The document summarizes a model of optimal shot selection in the NBA proposed by Matt Goldman and Justin Rao. The model treats each possession as a dynamic optimal stopping problem, with two requirements: 1) shots must have expected value exceeding continuing the possession, and 2) players' shot frequencies must generate equal marginal productivity. The model represents potential shot value as uniformly distributed, with players shooting if the value exceeds a cut threshold. It generates predictions about efficiency, usage rates, and their relationship that can be tested with data.

1. Dynamic and Allocative Efficiency in NBA
Decision Making
Part 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. Introduction and Overview
Winning 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. The Anatomy of a Possession
Optimal 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. 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. Intuition
Choosing 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. Intuition
Choosing 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. Intuition
Choosing 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. Intuition
Choosing 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. Intuition
Choosing 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. Intuition
Choosing 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. Intuition
Choosing 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. Intuition
Choosing 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. Model
Stochastic 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. Model
Stochastic 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. Model
Stochastic 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. Model
Stochastic 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. 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. 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. 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. 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. 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. 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. 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. Allocative Efficiency
Our 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. Allocative Efficiency
Results
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. Allocative Efficiency
Results
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. Allocative Efficiency
Results
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. 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. 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. 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. 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