Indices Development for Player’s Performance Evaluation through the...

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

Citation: Yin, W.; Ye, Z.; Shah, W.U.H Indices Development for Player’s Performance Evaluation through the Super-SBM Approach in Each Department for All Three Formats of Cricket Sustainability 2023 , 15 , 3201 https://doi.org/10.3390/su 15043201 Academic Editor: Giuseppe Battaglia Received: 29 December 2022 Revised: 30 January 2023 Accepted: 4 February 2023 Published: 9 February 2023 Copyright: © 2023 by the authors Licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/) sustainability Article Indices Development for Player’s Performance Evaluation through the Super-SBM Approach in Each Department for All Three Formats of Cricket Wei Yin 1 , Zhixiao Ye 2 and Wasi Ul Hassan Shah 3, * 1 Physical Education College, Taishan University, 525 Dongyue Street, Daiyue District, Tai’an 271000, China 2 Property Management Department, School of Management, Zhejiang Shuren University, Hangzhou 310015, China 3 School of Management, Zhejiang Shuren University, Hangzhou 310015, China * Correspondence: wasi 450@yahoo.com; Tel.: +86-1552-0742-040 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 Keywords: player’s efficiency; DEA super-SBM; cricket; efficient teams 1. Introduction Cricket has been a worldwide phenomenon since the late 19 th 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 20 th and early 21 st 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. T 20 s, on the other hand, were founded in June of 2003, not long after the millennium change. Rediff reports that T 20 s 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 Sustainability 2023 , 15 , 3201. https://doi.org/10.3390/su 15043201 https://www.mdpi.com/journal/sustainability

Sustainability 2023 , 15 , 3201 2 of 20 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 T 20 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 100 s and 50 s 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 – 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 – 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 – 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 Sections 4 and 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

Sustainability 2023 , 15 , 3201 3 of 20 unchanged, and the original effective value evaluation can be greater than 1, after which they can be compared [ 33 – 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 ∈ R M , y g ∈ R S 1 , y b ∈ R S 1 indicate the expected output of S 1 with m units of input. Assuming X > 0 , Y g > 0, Y b > 0, the production possibility set is defined as: X = [ x 1 , x 2 , . . . , x N ] ∈ R N × M , and the output matrix is expressed as Y g = y g 1 , y g 2 , . . . , y g N ∈ R S 1 × N P= n x , y g , y b | , x ≥ X η | , y g ≤ Y η , y b ≥ Y η , η ≥ 0 o (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 0 by considering the production possibility set γ = min     1 − 1 M ∑ M i= 1 S − i x io 1 + 1 S 1+S 2 ∑ S 1 r=1 S g r y g r 0 + ∑ S 2 r=1 S b r y b r 0         (2) s.t        x 0 =X η + S − y g 0 =Y g η − S g y b 0 =Y b η + S b S − ≥ 0, S g ≥ 0 , S b ≥ 0 , η ≥ 0 In Equation (2), it denotes the DMU’s efficiency, and the change value ranges from 0 to 1. The symbols ( S − S − ,S g , S b ) denote input, output, and slack, respectively. The DMU is at the vanguard of production only when the technical efficiency γ is 1 and S − , S g , S b are all 0; if γ < 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 κ = m T − 1 M ∑ M i=1 S − i x io ! (3)                  1=T+ 1 S 1+S 2 S 1 ∑ r = 1 S g r y g ro + S 2 ∑ r=1 S b r y b ro x 0 T=X β +S − y g 0 T=Y g β − S g y b 0 T=Y b β +S b S − ≥ 0 , S g ≥ 0 , S b ≥ 0 , β ≥ 0 , T ≥ 0 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 γ ∗ = m     1 M ∑ M i = 1 xi x 0 0 1 S 1+S 2 ∑ S 1 r=1 y s r y g ro + ∑ S 2 r = 1 → y b r y b ro     (4)

Sustainability 2023 , 15 , 3201 4 of 20                        x ≥ N ∑ j = 1 , 6 = 0 η j x j y g ≤ N ∑ j = 1 , 6 = 0 η j y g j → y b ≥ N ∑ j = 1 , 6 = 0 η j y b j x ≥ x 0 , y g ≤ y g 0 , → y b ≥ y b 0 , y g ≥ 0, η ≥ 0 , (5) Super-efficiency of DMU is denoted by γ ∗ 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 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), 100 s, and 50 s 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 T 20 batting efficiency Table 1. Inputs–outputs used for player’s efficiency Inputs–Outputs Used to Evaluate the Batsman’s Efficiency. Test Cricket ODI’s T 20’s Inputs Outputs Inputs Outputs Inputs Outputs Inns Played Total Runs Scored Batting Ave 100 s Scored 50 s Scored Inns played Balls faced Total Runs Scored Batting Ave Strike rate 100 s Scored 50 s Scored Inns played Balls faced Total Runs ScoredBatting Ave Strike rate 50 s Scored 6 s hit 4 s hit Inputs–outputs Used to Evaluate the Bowler’s Efficiency. Test Cricket ODIs T 20 s 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 T 20 s 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 Similarly, in T 20 games, more 6 s and 4 s are also considered good output. Therefore, they are included in the T 20 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 10 s and 5 s 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 T 20 s 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 – 39 ]. The dataset includes all the international cricket

Sustainability 2023 , 15 , 3201 5 of 20 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 T 20 s, 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 T 20 s 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 T 20 s. 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 T 20 s 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 runouts and catches taken by a fielder. Figures 1 – 3 show the top 20 performers in all three departments of cricket (batting, bowling, and fielding) for Test, ODIs, and T 20 s (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 ]. Sustainability 2023 , 15 , x FOR PEER REVIEW 6 of 22 Figure 1. Top 20 run ‐ scorers in all three formats of cricket, according to traditional criteria Figure 1. Top 20 run-scorers in all three formats of cricket, according to traditional criteria.

Sustainability 2023 , 15 , 3201 6 of 20 Sustainability 2023 , 15 , x FOR PEER REVIEW 7 of 22 Figure 2. Top 20 run wicket takers in all three formats of cricket, according to traditional criteria Figure 2. Top 20 run wicket takers in all three formats of cricket, according to traditional criteria 4.1. Player’s Efficiency in the Batting Department Table 2 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 T 20 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 50 s, 6 s, and 4 s hit during an inning are also important indicators of performance evaluation.

Sustainability 2023 , 15 , 3201 7 of 20 Sustainability 2023 , 15 , x FOR PEER REVIEW 8 of 22 Figure 3. Top 20 Fielders in all three formats of cricket, according to traditional criteria 4.1. Player’s Efficiency in the Batting Department Table 2 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 in ‐ dicate 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 T 20 scorers, only five are efficient These results prove that only scoring runs is not an accurate indicator of choosing Figure 3. Top 20 Fielders in all three formats of cricket, according to traditional criteria 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 T 20 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 ].

Sustainability 2023 , 15 , 3201 8 of 20 Table 2. Player’s efficiency in the batting department of cricket for all three formats Tests ODIs T 20 s 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 4.2. Player’s Efficiency in the Bowling Department Table 3 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 T 20 s, 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 T 20 cricketing history (2003–2019).

Sustainability 2023 , 15 , 3201 9 of 20 Table 3. Player’s efficiency in the bowling department for all three formats Tests ODIs T 20 s 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 4.3. Player’s Efficiency in the Fielding Department Table 4 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 T 20 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 T 20 s 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.

Sustainability 2023 , 15 , 3201 10 of 20 Table 4. Player’s efficiency in the fielding department for all three formats Tests ODIs T 20 s 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 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 T 20 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 T 20 s. 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 T 20 s, 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 – 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

Sustainability 2023 , 15 , 3201 11 of 20 Bradman. In 2009, he was the first to be inducted posthumously into the ICC Cricket Hall of Fame [ 43 – 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 – 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 200 th 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 Twenty 20 (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 T 20 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

Sustainability 2023 , 15 , 3201 12 of 20 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 800 th 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 Twenty 20 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 – 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 Twenty 20 World Championship tournaments regarding wickets taken and bowled. After Saeed Ajmal, Umar Gul had 74 dismissals in Twenty 20 Internationals, making him the second-highest wicket-taker of all time. In 2013, he gave the best international Twenty 20 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 T 20 Cup [ 67 – 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

Sustainability 2023 , 15 , 3201 13 of 20 Zealand Twenty 20 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 T 20 s). England’s average batting efficiency score is 0.9362, higher than all testplaying 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 T 20 s, 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 T 20 Batting, India has played in One-Day Internationals (ODIs) and Twenty-Over Internationals (T 20 Is) since 1974 and 2006, respectively (see Table 5 ). Five major I.C.C. events have been won by the team: the Cricket World Cup (1983 and 2011), the ICC T 20 World Cup (2007), and the ICC Champions Trophy (2002) and (2013) They have also placed second in the World Cup (2003), the T 20 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 ].

Sustainability 2023 , 15 , 3201 14 of 20 Table 5. Average player’s batting efficiency scores for each country Batting Test ODIs T 20 s 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 Table 6 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 T 20 s, 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 1970 s to the early 1990 s. 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 T 20 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 Twenty 20 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 T 20 World Cups. Pakistan is the only team to have ever won the ICC Cricket World Cup (1992), ICC T 20 World Cup (2009), ICC Champions Trophy (2017), and ICC Test Championship (2016) [ 80 ].

Sustainability 2023 , 15 , 3201 15 of 20 Table 6. Average player’s bowling efficiency scores for each country Bowling Test ODIs T 20 s 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 Table 7 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 T 20 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 Table 7. Average player’s fielding efficiency scores for each country Fielding Test ODIs T 20 s 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 In contrast, the most efficient team in T 20 s 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 20 th 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 1960 s, 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

Sustainability 2023 , 15 , 3201 16 of 20 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 – 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, 100 s scored, and 50 s 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, 100 s scored, and 50 s scored are used as outputs in the ODIs. For T 20 s, we used inns played, balls faced as inputs and total runs scored, batting average, strike rate, the 50 s Scored, 6 s hit, and 4 s 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 T 20 s, 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 T 20 s 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

Sustainability 2023 , 15 , 3201 17 of 20 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 T 20 s, 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 T 20 s, 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 T 20 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 Author Contributions: Conceptualization, W.Y. and W.U.H.S.; methodology, Z.Y.; software, W.U.H.S All authors have read and agreed to the published version of the manuscript Funding: This research received no external funding Institutional Review Board Statement: Not applicabe Informed Consent Statement: Not applicabe Data Availability Statement: Player’s data is available at Kaggle (Cricket data) Website: https://www. kaggle.com/datasets/mahendran 1/icc-cricket?resource=download (accessed on 1 January 2022) Conflicts of Interest: The authors declare no conflict of interest References 1 International Cricket Council. Available online: https://www.icc-cricket.com/about/the-icc/our-vision (accessed on 28 January 2023) 2 Bandyopadhyay, K. Cricket, terrorism and security in contemporary South Asia Sport Soc 2021 , 24 , 1352–1371. [ CrossRef ] 3 Malcolm, D.; Naha, S. Cricket at the beginning of the long twenty-first century Sport Soc 2021 , 24 , 1267–1273. [ CrossRef ] 4 Bhattacharjee, D.; Saikia, H. On Performance Measurement of Cricketers and Selecting an Optimum Balanced Team Int. J Perform. Anal. Sport 2014 , 14 , 262–275. [ CrossRef ] 5 Bhattacharjee, D.; Saikia, H. An Objective Approach of Balanced Cricket Team Selection Using Binary Integer Programming Method OPSEARCH 2016 , 53 , 225–247. [ CrossRef ] 6 ICC Men’s Test Player Rankings|ICC. Available online: https://www.icc-cricket.com/rankings/mens/player-rankings/test (accessed on 28 January 2023) 7 Barr, G.D.I.; Kantor, B.S. A Criterion for Comparing and Selecting Batsmen in Limited Overs Cricket J. Oper. Res. Soc 2004 , 55 , 1266–1274. [ CrossRef ] 8 Gerber, H.; Sharp, G.D. Selecting a Limited Overs Cricket Squad Using an Integer Programming Model South Afr. J. Res. Sport Phys. Educ. Recreat 2006 , 28 , 81–90. [ CrossRef ] 9 Norman, J.M.; Clarke, S.R. Optimal Batting Orders in Cricket J. Oper. Res. Soc 2010 , 61 , 980–986. [ CrossRef ] 10 Shanto, S.I.; Awan, N. A Sequential Principal Component-Based Algorithm for Optimal Lineup and Batting Order Selection in One Day International Cricket for Bangladesh Int. J. Perform. Anal. Sport 2019 , 19 , 567–583. [ CrossRef ] 11 Perera, H.; Davis, J.; Swartz, T.B. Optimal Lineups in Twenty 20 Cricket J. Stat. Comput. Simul 2016 , 86 , 2888–2900. [ CrossRef ] 12 Bukiet, B.; Ovens, M. A Mathematical Modelling Approach to One-Day Cricket Batting Orders J. Sports Sci. Med 2006 , 5 , 495–502 Available online: https://pubmed.ncbi.nlm.nih.gov/24357943/ (accessed on 28 January 2023) 13 Norman, J.M.; Clarke, S.R. Dynamic Programming in Cricket: Optimizing Batting Order for a Sticky Wicket J. Oper. Res. Soc 2007 , 58 , 1678–1682. [ CrossRef ] 14 Clarke, S.R.; Norman, J.M. Dynamic Programming in Cricket: Choosing a Night Watchman J. Oper. Res. Soc 2003 , 54 , 838–845 [ CrossRef ]

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