begin{array}{c}\mathit{1}\mathit{=}\mathit{T}\mathit{+}\frac{\mathit{1}}{\mathit{S}\mathit{1}\mathit{+}\mathit{S}\mathit{2}}\left({\displaystyle \sum }_{\mathit{r}=\mathit{1}}^{\mathit{S}\mathit{1}}\frac{{\mathit{S}}_{\mathit{r}}^{\mathit{g}}}{{\mathit{y}}_{\mathit{r}\mathit{o}}^{\mathit{g}}}+{\displaystyle \sum }_{\mathit{r}\mathit{=}\mathit{1}}^{\mathit{S}\mathit{2}}\frac{{\mathit{S}}_{\mathit{r}}^{\mathit{b}}}{{\mathit{y}}_{\mathit{r}\mathit{o}}^{\mathit{b}}}\right)\\ \mathit{x}\mathit{0}\mathit{T}\mathit{=}\mathit{X}\mathit{\beta }\mathit{+}{\mathit{S}}^{\mathit{-}}\\ {\mathit{y}}_{\mathit{0}}^{\mathit{g}}\mathit{T}\mathit{=}{\mathit{Y}}^{\mathit{g}}\mathit{\beta }\mathit{-}{\mathit{S}}^{\mathit{g}}\\ {\mathit{y}}_{\mathit{0}}^{\mathit{b}}\mathit{T}\mathit{=}{\mathit{Y}}^{\mathit{b}}\mathit{\beta }\mathit{+}{\mathit{S}}^{\mathit{b}}\\ {\mathit{S}}^{\mathit{-}}\mathit{\ge }\mathit{0},{\mathit{S}}^{\mathit{g}}\mathit{\ge }\mathit{0},{\mathit{S}}^{\mathit{b}}\mathit{\ge }\mathit{0},\mathit{\beta }\mathit{\ge }\mathit{0},\mathit{T}\mathit{\ge }\mathit{0}\end{array}$$

However, there are instances where certain decision-making units are also efficient in gauging the technological efficacy of alternatives. The super-efficiency SBM model (Super SBM model) was developed by expanding upon existing work in order to produce a fair approach of efficiency measurement.

$${\mathit{\gamma }}^{\mathit{\ast }}\mathit{=}\mathbf{m} \left[\frac{\frac{\mathit{1}}{\mathit{M}}{{\displaystyle \sum }}_{\mathit{i}=\mathit{1}}^{\mathit{M}}\frac{\overline{\mathit{x}}\mathit{i}}{\frac{\mathit{x}}{\mathit{0}}\mathit{0}}}{\frac{\mathit{1}}{\mathit{S}\mathit{1}\mathit{+}\mathit{S}\mathit{2}}\left({{\displaystyle \mathit{\sum }}}_{\mathit{r}\mathit{=}\mathit{1}}^{\mathit{S}\mathit{1}}\frac{{\overline{\mathit{y}}}_{\mathit{r}}^{\mathit{s}}}{{\mathit{y}}_{\mathit{r}\mathit{o}}^{\mathit{g}}}\mathit{+}{{\displaystyle \sum }}_{\mathit{r}=\mathit{1}}^{\mathit{S}\mathit{2}}\frac{{\overrightarrow{\mathit{y}}}_{\mathit{r}}^{\mathit{b}}}{{\mathit{y}}_{\mathit{r}\mathit{o}}^{\mathit{b}}}\right)}\right]$$

(4)

$$\{\begin{array}{c}\overline{\mathit{x}}\mathit{\ge }{\displaystyle \sum }_{\mathit{j}=\mathit{1},\mathit{\ne }\mathit{0}}^N{\mathit{\eta }}_{\mathit{j}}{\mathit{x}}_{\mathit{j}}\\ {\overline{\mathit{y}}}^{\mathit{g}}\mathit{\le }{\displaystyle \sum }_{\mathit{j}=\mathit{1},\mathit{\ne }\mathit{0}}^{\mathit{N}}{\mathit{\eta }}_{\mathit{j}}{\mathit{y}}^{\mathit{g}}{ }_{\mathit{j}}\\ {\overrightarrow{\mathit{y}}}^{\mathit{b}}\mathit{\ge }{\displaystyle \sum }_{\mathit{j}=\mathit{1},\mathit{\ne }\mathit{0}}^N{\mathit{\eta }}_{\mathit{j}}{\mathit{y}}^{\mathit{b}}{ }_{\mathit{j}}\\ \overline{\mathit{x}}\mathit{\ge }\mathit{x}\mathit{0},{\overline{\mathit{y}}}^{\mathit{g}}\mathit{\le }{\mathit{y}}_{\mathit{0}}^{\mathit{g}},\overrightarrow{{\mathit{y}}^{\mathit{b}}}\mathit{\ge }{\mathit{y}}_{\mathit{0}}^{\mathit{b}},{\overline{\mathit{y}}}^{\mathit{g}}\mathit{\ge }\mathit{0}\mathit{,}\mathit{\eta }\mathit{\ge }\mathit{0},\end{array}$$

(5)

Super-efficiency of DMU is denoted by ${\mathit{\gamma }}^{\ast }$ in Equation (4), and it can be more than 1.

3. Inputs–Outputs Selection and Data Sources

The input–output selection in DEA efficiency evaluation is a topic of great concern because it impacts the final efficiency scores of each DMU, which could create bias in the estimation process. The studies employed several input–output bundles to evaluate the players’ efficiency.

Table 1. Inputs–outputs used for player’s efficiency.

Inputs–Outputs Used to Evaluate the Batsman’s Efficiency.
Test Cricket		ODI’s		T20’s
Inputs	Outputs	Inputs	Outputs	Inputs	Outputs
Inns Played	Total Runs Scored Batting Ave 100s Scored 50s Scored	Inns played Balls faced	Total Runs Scored Batting Ave Strike rate 100s Scored 50s Scored	Inns played Balls faced	Total Runs ScoredBatting Ave Strike rate 50s Scored 6s hit4s hit
Inputs–outputs Used to Evaluate the Bowler’s Efficiency.
Test Cricket		ODIs		T20s
Inputs	Outputs	Inputs	Outputs	Inputs	Outputs
Inns Played Balls Bowled Runs given	Wickets taken Ave.t Econ.t SR.t 5 wickets 10 wickets	Inns Played Balls Bowled Runs given	Wickets taken Ave.t Econ.t SR.t 4 wickets 5 wickets	Inns Played Balls Bowled Runs given	Wickets taken Ave.t Econ.t SR.t 4 wickets 5 wickets
Inputs-Outputs Used to Evaluate the Fielder’s Efficiency.
Test Cricket		ODIs		T20s
Inputs	Outputs	Inputs	Inputs	Outputs	Inputs
Inns Played	Stumps Ct Wk. Ct Fi	Inns Played	Stumps Ct Wk. Ct Fi	Inns Played	Stumps Ct Wk. Ct Fi

presents the input and output bundles taken for efficiency evaluation in each format of cricket in all three departments (batting, bowling, and field). In the test format for batsman efficiency, we used total innings (inns) played as an input, while total runs scored, batting average (Ave), 100s, and 50s scored as outputs. As in limited-over games, balls faced and strike rate matter a lot. Therefore, we included these variables in one day and T20 batting efficiency.

Similarly, in T20 games, more 6s and 4s are also considered good output. Therefore, they are included in the T20 batsman efficiency evaluation. In bowlers’ efficiency innings played, balls bowled and runs given are used as inputs while wickets taken, average transformed (Avg.t), economy transformed (Econ. t), strike Rate transformed (ST.t), and 10s and 5s wickets taken are used as outputs. The bowling department’s lower average, strike rate, and economy are considered better. Therefore, to include it as a good output, we transformed the data by dividing 1 with each value as the denominator and named ave.t, SR.t, and Econ.t. ODIs and T20s output are changed with 5 and 4 wickets taken in an inning. Finally, a fielder’s efficiency is estimated through innings played as inputs; stumps, catches taken as a wicket-keeper (Ct. wk.), and catches taken as a fielder (Ct. fi.) are considered to be outputs. The data was taken from cricket data Kaggle, the crick.info website, and the ICC website [37,38,39]. The dataset includes all the international cricket players from 1877 to 2019. However, a minimum limit was set to select the DUMs for each format and department. For Tests 2000, ODIs, 1000, and T20s, 500 scores were the minimum limit to select for the players. Therefore, 313, 388, and 129 players were selected for batting efficiency valuation. Furthermore, it consists of the players who played Test cricket, ODIs, and T20s between 1877 and 2019. Similarly, 240, 100, and 30 wickets were the minimum limit to select the DMU for bower’s efficiency in Tests, ODIs, and T20s. Fifty, 150, and 94 DMUs were selected to evaluate the bowing efficiency in all three formats, respectively. In the fielding department, 100, 100, and 30 dismissals were selected as a minimum limit to select the DMU. Therefore, 86, 50, and 50 DMUs were selected for Tests, ODIs, and T20s.

4. Results and Discussions

Cricket analysts and the ICC often compile and maintain a list of top performers at each player’s career end. It is based on absolute performance measurements like the number of runs scored by batters, the number of wickets taken by bowlers, and run-outs and catches taken by a fielder. Figure 1, Figure 2 and Figure 3 show the top 20 performers in all three departments of cricket (batting, bowling, and fielding) for Test, ODIs, and T20s (from 1877–2019). However, these performances are only output-oriented and do not show the actual depth of a player’s efficiency, as it does not count the other input indexes like balls faced or bowled, averages, economy, and the strike rates of a player throughout his career. Therefore, our study employed multiple input–output indexes (described in Table 1) to evaluate the actual Efficiency of Players [29].

4.1. Player’s Efficiency in the Batting Department

Table 2. Player’s efficiency in the batting department of cricket for all three formats.

Tests		ODIs		T20s
Player	Score	Player	Score	Player	Score
DG Bradman (AUS)	1.6948	SR Tendulkar (INDIA)	1.2004	V Kohli (INDIA)	1.5031
CJL Rogers (AUS)	1.4557	V Kohli (INDIA)	1.1411	CH Gayle (WI)	1.1843
GA Headley (WI)	1.4502	SO Hetmyer (WI)	1.1135	MDKJ Perera (SL)	1.1235
SR Tendulkar (INDIA)	1.1553	RR Rossouw (SA)	1.0776	Babar Azam (PAK)	1.1032
LG Rowe (WI)	1.1534	JC Buttler (ENG)	1.0596	AJ Finch (AUS)	1.0987
AC Hudson (SA.)	1.1471	GJ Maxwell (AUS)	1.0573	Imran Nazir (PAK.)	1.0972
Misbah-ul-Haq (PAK)	1.1184	AB de Villiers (SA)	1.042	C Munro (NZ)	1.0892
H Sutcliffe (ENG)	1.0909	AD Russell (WI)	1.042	GC Smith (SA)	1.073
DN Sardesai (INDIA)	1.0755	Shahid Afridi (PAK.)	1.0363	MJ Guptill (NZ.)	1.0411
VVS Laxman (INDIA)	1.0475	IJL Trott (ENG)	1.0303	M. Shahzad (AFG)	1.0123
IR Bell (ENG)	1.0364	JJ Roy (ENG)	1.0278	CPS Chauhan (INDIA)	1.0455
S Chanderpaul (WI)	1.0356	MS Dhoni (INDIA)	1.0200	GJ Maxwell (AUS)	1.0091
JH Kallis (ICC/SA)	1.0326	Fakhar Zaman (PAK)	1.019	KL Rahul (INDIA)	0.9924
Habibul Bashar (BDESH)	1.0312	BC Broad (ENG)	1.0185	DA Warner (AUS)	0.9721
KF Barrington (ENG)	1.025	SM Patil (INDIA)	1.0166	KP Pietersen (ENG)	0.9693
JE Root (ENG)	1.0233	Haris Sohail (PAK)	1.0162	HM Amla (SA/World)	0.962
KC Sangakkara (SL)	1.0200	JH Kallis (SA)	1.0158	SR Watson (AUS)	0.9556
KL Rahul (INDIA)	1.0107	TM Head (AUS)	1.0156	BB McCullum (NZ.)	0.9537
CL Walcott (WI)	1.0082	IVA Richards (WI)	1.008	RR Hendricks (SA.)	0.9515
JC Buttler (ENG)	1.0068	JM Bairstow (ENG)	1.0077	SPD Smith (AUS)	0.9511

presents the results for the batting department and digs out the top 20 most efficient batters in cricketing history for all three game formats. The results of Table 2 indicate that, by applying the DEA Super-SBM approach, we found the 20 most efficient batsmen in cricketing history. They are different from the top 20 chosen on traditional criteria of runs scored in each format. Out of the top 20 run scorers in Test cricket, only four are in the top 20 efficient players, while out of the top 20 run-scorers in ODIs, only three are in the top 20 efficient players. Further, in the top 20 T20 scorers, only five are efficient. These results prove that only scoring runs is not an accurate indicator of choosing the best performer, but that the average strike rate and centuries, the 50s, 6s, and 4s hit during an inning are also important indicators of performance evaluation.

Further elaborating on the results, we found that DG Bradman and C.J.L. Rogers from Australia, GA Headley from the West Indies, SR Tendulkar from India, and LG Rowe from the West Indies are the five most efficient batters in cricketing Test history (1877–2019). In other words, these batters get runs at the best of averages and strike rates and scored more centuries and hundreds in their career. SR Tendulkar & V Kohli from India, SO Hetmyer from the West Indies, RR Rossouw from South Africa, and JC Buttler from England are the top five most efficient ODI batters from 1977–2019. V Kohli from India, CH Gayle from the West Indies, MDKJ Perera from Sri Lanka, Babar Azam from Pakistan, and AJ Finch from Australia are the top five most efficient batsmen in T20 cricket history (2003–2019). To some extent, our results are aligned with the ICC ranking, but not all the top-ranked players are efficient [6].

4.2. Player’s Efficiency in the Bowling Department

Table 3. Player’s efficiency in the bowling department for all three formats.

Tests		ODIs		T20s
Player	Score	Player	Score	Player	Score
M Muralitharan (ICC/SL)	1.2797	MA Starc (AUS)	1.6209	Umar Gul (PAK.)	1.2500
DL Underwood (ENG)	1.2201	Waqar Younis (P.A.K.)	1.4133	SL Malinga (SL)	1.2121
Sir RJ Hadlee (NZ)	1.2094	JJ Bumrah (INDIA)	1.3373	Rashid Khan (AFG)	1.2022
Wasim Akram (PAK)	1.1787	DK Lillee (AUS)	1.2577	BAW Mendis (SL)	1.1661
R Benaud (AUS)	1.1645	Rashid Khan (A.F.G.)	1.1528	Saeed Ajmal (PAK.)	1.0776
J Garner (WI)	1.1303	Wasim Akram (PAK)	1.1447	DL Vettori (NZ)	1.0605
Waqar Younis (P.A.K.)	1.1116	M Muralitharan (SL)	1.1409	Shahid Afridi (PAK.)	1.0531
DK Lillee (AUS)	1.1046	J Garner (WI)	1.1201	Imran Tahir (SA.)	1.0468
DW Steyn (SA)	1.0718	Sir RJ Hadlee (NZ)	1.0694	Imad Wasim (PAK)	0.9956
LR Gibbs (WI)	1.0609	B Lee (AUS)	1.0604	Shakib Al Hasan (B.D.)	0.9894
CEL Ambrose (WI)	1.0545	MA Holding (WI)	1.0299	SP Narine (WI)	0.9726
AA Donald (SA.)	1.0454	GD McGrath (AUS)	1.0177	S Badree (W.I./World)	0.9554
MD Marshall (WI)	1.0439	Saqlain Mushtaq (PAK.)	1.0160	DT Johnston (IRE.)	0.9413
R Ashwin (INDIA)	1.0411	AA Donald (SA.)	1.0111	GP Swann (ENG)	0.9411
FS Trueman (ENG)	1.0364	M. Shami (INDIA)	0.9879	RE van der Merwe (SA)	0.9408
GP Swann (ENG)	1.0263	SM Pollock (SA.)	0.9845	KOK Williams (WI)	0.9283
GD McGrath (AUS)	1.0162	SK Warne (AUS)	0.9812	MA Starc (AUS)	0.9175
BS Bedi (INDIA)	1.0045	SE Bond (NZ)	0.9795	PJ Cummins (AUS)	0.9131
GD McKenzie (AUS)	0.9952	MD Marshall (WI)	0.9525	J Botha (SA.)	0.9064
Imran Khan (PAK)	0.9871	BAW Mendis (SL)	0.9523	M. Hafeez (PAK)	0.8981

explains the player’s efficiency scores in the bowling department of cricket for all formats. It indicates that by applying the DEA Super-SBM approach, we found the 20 most efficient bowlers in cricketing history. Similar to the batting efficiency results, the top 20 most efficient bowlers are different from the top 20 chosen on traditional criteria of wickets taken in each format. Out of the top 20 wickets takers in Test cricket, only seven are in the top 20 efficient bowlers list, while out of the top 20 wicket takers in ODIs, only eight are in the top 20 efficient players. Further, in the top 20 T20s, for wicket-takers, only 10 are efficient. These results prove that only taking wickets is not the accurate indicator of choosing the best performer; average, strike rate, economy, and more wickets in each format are also important performance evaluation indicators. In other words, these bowlers get more wickets with the best averages, strike rate, and economy in their cricketing careers. M Muralitharan from Sri Lanka, DL Underwood from England, Sir RJ Hadlee from New Zealand, Wasim Akram from Pakistan, and R Benaud from Australia are the five most efficient Test bowlers in cricket. According to our estimation, MA Starc from Australia, Waqar Younis from Pakistan, JJ Bumrah from India, DK Lillee from Australia, and Rashid Khan from Afghanistan are the top five most efficient bowlers in cricketing ODI history (1977–2019). Umar Gul from Pakistan, SL Malinga from Sri Lanka, Rashid Khan from Afghanistan, BAW Mendis from Sri Lanka, and Saeed Ajmal from Pakistan are the top five most efficient blowers of T20 cricketing history (2003–2019).

4.3. Player’s Efficiency in the Fielding Department

Table 4. Player’s efficiency in the fielding department for all three formats.

Tests		ODIs		T20s
Player	Score	Player	Score	Player	Score
Saleem Yousuf (PAK.)	1.6670	L Ronchi (AUS/NZ)	1.4036	SA Edwards (NL)	1.600
WAS Oldfield (AUS)	1.6408	DPMD Jayawardene (SL)	1.3625	JM Bairstow (ENG)	1.5714
TD Paine (AUS)	1.4396	MS Dhoni (Asia/INDIA)	1.3222	MS Dhoni (INDIA)	1.4632
MV Boucher (ICC/SA)	1.4037	RS Kaluwitharana (SL)	1.2033	Kamran Akmal (PAK)	1.3011
RB Simpson (AUS)	1.2589	AC Gilchrist (AUS)	1.1714	Q de Kock (SA)	1.2508
N Dickwella (SL)	1.1751	LRPL Taylor (NZ.)	1.1498	AB de Villiers (S.A.)	1.2431
AC Gilchrist (AUS)	1.1614	KC Sangakkara (SL)	1.144	DA Miller (SA)	1.2025
SP Fleming (NZ)	1.0627	Q de Kock (SA)	1.0555	PW Borren (NL)	1.0310
Jayawardene (SL)	1.0346	Moin Khan (P.A.K.)	1.0215	Shoaib Malik (ICC/PAK)	1.000
R Dravid (ICC/INDIA)	1.0244	JC Buttler (ENG)	1.0128	MJ Guptill (NZ.)	0.92
Kamran Akmal (PAK)	1.0015	RW Marsh (AUS)	0.9833	BN Cooper (NL)	0.9015
Q de Kock (SA)	1.0013	MV Boucher (Afr/SA)	0.9684	D Ramdin (WI)	0.9006
BJ Haddin (AUS)	1.0013	NR Mongia (INDIA)	0.9649	LRPL Taylor (NZ.)	0.88
TG Evans (ENG)	0.9774	BJ Haddin (AUS)	0.9592	M.Shahzad (AFG)	0.8653
ATW Grout (AUS)	0.9588	IA Healy (AUS)	0.9472	Mushfiqur Rahim (BD.)	0.8604
JH Kallis (ICC/SA)	0.9524	Khaled Mashud (BDESH)	0.9259	DA Warner (AUS)	0.8600
RW Marsh (AUS)	0.9468	Rashid Latif (PAK)	0.9258	Umar Akmal (PAK)	0.841
ME Waugh (AUS)	0.9377	D Ramdin (WI)	0.9212	SK Raina (INDIA)	0.8400
SPD Smith (AUS)	0.9369	Sarfaraz Ahmed (P.A.K.)	0.9033	M Nabi (AFG.)	0.8200
RT Ponting (AUS)	0.9333	DJ Richardson (SA.)	0.8994	L Ronchi (AUS/ICC/NZ)	0.8148

explains the player’s efficiency scores in the fielding department of cricket for all formats. It indicates that by applying the DEA Super-SBM approach, we found the 20 most efficient fielders in cricketing history. Similar to the batting and bowling efficiency results, the top 20 most efficient Fielders differ from the top 20 chosen on traditional criteria of catches taken or run-outs in each format. Out of the top 20 fielders in Test cricket, only six are in the top 20 efficient fielders list, while out of the top 20 wicket-takers in ODIs, only five are in the top 20 efficient players. Further, in the top 20 T20 fielders, only six are efficient. These results prove that only dismissals are not an accurate indicator of choosing the best performer, but there are also important indicators of performance evaluation. In other words, these fielders get more dismissals with fewer innings played in their careers. Saleem Yousuf from Pakistan, WAS Oldfield from Australia, TD Paine from Australia, MV Boucher from South Africa, and RB Simpson from Australia are the five most efficient Test fielders in cricketing history. L Ronchi (AUS/NZ), DPMD Jayawardene from Sri Lanka, MS Dhoni from India, RS Kaluwitharana from Sri Lanka, and AC Gilchrist from Australia are the five most efficient fielders in ODI cricketing history (1977–2019). SA Edwards from the Netherlands, JM Bairstow from England, MS Dhoni from India, and Kamran Akmal from Pakistan are the most efficient fielders in T20s from 2003–2019.

The results concluded that the new indices for player performance evaluations are comprehensive and more accurate because they incorporate all the input–outputs associated with cricketer performance in all cricket departments (batting, balling, and fielding). The efficiency scores of players estimated through proposed indices differ from results extracted from traditional performance indicators, demonstrating that those outdated measures do not show the overall efficiency of cricket players. Results proved that these indices could be used to evaluate cricketers’ career efficiency further. The ICC and sports analysts could use these indices to measure any player’s overall efficiency in any cricket format.

4.4. Efficient Players Description in All Three Departments of Cricket

Through DEA Super-SBM, we found the most efficient cricketers in all three formats of cricket. This section discusses some of the most efficient cricketers and their achievements. Starting from the Test batting, we found that DG Bradman from Australia is the most efficient Test batsman. SR Tendulkar from India is the most efficient ODI batsman, while V Kohli from India is the most efficient T20 batsman. In the bowling department, we found that M Muralitharan from Sri Lanka is the most efficient bowler in Test cricket. MA Starc from Australia is the most efficient bowler in ODIs. Umar Gul from Pakistan was found to be the most efficient bowler in the T20s. Saleem Yousuf from Pakistan, L Ronchi (Australia/New Zealand), and SA Edwards from the Netherlands are the most efficient fielders in cricketing history in Tests, ODIs, and T20s, respectively.

4.4.1. Sir Donald George Bradman

The Australian cricketer Sir Donald George Bradman, AC (27 August 1908–25 February 2001), also known as “The Don,” is usually regarded as the game’s greatest batsman in history. Many believe Bradman’s 99.94 Test lifetime batting average to be the best achievement in the history of any major sport. According to an Australian legend, a young Bradman would practice his swing using a cricket stump and a golf ball. In under two years, he moved from playing bush cricket to playing for the Australian Test team. Before he reached 22, he had already set numerous scoring records, some of which are still in existence today. He was acclaimed as Australia’s sporting hero during the darkest days of the Great Depression [40,41,42]. Bradman had a career spanning 20 years, and throughout that time, he maintained a scoring average that ensured his success. The England team developed a controversial strategy called “Bodyline” to limit his run-scoring. Bradman was a captain and administrator whose commitment to entertaining and attacking cricket brought record crowds. However, he could not tolerate the continual adulation, and his interactions with others reflected this. John Howard, the Australian prime minister, declared him the “greatest living Australian” nearly 50 years after his retirement from Test cricket. A museum honoring Bradman’s life was opened while he was still alive, and his image appeared on stamps and coins. On the centennial of his birth, 27 August 2008, the Royal Australian Mint issued a commemorative AUD 5 gold coin portraying Bradman. In 2009, he was the first to be inducted posthumously into the ICC Cricket Hall of Fame [43,44,45,46,47,48].

4.4.2. Sachin Ramesh Tendulkar

Indian former international cricket player and team captain Sachin Ramesh Tendulkar was born on 24 April 1973. There is a consensus that he ranks among cricket’s all-time great batsmen. He batted at the highest level for the Indian Cricket Team as a right-handed batter. His at-bat abilities, technique, vision, and game-reading are all well-known. He has scored over 18,000 runs in ODIs and over 15,000 runs in Tests, making him the all-time leader in both categories. The number of Man of the Match accolades he has collected across all formats of international cricket is also a record. Tendulkar began playing cricket at age 11. On 15 November 1989, at 16, he made his Test match debut against Pakistan in Karachi. He went on to play for Mumbai in India’s domestic league and India for over 24 years in international competition. In 2002, halfway through his career, Wisden placed him as the second-greatest Test batsman of all time, behind Don Bradman, and the second-greatest ODI batsman, behind Viv Richards [49,50,51,52,53]. Later in his career, Tendulkar helped India win its first-ever World Cup at the 2011 Cricket World Cup. In 1994, Tendulkar was honored with the Arjuna Award for his achievements in sports. In 1997, he was honored with India’s highest sporting honor, the Khel Ratna Award. In 1999 and 2008, he was honored with India’s two highest civilian honors, the Padma Shri and Padma Vibhushan awards. The Prime Minister’s Office announced the Bharat Ratna, India’s highest civilian award, just a few hours after his final match in November 2013. After playing in his 200th Test match in November 2013, he announced his retirement from all forms of cricket, having already left the sport after a career spanning more than two-and-a-half decades. Tendulkar amassed a total of 34,357 runs in 664 international cricket matches. He joined the ICC Hall of Fame in 2019 due to his achievements in cricket [54,55].

4.4.3. Virat Kohli

Virat Kohli, an Indian cricketer born on 5 November 1988, is a former national team captain. He bats right-handed for Delhi in local competition and for Royal Challengers Bangalore in the Indian Premier League. In 2011, Kohli participated in his debut test. In 2013, he became the first batsman in ICC history to top the batting rankings for ODIs. Twice at the ICC World Twenty20 (in 2014 and 2016), he was named the tournament’s Man of the Match [56,57]. In addition, he holds the record for becoming the player to score 24,000 runs in his international career the quickest. He holds the record for the most runs ever scored in a T20 World Cup competition. Many organizations have recognized Kohli for his outstanding play, including the International Cricket Council (ICC), which has awarded him the Sir Garfield Sobers Trophy (ICC Men’s Cricketer of the Decade) for the period from 2011–2020, the ICC Cricketer of the Year award in 2017 and 2018, the ICC Test Player of the Year award in 2018, the ICC One Day International Player of the Year award in 2012, 2017, and 2018, and the Wisden Leading Cricketer in the World (2016, 2017, and 2018). In 2018, he was awarded the Rajiv Gandhi Khel Ratna, India’s highest sporting accolade. In 2013, he was also awarded India’s Arjuna Award, the country’s highest civilian accolade. ESPN labeled him one of the most famous athletes in the world in 2016, while Forbes included him in their list of the most valuable athlete brands. According to Time, he is one of 2018’s 100 most influential individuals. The 2020 season is expected to make him approximately USD 26 million, putting him at #66 on Forbes’ ranking of the 100 highest-paid athletes in the world [58,59].

4.4.4. Deshabandu Muttiah Muralitharan

Sri Lankan cricket coach, former professional cricketer, and businessman Deshabandu Muttiah Muralitharan (born 17 April 1972) is a member of the ICC Cricket Hall of Fame. Muralitharan, who took more than six wickets per Test match on average, is considered one of the best bowlers ever. He has more ODI wickets (53) than anyone else in cricket history and holds the Test record (800) for most wickets taken by a bowler. As of 2022, he is the best bowler in international cricket history in terms of wickets. For a record 1711 days, covering 214 Test matches, Muralitharan was ranked as the best bowler in the world by the International Cricket Council. On December 3, 2007, he passed Shane Warne and became the Test cricket all-time leader in wickets taken [60,61]. Muralitharan had the record until Warne overtook him in the latter part of 2004 when a shoulder injury prevented him from continuing to bowl. Muralitharan surpassed Wasim Akram’s previous record of 502 ODI wickets with the dismissal of Gautam Gambhir on 5 February 2009, in Colombo. After capturing his 800th and final wicket with the last test match’s final delivery on 22 July 2010, he announced his retirement from Test cricket. In addition to being crowned the greatest Test match bowler by Wisden’s Cricketers’ Almanack in 2002, Muralitharan became the first Sri Lankan player to be inducted into the ICC Cricket Hall of Fame in 2017. Ada Derana honored him as Sri Lanka’s Top Young Achiever in 2017 [62,63].

4.4.5. Mitchell Aaron Starc

Born on 30 January 1990, Mitchell Aaron Starc is a professional cricketer for the Australian national and New South Wales state teams. Starc represents Australia in all three major international cricket formats (Test cricket, One Day Internationals (ODI), and Twenty20 Internationals) as a left-arm fast bowler and a lower-order left-handed batsman. He is widely considered the best bowler of all time, and in 2015, he had the best overall rating of any bowler in One-Day International matches. Starc has been playing international cricket since 2010, but a series of injuries derailed his career early. While playing for the Australian team that eventually won the 2015 Cricket World Cup, he rose to prominence. He was named the tournament’s Most Valuable Player for outstanding play. Starc’s ability to bowl at a high speed — his fastest delivery has been clocked at over 160 km/h—and to induce reverse swing with his deliveries has earned him widespread acclaim. For Australia in Test cricket, he has taken the eighth-most wickets all time [64,65,66].

4.4.6. Umar Gul

Umar Gul, born on 15 October 1982, is a bowling coach for the Quetta Gladiators of the Pakistan Super League. He was a right-arm fast-medium bowler for the Pakistani cricket team and played in all three formats. He became well-known as one of the game’s best bowlers after he finished first in the 2007 and 2009 Twenty20 World Championship tournaments regarding wickets taken and bowled. After Saeed Ajmal, Umar Gul had 74 dismissals in Twenty20 Internationals, making him the second-highest wicket-taker of all time. In 2013, he gave the best international Twenty20 performance and won the award. Gul called it quits after a 20-year cricket career on 16 October 2020, following the final group stage match of the 2020–21 National T20 Cup [67,68,69].

4.4.7. Saleem Yousuf

Pakistani cricketer Saleem Yousuf (born 7 December 1959) played in 32 Test matches and 86 One Day Internationals (ODIs) between 1982 and 1990. He was a wicket-keeper. In a match against England at Edgbaston in 1987, he scored 91 not out. He batted brilliantly in the 1987 World Cup as Pakistan returned from an apparent loss to beat the West Indies. The first wicket-keeper in One-Day International history to register three stumpings in an inning was Saleem Yousuf in 1990. To this day, he shares this record with two other players [70,71].

4.4.8. Luke Ronchi

New Zealand–Australian cricket coach and player Luke Ronchi was born on 23 April 1981. He played for two national cricket teams in international competitions: Australia and New Zealand. Ronchi is the only cricketer in history representing Australia and New Zealand. He was a member of the New Zealand team that came in second place at the 2015 Cricket World Cup, losing to Australia in the final. He has represented several New Zealand Twenty20 teams in domestic matches, including Wellington. In June 2017, he officially ended his international cricket career [72,73].

4.4.9. Scott Andrew Edwards

Scott Andrew Edwards, born in Australia on 23 August 1996, is a professional cricket player for the Netherlands. On 29 November 2017, he made his debut for the Netherlands in professional cricket against Namibia in the 2015–17 ICC Intercontinental Cup. He played his maiden list A match for the Netherlands on December 8, 2017, against Namibia in the 2015–17 ICC World Cricket League Championship. In June 2022, when Pieter Seelaar was forced to retire from international cricket due to a chronic back injury, Edwards was named the new captain of the Dutch cricket team. Edwards is the Netherlands’ ninth ODI captain [74].

4.5. Aggregate Country- Wise Player’s Efficiency

Section 4.5 evaluates the aggregate efficiency scores of all three departments (batting, bowling, and fielding) for each country. It includes all three formats of cricket (Test, ODIs, and T20s). England’s average batting efficiency score is 0.9362, higher than all test-playing nations. It demonstrates that English batters are most efficient in Test batting. In contrast, the Australian batters are most efficient in ODI batting, with an efficiency score of 0.8183. Finally, Indian batters are most efficient in T20s, with an efficiency score of 0.9896. These results show that English batters scored with a high average and strike rate, while Australian Batman scored more with a leading batting average and strike rates in ODIs. At the same time, Indian batters can score with a fast strike rate and average and hit more bourrides to get higher efficiency than their counterparts.

Through September 2022, England, with the most efficient batters, will have played 1055 Test matches, with a record of 384 wins and 317 losses (with 354 draws). One of the most prestigious sporting trophies is the Ashes, which England has won 32 times in the Test series versus Australia [75,76]. The most efficient in ODI batting throughout 973 One-Day Internationals (ODIs), the Australian cricket team has won 590, lost 340, tied nine times, and had 34 games finish in a no-result. Despite being ranked first for 141 of the 185 months since the tournament’s inception in 2002, Australia is currently ranked third in the ICC ODI Championship with 107 rating points as of May 2022. Australia is the most successful team in the history of one-day international cricket, having won more than 60 percent of their matches and having reached the World Cup finals a record seven times (1975, 1987, 1996, 1999, 2003, 2007, and 2015). The West Indies previously held the record for most consecutive World Cup appearances with three (1975, 1979, and 1983). However, Australia is the first and only team to win three consecutive World Cups (1996, 1999, 2003, and 2007) (1999, 2003, and 2007). Before losing to Pakistan by four wickets in the group stage of the 2011 Cricket World Cup, the squad had won 34 straight matches at the World Cup. Moreover, it is the second team, after India in 2011, to win the World Cup on its turf in 2015. The Australian cricket team is the only one to have ever repeated as champions of the ICC Champions Trophy, which they did in 2006 and 2009. Until 2021, no other team had won more than two Cricket World Cups, with Australia being the sole winner with five victories [77].

The most efficient team in T20 Batting, India has played in One-Day Internationals (ODIs) and Twenty-Over Internationals (T20Is) since 1974 and 2006, respectively (see

Table 5. Average player’s batting efficiency scores for each country.

Batting
Test		ODIs		T20s
Country	Score	Country	Score	Country	Score
Australia	0.9266	Afghanistan	0.5580	Afghanistan	0.8115
Bangladesh	0.7812	Australia	0.8183	Australia	0.9328
England	0.9362	Bangladesh	0.6493	Bangladesh	0.7848
India	0.8899	England	0.6574	England	0.8696
New Zealand	0.8572	India	0.6721	Hongkong	0.7609
Pakistan	0.8776	Ireland	0.5129	India	0.9896
South Africa	0.8650	Kenya	0.5400	Ireland	0.8616
Sri Lanka	0.82998	Netherlands	0.6100	Kenya	0.7599
West Indies	0.8581	New Zealand	0.7604	Netherlands	0.8133
Zimbabwe	0.7538	Pakistan	0.6841	New Zealand	0.9168
		South Africa	0.6440	Oman	0.7821
		Scotland	0.6310	Pakistan	0.8986
		Sri Lanka	0.6007	South Africa	0.9075
		Zimbabwe	0.5458	Scotland	0.8730
				Sri Lanka	0.8723
				UAE	0.7868
				West Indies	0.8546
				Zimbabwe	0.8151

). Five major I.C.C. events have been won by the team: the Cricket World Cup (1983 and 2011), the ICC T20 World Cup (2007), and the ICC Champions Trophy (2002) and (2013). They have also placed second in the World Cup (2003), the T20 World Cup (2014), and the Champions Trophy (2012, 2000, and 2017). The squad finished second in the very first ICC World Test Championship from 2019–2021. When they won the 2011 Cricket World Cup, they became the second side to do so, following the West Indies, and the first to do it on their home soil [78].

Table 6. Average player’s bowling efficiency scores for each country.

Bowling
Test		ODIs		T20s
Country	Score	Country	Score	Country	Score
Australia	0.9607	Afghanistan	0.766	Afghanistan	0.8233
England	0.9702	Australia	0.9452	Australia	0.8982
India	0.9265	Bangladesh	0.8307	Bangladesh	0.8496
New Zealand	0.9412	England	0.8263	England	0.8672
Pakistan	0.9843	India	0.8291	India	0.8185
South Africa	0.9414	New Zealand	0.8524	Ireland	0.9537
Sri Lanka	0.9639	Pakistan	0.8684	Netherlands	0.8609
West Indies	1.0089	South Africa	0.8742	New Zealand	0.8864
		Sri Lanka	0.8100	Pakistan	0.9817
		West Indies	0.8544	South Africa	0.8661
		Zimbabwe	0.7461	Scotland	0.8227
				Sri Lanka	0.9101
				UAE	0.8045
				West Indies	0.8323
				Zimbabwe	0.7944

explains the best bowling team in terms of aggregate players’ efficiency scores. Results show that West Indies bowlers are most efficient in Test cricket, with efficiency scores of 1.0089. They also indicate that West-Indies bowlers get more wickets with the best economy, strike rate, and average. In contrast, Australian bowlers are most efficient in ODIs with 0.9452 average efficiency scores. Finally, the Pakistani bowling attack is most efficient in T20s, with an average efficiency score of 0.9817. The bowlers from these three countries cost fewer runs with more wickets with the best bowling average, economy, and strike rate. The most efficient bowlers in Test cricket history, the Windies, as the West Indies cricket team is affectionately known, are a men’s international cricket team that represents the English-speaking nations and territories of the Caribbean under the auspices of Cricket West Indies. The members of this combined team come from 15 different Caribbean countries and territories. The West Indies cricket team dominated both Test and One-Day International competition from the mid-to-late 1970s to the early 1990s. Cricket has produced several world-class players, many of whom have called the West Indies home [79]. The most efficient ODI bowlers are from the Australian cricket team; the history and performance of the Australian cricket team are already described above. With the best T20 bowling attack, Team Pakistan has played 446 Test matches, winning 146, losing 136, and drawing 164. After being granted Test status on 28 July 1952, Pakistan’s first match was a loss to India by an inning and 70 runs at Delhi’s Feroz Shah Kotla Ground that October. The team has played 945 One-Day Internationals, with 498 wins, 418 losses, 9 draws, and 20 no-outcomes. Pakistan won the World Cup in 1992 and second place in 1999. Pakistan co-hosted the 1987 and 1996 World Cups with other South Asian countries; the 1996 final was played at Lahore’s Gaddafi Stadium. The squad has competed in 215 Twenty20 Internationals, coming out on top 131 times while also suffering 76 defeats and 3 draws. Pakistan has placed second in the 2007 and 2022 ICC Men’s T20 World Cups. Pakistan is the only team to have ever won the ICC Cricket World Cup (1992), ICC T20 World Cup (2009), ICC Champions Trophy (2017), and ICC Test Championship (2016) [80].

Table 7. Average player’s fielding efficiency scores for each country.

Fielding
Test		ODIs		T20s
Country	Score	Country	Score	Country	Score
Australia	0.9486	Australia	0.9578	Afghanistan	0.7905
Bangladesh	0.5292	Bangladesh	0.8331	Australia	0.7993
England	0.7417	England	0.8657	Bangladesh	0.7402
India	0.7374	India	0.9229	England	0.8964
New Zealand	0.8294	New Zealand	0.8522	India	0.9508
Pakistan	0.9387	Pakistan	0.837	Ireland	0.7113
South Africa	0.9244	South Africa	0.9191	Netherlands	0.9687
Sri Lanka	0.9061	Sri Lanka	0.9463	New Zealand	0.7347
West Indies	0.7632	West Indies	0.8238	Pakistan	0.9108
Zimbabwe	0.7087	Zimbabwe	0.6985	South Africa	1.0991
			+	Scotland	0.7622
				Sri Lanka	0.6543
				West Indies	0.761

explains the player’s aggregate efficiency score in the fielding department of all cricket-playing countries for all three formats of cricket. Results demonstrate that Australian fielders are most efficient in test and ODIs formats, with a mean efficiency score of 0.9486 and 0.9578, respectively, while South African players are most efficient in the fielding department of T20 cricket, with a mean efficiency score of 1.0991. These results indicate that these cricket team fielders played fewer innings to take more catches and run-outs. The history and performance of the Australian cricket team (efficient in test and ODI fielding) have already been described in the above sections.

In contrast, the most efficient team in T20s fielding is South Africa. They hosted an England cricket team in the 1888–89 season; South Africa also began playing international cricket at the highest level. While they couldn’t compete with Australia or England at the start of the 20th century, the team improved through experience and training. They continued to play regular games against teams from Australia, England, and New Zealand well into the 1960s, when opposition to apartheid was at its height. In line with the actions of other international sports governing bodies, the International Cricket Council (ICC) banned the team internationally. By the time the ban was instituted, South Africa had developed into a team that could legitimately be considered the best in the world. They had even beaten Australia. It wasn’t until 1991 that South Africa could play against India, Pakistan, Sri Lanka, and the West Indies without a ban. Since its reinstatement, the team has been dominant, reaching the top of international polls on multiple occasions. With a winning percentage of over 60% in one-day internationals, South Africa is among the best teams in the sport. Its only victory in an ICC-sanctioned tournament was the Champions Trophy in 1998. In 1998, South Africa triumphed at the Commonwealth Games, taking home the gold medal [81]. Results proved that the study proposed indices are comprehensive in nature and superior to existing techniques due to multiple inputs–outputs used to evaluate the performance of each department in all formats of cricket. Existing literature advocate the results of this study [82,83,84,85,86,87,88,89,90].

5. Conclusions

To evaluate the player’s efficiency in all three formats of cricket, this study proposed the input–output indices for the game’s batting, bowling, and fielding departments. The research employed DEA Super-SBM to analyze the player’s efficiency in all three formats of cricket. Further, it evaluates the most efficient player and cricket team in the game’s history. Super-SBM has this property to differentiate the most efficient team and player among efficient DMUs. To measure the player’s efficiency in the batting department, we used innings played as a single input and total runs scored batting average, 100s scored, and 50s scored as outputs in the test format.

Similarly, innings played and balls faced are used as inputs, while total runs scored, batting average, strike rate, 100s scored, and 50s scored are used as outputs in the ODIs. For T20s, we used inns played, balls faced as inputs and total runs scored, batting average, strike rate, the 50s Scored, 6s hit, and 4s hit as output indices. To measure the bowler’s efficiency, innings played, balls bowled, and runs given are used as inputs, and wickets taken, transformed (average, economy, and strike rate), five wickets, and 10 wickets in a single game are used as output variables in Test cricket. In ODIs, innings played, balls bowled, and run given are used as inputs, wickets taken and transformed (average, economy, and strike rate), and four wickets in a single match and five wickets in a single match are used as outputs. In T20s, the innings played, balls bowled, and runs given are used as inputs, while wickets taken, transformed bowling average, economy, and strike rate, four wickets in one game and five wickets in one game are used as outputs. In the fielding department, innings played are used as input, while stumps, catches taken as a wicket-keeper, and catches taken in the fielding are used as outputs in all three formats of cricket.

The results concluded that the new indices for player performance evaluations are comprehensive and more accurate because they incorporate all the input–outputs associated with cricketer performance in all cricket departments (batting, balling, and fielding). The efficiency scores of players estimated through the proposed indices differ from results extracted from traditional performance indicators, demonstrating that those outdated measures don’t show the overall efficiency of cricket players. Compared to traditional indicators for players’ performance, the proposed indices include all affecting factors as inputs and outputs. Further, DEA Super-SBM employed all the inputs and outputs to give comprehensive efficiency scores, which is ultimately more effective than single input–output indicators. The results proved that these indices could be used to evaluate cricketers’ career efficiency further. The ICC and sports analysts could use these indices to measure any player’s overall efficiency in any cricket format.

Moreover, the study evaluates the results for players in each department (batting, bowling, and fielding) and ranks the player and teams in each format. Sir Bradman from Australia, Sachin Tendulkar, and Virat Kohli from India are found to be the most efficient batters in Test, ODIs, and T20s history, respectively. Muralitharan from Sri Lanka, Mitchell Starc from Australia, and Umar Gul from Pakistan are super-efficient bowlers in all three formats, respectively. Saleem Yousuf from Pakistan, New Zealand-Australian cricketer Luke Ronchi, and Scott Edwards from the Netherlands were found to be the most efficient fielders in cricketing history for all three formats. Further study ranks the top 20 most efficient players in the cricketing history in each department for all three formats. England’s average batting efficiency score is 0.9362, higher than all test-playing nations. It demonstrates that English batters are most efficient in test batting.

In contrast, the Australian batsmen are the most efficient in ODI batting with an efficiency score of 0.8183. Finally, Indian batsmen are most efficient in T20s, with an efficiency score of 0.9896. West Indies bowlers are most efficient in Test cricket, with efficiency scores of 1.0089. It indicates that West Indies bowlers get more wickets with the best economy, strike rate, and average. In contrast, Australian bowlers are most efficient in ODIs with 0.9452 average efficiency scores. Finally, the Pakistani bowling attack is most efficient in T20s, with an average efficiency score of 0.9817. The results conclude that Australian fielders are most efficient in test and ODIs formats, with a mean efficiency score of 0.9486 and 0.9578, respectively, while South African players are most efficient in the fielding department of T20 cricket, with a mean efficiency score of 1.0991.

With all its contribution to the literature, this study has some limitations, which are described as follows: The data availability is a constraint, as data for cricket players’ indices were available until 2019. Further, as time passes, a few game rules and regulations changes could change the player’s performance; this is another study limitation.