# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021053
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## Efficiency, RTS, and marginal returns from salary on the performance of the NBA players: A parallel DEA network with shared inputs

 1 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China 2 School of Management, University of Science and Technology of China, Hefei 230026, China 3 Government College University Faisalabad, Punjab, Pakistan

* Corresponding author: Muhammad Salman Mansoor

Received  August 2020 Revised  December 2020 Early access March 2021

National Basketball Association (NBA) is one of the popular sports leagues worldwide and is also a business source that generates enormous financial resources. Generally, the salary of sports players is associated with their performance in the field. However, the NBA players' performance in the game is related to specific technical features in the offensive and defensive activities. This paper aims to measure the impact of NBA players' salary on their efficiency levels using a big data set of eleven seasons (2604 players from 2005 to 2016) by considering the players' performance in offensive and defensive activities. First, we propose models to measure players' overall, offensive, and defensive efficiencies based on a non-homogeneous parallel data envelopment analysis (DEA) network. Then, we introduce input-output oriented network models to estimate the marginal returns from salary on the outcomes of both offensive and defensive activities. Results indicated that all players' average overall efficiency is low (63.5%), with 17 efficient players. The offensive efficiency is 12.8% higher than the defensive efficiency. When the impact of salary on offensive (defensive) activity is considered, about 73% (47%) of the players' observations indicate increasing marginal returns, respectively.

Citation: Saeed Assani, Muhammad Salman Mansoor, Faisal Asghar, Yongjun Li, Feng Yang. Efficiency, RTS, and marginal returns from salary on the performance of the NBA players: A parallel DEA network with shared inputs. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021053
##### References:
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Zajac, Game indicators determining sports performance in the NBA, Journal of Human Kinetics, 37 (2013), 145-151.  doi: 10.2478/hukin-2013-0035.  Google Scholar [22] P. Moreno and S. Lozano, A network DEA assessment of team efficiency in the NBA, Annals of Operations Research, 214 (2014), 99-124.  doi: 10.1007/s10479-012-1074-9.  Google Scholar [23] A. Stefaniec, K. Hosseini, J. Xie and Y. Li, Sustainability assessment of inland transportation in China: A triple bottom line-based network DEA approach, Transportation Research Part D: Transport and Environment, 80 (2020), 102258. doi: 10.1016/j. trd. 2020.102258.  Google Scholar [24] G. Villa and S. Lozano, Dynamic network DEA approach to basketball games efficiency, Journal of the Operational Research Society, 69 (2018), 1738-1750.  doi: 10.1080/01605682.2017.1409158.  Google Scholar [25] M. Yang, Y. Wei, L. Liang, J. Ding and X. Wang, Performance evaluation of NBA teams: A non-homogeneous DEA approach, Journal of the Operational Research Society, (2020), 1–12. doi: 10.1080/01605682.2020.1718560.  Google Scholar [26] L. Zhang and K. Chen, Hierarchical network systems: An application to high-technology industry in China, Omega, 82 (2019), 118-131.  doi: 10.1016/j.omega.2017.12.007.  Google Scholar

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##### References:
 [1] S. Assani, J. Jiang, A. Assani and F. Yang, Scale efficiency of China's regional R & D value chain: A double frontier network DEA approach, Journal of Industrial & Management Optimization, 17 (2021), 1357-1382.  doi: 10.3934/jimo.2020025.  Google Scholar [2] S. Assani, J. Jiang, A. Assani and F. Yang, Most productive scale size of China's regional R & D value chain: A mixed structure network, preprint, arXiv: 1910.03805. Google Scholar [3] S. Assani, J. Jiang, R. Cao and F. Yang, Most productive scale size decomposition for multi-stage systems in data envelopment analysis, Computers and Industrial Engineering, 120 (2018), 279-287.  doi: 10.1016/j.cie.2018.04.043.  Google Scholar [4] S. Assani and M. S. Mansoor, Salary, offensive, and defensive stats of 2604 NBA players over 11 seasons (2005-2016), Mendeley Data, V1 (2020). doi: 10.17632/fm86gnkw6x. 1.  Google Scholar [5] J. E. Boscá, V. Liern, A. Martínez and R. Sala, Increasing offensive or defensive efficiency? An analysis of Italian and Spanish football, Omega, 37 (2009), 63-78.  doi: 10.1016/j.omega.2006.08.002.  Google Scholar [6] A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8.  Google Scholar [7] Y. Chen, Y. Gong and X. Li, Evaluating NBA player performance using bounded integer data envelopment analysis, INFOR: Information Systems and Operational Research, 55 (2017), 38-51.  doi: 10.1080/03155986.2016.1262581.  Google Scholar [8] W. W. Cooper, J. L. Ruiz and I. Sirvent, Selecting non-zero weights to evaluate effectiveness of basketball players with DEA, European Journal of Operational Research, 195 (2009), 563-574.  doi: 10.1016/j.ejor.2008.02.012.  Google Scholar [9] Ó. Gutiérrez and J. L. Ruiz, Data envelopment analysis and cross-efficiency evaluation in the management of sports teams: The assessment of game performance of players in the Spanish handball league, Journal of Sport Management, 27 (2013), 217-229.  doi: 10.1123/jsm.27.3.217.  Google Scholar [10] C. -K. Hu, F. -B. Liu, H. -M. Chen and C. -F. Hu, Network data envelopment analysis with fuzzy non-discretionary factors, Journal of Industrial & Management Optimization. doi: 10.3934/jimo. 2020046.  Google Scholar [11] C. Kao, Efficiency decomposition for general multi-stage systems in data envelopment analysis, European Journal of Operational Research, 232 (2014), 117-124.  doi: 10.1016/j.ejor.2013.07.012.  Google Scholar [12] C. Kao, Network data envelopment analysis: A review, European Journal of Operational Research, 239 (2014), 1-16.  doi: 10.1016/j.ejor.2014.02.039.  Google Scholar [13] C. Kao and S.-T. Liu, Cross efficiency measurement and decomposition in two basic network systems, Omega, 83 (2019), 70-79.  doi: 10.1016/j.omega.2018.02.004.  Google Scholar [14] C. Kao and S.-N. Hwang, Decomposition of technical and scale efficiencies in two-stage production systems, European Journal of Operational Research, 211 (2011), 515-519.  doi: 10.1016/j.ejor.2011.01.010.  Google Scholar [15] H. Katayama and H. Nuch, A game-level analysis of salary dispersion and team performance in the national basketball association, Applied Economics, 43 (2011), 1193-1207.  doi: 10.1080/00036840802600335.  Google Scholar [16] B. L. Lee and A. C. Worthington, A note on the 'Linsanity' of measuring the relative efficiency of National Basketball association guards, Applied Economics, 45 (2013), 4193-4202.  doi: 10.1080/00036846.2013.770125.  Google Scholar [17] Y. Li, L. Wang and F. Li, A data-driven prediction approach for sports team performance and its application to National Basketball Association, Omega, 98 (2021), 102-123.  doi: 10.1016/j.omega.2019.102123.  Google Scholar [18] Y. Li, X. Lei and A. Morton, Performance evaluation of nonhomogeneous hospitals: the case of Hong Kong hospitals, Health Care Management Science, 22 (2019), 215-228.  doi: 10.1007/s10729-018-9433-y.  Google Scholar [19] Y. Li, X. Shi, A. Emrouznejad and L. Liang, Environmental performance evaluation of Chinese industrial systems: A network SBM approach, Journal of the Operational Research Society, 69 (2018), 825-839.  doi: 10.1057/s41274-017-0257-9.  Google Scholar [20] R. Lyons, E. N. Jackson and A. Livingston, Determinants of NBA player salaries, The Sport Journal, 18 (2015). doi: 10.17682/sportjournal/2015.019.  Google Scholar [21] K. Mikolajec, A. Maszczyk and T. Zajac, Game indicators determining sports performance in the NBA, Journal of Human Kinetics, 37 (2013), 145-151.  doi: 10.2478/hukin-2013-0035.  Google Scholar [22] P. Moreno and S. Lozano, A network DEA assessment of team efficiency in the NBA, Annals of Operations Research, 214 (2014), 99-124.  doi: 10.1007/s10479-012-1074-9.  Google Scholar [23] A. Stefaniec, K. Hosseini, J. Xie and Y. Li, Sustainability assessment of inland transportation in China: A triple bottom line-based network DEA approach, Transportation Research Part D: Transport and Environment, 80 (2020), 102258. doi: 10.1016/j. trd. 2020.102258.  Google Scholar [24] G. Villa and S. Lozano, Dynamic network DEA approach to basketball games efficiency, Journal of the Operational Research Society, 69 (2018), 1738-1750.  doi: 10.1080/01605682.2017.1409158.  Google Scholar [25] M. Yang, Y. Wei, L. Liang, J. Ding and X. Wang, Performance evaluation of NBA teams: A non-homogeneous DEA approach, Journal of the Operational Research Society, (2020), 1–12. doi: 10.1080/01605682.2020.1718560.  Google Scholar [26] L. Zhang and K. Chen, Hierarchical network systems: An application to high-technology industry in China, Omega, 82 (2019), 118-131.  doi: 10.1016/j.omega.2017.12.007.  Google Scholar
Classical parallel structure
The non-homogeneous parallel network of NBA player activities
The tendency of efficiency values for LeBron James in three different seasons
Salary averages of players' activities based on efficiency's scores
Summary of inputs and outputs descriptive statistics of NBA players
 Variables Mean S.D. Min Max Inputs Minutes played 1923 573 1000 3424 Salary 6224124 5107788 160244 30453805 Outputs Offensive activity Assists 180 145 6 925 Offensive rebounds 85 65 5 440 Field goals 308 146 53 978 Free throws 153 110 9 756 Defensive activity Defensive rebounds 248 131 39 882 Steals 60 31 7 217 Blocks 38 37 1 285
 Variables Mean S.D. Min Max Inputs Minutes played 1923 573 1000 3424 Salary 6224124 5107788 160244 30453805 Outputs Offensive activity Assists 180 145 6 925 Offensive rebounds 85 65 5 440 Field goals 308 146 53 978 Free throws 153 110 9 756 Defensive activity Defensive rebounds 248 131 39 882 Steals 60 31 7 217 Blocks 38 37 1 285
Efficiency evaluation and RTS of NBA players from 2005-2016
 Season Network BCC Models (2) and (3) RTS Overall Offensive Defensive IRTS DRTS 05-06 0.6440 0.6610 0.6281 54.8% 45.2% 06-07 0.6387 0.6584 0.6168 52.3% 47.7% 07-08 0.6415 0.6620 0.6020 52.8% 47.2% 08-09 0.6413 0.6738 0.5957 53.1% 46.9% 09-10 0.6402 0.6850 0.5916 53.4% 46.6% 10-11 0.6277 0.6724 0.5768 55.9% 44.1% 11-12 0.6134 0.6690 0.5741 54.6% 45.4% 12-13 0.6148 0.6570 0.5764 55.6% 44.4% 13-14 0.6254 0.6846 0.5864 53.5% 46.5% 14-15 0.6430 0.6969 0.6068 66.7% 33.3% 15-16 0.6537 0.7033 0.6265 64.3% 35.7% Average 0.6349 0.6749 0.5983 58.6% 41.4%
 Season Network BCC Models (2) and (3) RTS Overall Offensive Defensive IRTS DRTS 05-06 0.6440 0.6610 0.6281 54.8% 45.2% 06-07 0.6387 0.6584 0.6168 52.3% 47.7% 07-08 0.6415 0.6620 0.6020 52.8% 47.2% 08-09 0.6413 0.6738 0.5957 53.1% 46.9% 09-10 0.6402 0.6850 0.5916 53.4% 46.6% 10-11 0.6277 0.6724 0.5768 55.9% 44.1% 11-12 0.6134 0.6690 0.5741 54.6% 45.4% 12-13 0.6148 0.6570 0.5764 55.6% 44.4% 13-14 0.6254 0.6846 0.5864 53.5% 46.5% 14-15 0.6430 0.6969 0.6068 66.7% 33.3% 15-16 0.6537 0.7033 0.6265 64.3% 35.7% Average 0.6349 0.6749 0.5983 58.6% 41.4%
Original and efficient inputs and outputs for LeBron James${}^{15-16}$
 MP SLR AST ORB FG FT DRB STL BLK Original 2709 22970500 514 111 737 359 454 104 49 Efficient 2581 21883995 545 118 781 381 481 110 52 Referent players for offensive process Kevin Durant${}^{09-10}$ $(\lambda =0.7412)$, Kobe Bryant${}^{05-06}$ $(\lambda =0.2588)$ Referent players for defensive process Andre Drummond${}^{15-16}$ $(\lambda =0.6834)$, Chris Paul${}^{07-08}$ $(\lambda =0.3166)$
 MP SLR AST ORB FG FT DRB STL BLK Original 2709 22970500 514 111 737 359 454 104 49 Efficient 2581 21883995 545 118 781 381 481 110 52 Referent players for offensive process Kevin Durant${}^{09-10}$ $(\lambda =0.7412)$, Kobe Bryant${}^{05-06}$ $(\lambda =0.2588)$ Referent players for defensive process Andre Drummond${}^{15-16}$ $(\lambda =0.6834)$, Chris Paul${}^{07-08}$ $(\lambda =0.3166)$
Marginal returns from salary on offensive and defensive activities of NBA players
 Season Impact of salary on offensive Impact of salary on defensive Increase Constant Decrease Increase Constant Decrease 05-06 64.00% 5.00% 31.00% 46.00% 6.00% 48.00% 06-07 70.00% 8.00% 22.00% 51.00% 4.00% 45.00% 07-08 72.00% 3.00% 25.00% 48.00% 7.00% 45.00% 08-09 71.00% 1.00% 28.00% 44.57% 1.00% 54.43% 09-10 73.00% 0.00% 27.00% 51.00% 0.00% 49.00% 10-11 77.78% 0.00% 22.22% 52.00% 0.00% 48.00% 11-12 75.00% 0.00% 25.00% 49.00% 0.00% 51.00% 12-13 72.00% 0.00% 28.00% 43.00% 0.00% 57.00% 13-14 73.00% 0.00% 27.00% 42.00% 0.00% 58.00% 14-15 76.00% 0.00% 24.00% 45.00% 0.00% 55.00% 15-16 77.00% 0.00% 23.00% 47.00% 0.00% 53.00% Mean 72.80% 1.55% 25.66% 47.14% 1.64% 51.22%
 Season Impact of salary on offensive Impact of salary on defensive Increase Constant Decrease Increase Constant Decrease 05-06 64.00% 5.00% 31.00% 46.00% 6.00% 48.00% 06-07 70.00% 8.00% 22.00% 51.00% 4.00% 45.00% 07-08 72.00% 3.00% 25.00% 48.00% 7.00% 45.00% 08-09 71.00% 1.00% 28.00% 44.57% 1.00% 54.43% 09-10 73.00% 0.00% 27.00% 51.00% 0.00% 49.00% 10-11 77.78% 0.00% 22.22% 52.00% 0.00% 48.00% 11-12 75.00% 0.00% 25.00% 49.00% 0.00% 51.00% 12-13 72.00% 0.00% 28.00% 43.00% 0.00% 57.00% 13-14 73.00% 0.00% 27.00% 42.00% 0.00% 58.00% 14-15 76.00% 0.00% 24.00% 45.00% 0.00% 55.00% 15-16 77.00% 0.00% 23.00% 47.00% 0.00% 53.00% Mean 72.80% 1.55% 25.66% 47.14% 1.64% 51.22%
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