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# Allocation and Dynamic Efficiency in NBA Decision Making

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2011 5th MIT Sloan Sports Analytics Conference

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### Allocation and Dynamic Efficiency in NBA Decision Making

1. 1. Dynamic and Allocative Eﬃciency in NBA Decision MakingPart of a longer, forthcoming paper entitled:“He Got Game Theory” Matt Goldman1 and Justin M. Rao2 1 Department of Economics University of California, San Diego 2 Yahoo! Research Labs March 5, 2005 MIT Sloan Sports Analytics Conference Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
2. 2. Introduction and OverviewWinning is All About Eﬃciency Teams have roughly equal possessions per game. The more eﬃcient team will always win (Oliver (2002), James(1977)). Team’s problem: allocate scarce possessions between teammates and across the shot clock to maximize points. We attack this problem using game theory and optimal stopping. To do so, we need estimates of what would happen to a particular player’s eﬃciency if he chose to increase or decrease his usage. Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
3. 3. The Anatomy of a PossessionOptimal Stopping Under the Pressure of the Shot Clock At each shot clock interval the choice is between using and waiting for a better opportunity As shot clock goes to 0, the value of continuing the possession declines and players must shoot much more frequently. Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
4. 4. Two Requirements of Optimal Shot Selection We model a half-court possession as a dynamic optimal stopping problem, this leads to two fundamental requirements: 1 Dynamic eﬃciency : a shot is realized only if its expected value exceeds the continuation value of a possession 2 Allocative eﬃciency : The frequency at which each player shoots generates equal marginal productivity Eﬀective randomization over who shoots is a best-response to selective defensive pressure It ensures the team could not reallocate to more eﬃcient players on the margin (which would increase output) Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
5. 5. IntuitionChoosing From Feasible Combinations of Eﬃciency and Usage Intuition: We observe ei,t and ui,t for each player in 18 diﬀerent periods of the shot clock. The ﬁtted red line is an estimate of what eﬃciency-usage combinations are possible for the given player. The green line indicates the value of the marginal shot a player must be willing to take to reach any given level of usage. Figure 1: ’Usage Curve’ for the aggregate of all NBA Centers Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
6. 6. IntuitionChoosing From Feasible Combinations of Eﬃciency and Usage Intuition: We observe ei,t and ui,t for each player in 18 diﬀerent periods of the shot clock. The ﬁtted red line is an estimate of what eﬃciency-usage combinations are possible for the given player. The green line indicates the value of the marginal shot a player must be willing to take to reach any given level of usage. Figure 1: ’Usage Curve’ for the aggregate of all NBA Centers Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
7. 7. IntuitionChoosing From Feasible Combinations of Eﬃciency and Usage Intuition: We observe ei,t and ui,t for each player in 18 diﬀerent periods of the shot clock. The ﬁtted red line is an estimate of what eﬃciency-usage combinations are possible for the given player. The green line indicates the value of the marginal shot a player must be willing to take to reach any given level of usage. Figure 1: ’Usage Curve’ for the aggregate of all NBA Centers Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
8. 8. IntuitionChoosing From Feasible Combinations of Eﬃciency and Usage Intuition: We observe ei,t and ui,t for each player in 18 diﬀerent periods of the shot clock. The ﬁtted red line is an estimate of what eﬃciency-usage combinations are possible for the given player. The green line indicates the value of the marginal shot a player must be willing to take to reach any given level of usage. Figure 1: ’Usage Curve’ for the aggregate of all NBA Centers Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
9. 9. IntuitionChoosing From Feasible Combinations of Eﬃciency and Usage Intuition: We observe ei,t and ui,t for each player in 18 diﬀerent periods of the shot clock. The ﬁtted red line is an estimate of what eﬃciency-usage combinations are possible for the given player. The green line indicates the value of the marginal shot a player must be willing to take to reach any given level of usage. Figure 1: ’Usage Curve’ for the aggregate of all NBA Power Forwards Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
10. 10. IntuitionChoosing From Feasible Combinations of Eﬃciency and Usage Intuition: We observe ei,t and ui,t for each player in 18 diﬀerent periods of the shot clock. The ﬁtted red line is an estimate of what eﬃciency-usage combinations are possible for the given player. The green line indicates the value of the marginal shot a player must be willing to take to reach any given level of usage. Figure 1: ’Usage Curve’ for the aggregate of all NBA Small Forwards Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
11. 11. IntuitionChoosing From Feasible Combinations of Eﬃciency and Usage Intuition: We observe ei,t and ui,t for each player in 18 diﬀerent periods of the shot clock. The ﬁtted red line is an estimate of what eﬃciency-usage combinations are possible for the given player. The green line indicates the value of the marginal shot a player must be willing to take to reach any given level of usage. Figure 1: ’Usage Curve’ for the aggregate of all NBA Shooting Guards Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
12. 12. IntuitionChoosing From Feasible Combinations of Eﬃciency and Usage Intuition: We observe ei,t and ui,t for each player in 18 diﬀerent periods of the shot clock. The ﬁtted red line is an estimate of what eﬃciency-usage combinations are possible for the given player. The green line indicates the value of the marginal shot a player must be willing to take to reach any given level of usage. Figure 1: ’Usage Curve’ for the aggregate of all NBA Point Guards Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
13. 13. ModelStochastic Shot Arrivals In each one-second period, with t seconds remaining, player i observes potential shot value: η ∼ Uniform(Bi , Ai ). We assume this distribution is constant across all periods of the shot clock. Player i shoots if and only if η > ci,t (the cut threshold). From such a player we should observe: A+ci,t Eﬃciency: ei,t = 2 Usage Hazard Rate (Probability that player i uses the possession, A−ci,t given that his team has the ball): ui,t = Ai −Bi dei −(Ai −Bi ) ’Usage Curve’ Slope: dui = 2 Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
14. 14. ModelStochastic Shot Arrivals In each one-second period, with t seconds remaining, player i observes potential shot value: η ∼ Uniform(Bi , Ai ). We assume this distribution is constant across all periods of the shot clock. Player i shoots if and only if η > ci,t (the cut threshold). From such a player we should observe: A+ci,t Eﬃciency: ei,t = 2 Usage Hazard Rate (Probability that player i uses the possession, A−ci,t given that his team has the ball): ui,t = Ai −Bi dei −(Ai −Bi ) ’Usage Curve’ Slope: dui = 2 Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
15. 15. ModelStochastic Shot Arrivals In each one-second period, with t seconds remaining, player i observes potential shot value: η ∼ Uniform(Bi , Ai ). We assume this distribution is constant across all periods of the shot clock. Player i shoots if and only if η > ci,t (the cut threshold). From such a player we should observe: A+ci,t Eﬃciency: ei,t = 2 Usage Hazard Rate (Probability that player i uses the possession, A−ci,t given that his team has the ball): ui,t = Ai −Bi dei −(Ai −Bi ) ’Usage Curve’ Slope: dui = 2 Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
16. 16. ModelStochastic Shot Arrivals In each one-second period, with t seconds remaining, player i observes potential shot value: η ∼ Uniform(Bi , Ai ). We assume this distribution is constant across all periods of the shot clock. Player i shoots if and only if η > ci,t (the cut threshold). From such a player we should observe: A+ci,t Eﬃciency: ei,t = 2 Usage Hazard Rate (Probability that player i uses the possession, A−ci,t given that his team has the ball): ui,t = Ai −Bi dei −(Ai −Bi ) ’Usage Curve’ Slope: dui = 2 Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
17. 17. Dynamic Eﬃciency All player’s should chose ci,t equal to the value of continuing the possession. If they don’t, they are throwing away points for their team. Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
18. 18. Dynamic Eﬃciency All player’s should chose ci,t equal to the value of continuing the possession. If they don’t, they are throwing away points for their team. Figure 2: OVERSHOOTING! Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
19. 19. Dynamic Eﬃciency All player’s should chose ci,t equal to the value of continuing the possession. If they don’t, they are throwing away points for their team. Figure 2: UNDERSHOOTING! Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
20. 20. Dynamic Eﬃciency: Results Figure shows that, on average, this is nearly the case for ALL periods. NBA players understand Dynamic Eﬃciency very well. Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
21. 21. Undershooting and Overshooting We preformed a t-test of every player’s adherence to Dynamic Eﬃciency (negative is overshooting; positive is undershooting) Overshooting is very rare, undershooting is more common. Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
22. 22. Undershooting and Overshooting We preformed a t-test of every player’s adherence to Dynamic Eﬃciency (negative is overshooting; positive is undershooting) Overshooting is very rare, undershooting is more common. Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
23. 23. Who Overshoots? Who Undershoots? Top 7 Overshooters and Undershooters (by t-statistic) Overshooter t Undershooter t Russell Westbrook -2.97838 Chris Paul 5.341594 Tyrus Thomas -1.983194 Brandon Roy 5.127828 Lamar Odom -1.942873 LeBron James 4.931574 Monta Ellis -1.908993 Al Jeﬀerson 4.623833 Larry Hughes -1.802382 Joe Johnson 4.437453 Drew Gooden -1.790609 Amare Stoudemire 4.061478 Tracy McGrady -1.770439 Vince Carter 4.0012 Undershooters are primarily elite oﬀensive players who may be sacriﬁcing immediate preformance in order to conserve energy. High salary players are more likely to undershoot (t=2.75). Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
24. 24. Allocative EﬃciencyOur Statistical Test All teammates in a given line-up should choose a ’worst shot’ (ci,t ) of the same value in each shot clock period. This ensures the team cannot reallocate to more productive players (on the margin) and increase output. Deviation from this standard is measured by Spread (deﬁned in the paper). We use concept of 3-man (“most important 3 of 5-man line-up”) and 4-man cores to increase statistical power. Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
25. 25. Allocative EﬃciencyResults A team that Allocates perfectly will have Spread = 0 in expectation. The majority of line-ups achieve Allocative Eﬃciency, but it is very clear that some core’s are imperfect(t = 8.05). Additionally less experienced (t = 2.70) and higher salary (t = 3.96) ’3-man cores’ Figure 3: Histogram of observed spread amongst all ’3 man cores’ demonstrate larger deviations from Allocative Eﬃciency. Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
26. 26. Allocative EﬃciencyResults A team that Allocates perfectly will have Spread = 0 in expectation. The majority of line-ups achieve Allocative Eﬃciency, but it is very clear that some core’s are imperfect(t = 8.05). Additionally less experienced (t = 2.70) and higher salary (t = 3.96) ’3-man cores’ Figure 3: Histogram of observed spread amongst all ’3 man cores’ demonstrate larger deviations from Allocative Eﬃciency. Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
27. 27. Allocative EﬃciencyResults A team that Allocates perfectly will have Spread = 0 in expectation. The majority of line-ups achieve Allocative Eﬃciency, but it is very clear that some core’s are imperfect(t = 8.05). Additionally less experienced (t = 2.70) and higher salary (t = 3.96) ’3-man cores’ Figure 3: Histogram of observed spread amongst all ’3 man cores’ demonstrate larger deviations from Allocative Eﬃciency. Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
28. 28. Player Usage Curves The results of our parametric procedure yield estimates of each player’s abillity to create shots on the margin: ’Usage Curve’ Slope: duii = −(Ai2−Bi ) de What kinds of players have ﬂatter usage curves? Table: Robust OLS Regression Explaining the Usage Curve LHS/RHS PG SG SF PF C Usage % ˆ ˆ A i −B i 2 2.51* 2.87* 2.80* 3.15* 3.26* -4.32* s.e. 0.26 0.30 0.29 0.31 0.33 0.97 R 2 = 0.8465, ∗p <0.01 Perimeter players and high-usage players have signiﬁcantly ﬂatter usgae curves. A player who wants to use an additional 1% of his team’s half-court oﬀense possessions, will see his overall eﬃciency drop by .0025 − .006 points per possession. This estimate holds teammates constant. Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
29. 29. Player Usage Curves The results of our parametric procedure yield estimates of each player’s abillity to create shots on the margin: ’Usage Curve’ Slope: duii = −(Ai2−Bi ) de What kinds of players have ﬂatter usage curves? Table: Robust OLS Regression Explaining the Usage Curve LHS/RHS PG SG SF PF C Usage % ˆ ˆ A i −B i 2 2.51* 2.87* 2.80* 3.15* 3.26* -4.32* s.e. 0.26 0.30 0.29 0.31 0.33 0.97 R 2 = 0.8465, ∗p <0.01 Perimeter players and high-usage players have signiﬁcantly ﬂatter usgae curves. A player who wants to use an additional 1% of his team’s half-court oﬀense possessions, will see his overall eﬃciency drop by .0025 − .006 points per possession. This estimate holds teammates constant. Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
30. 30. Player Usage Curves The results of our parametric procedure yield estimates of each player’s abillity to create shots on the margin: ’Usage Curve’ Slope: duii = −(Ai2−Bi ) de What kinds of players have ﬂatter usage curves? Table: Robust OLS Regression Explaining the Usage Curve LHS/RHS PG SG SF PF C Usage % ˆ ˆ A i −B i 2 2.51* 2.87* 2.80* 3.15* 3.26* -4.32* s.e. 0.26 0.30 0.29 0.31 0.33 0.97 R 2 = 0.8465, ∗p <0.01 Perimeter players and high-usage players have signiﬁcantly ﬂatter usgae curves. A player who wants to use an additional 1% of his team’s half-court oﬀense possessions, will see his overall eﬃciency drop by .0025 − .006 points per possession. This estimate holds teammates constant. Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making
31. 31. Conclusion Thanks for your time! If you liked our presentation, come check out our blog at www.hooptheory.com Our contact info: Matt Goldman: mrgoldman@ucsd.edu Justin M. Rao jmrao@yahoo-inc.com Goldman and Rao Dynamic and Allocative Eﬃciency in NBA Decision Making