# 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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##### References:
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%