Tick-Tock Shot Clock: Optimal Stopping in the NBA - UC San Diego ...

Tick-Tock Shot Clock: Optimal Stopping in the NBA ∗
Matt Goldman 1 Justin M. Rao 2
1 Department of Economics, University of California, San Diego
2 Yahoo! Research, Santa Clara, CA
Fall 2011
Abstract
The paper examines the optimality of the shooting decisions of National Basketball
Association (NBA) players using a rich dataset of 1.4 million offensive possessions. The
decision to shoot is a complex problem that involves weighing the continuation value of
the possession and the outside option of a teammate shooting. We model this as a dynamic
mixed-strategy equilibrium. At each second of the shot clock, dynamic efficiency
requires that marginal shot value exceeds the continuation value of the possession. Allocative
efficiency is the additional requirement that at that “moment”, each player in
the line-up has equal marginal efficiency. To apply our abstract model to the data we
make assumptions about the distribution of potential shots. We first assume nothing
about the opportunity distribution and establish a strict necessary condition for optimality,
which nearly all players/teams pass. Adding distributional assumptions, we
establish sufficient conditions for optimality. In line with dynamic efficiency, we find
that the “cut threshold” declines monotonically with time remaining on the shot clock
at approximately the correct rate. Most line-ups show strong adherence to allocative
efficiency. We link departures in optimality to line-up experience, player salary and
overall ability.
∗ We would like to thank James Andreoni, David Eil, Pedro Rey Biel and Charlie Sprenger for extensive
notes. Nageeb Ali, Gordon Dahl, Uri Gneezy, Craig McKenzie, David Miller, William Peterman, Joel Sobel
and Joel Watson provided helpful comments. This paper is a merger of Goldman’s “Optimizing in the
Half-Court” and ideas presented in Rao’s “He Got Game Theory: Optimal Decision Making and the NBA”.
1

1 Introduction
Basketball is hard. Indeed one has to be tall, athletic and coordinated. In this paper we
focus on another way basketball is hard. It is a strategic interaction, involving a complex
optimization problem, in which good decision-making is paramount. Professional basketball
is played in a series of 24-second (or less) “possessions.” In each possession the offensive
team has to maximize the number of points scored (save end-of-game situations). In order
to optimize, players must effectively employ two workhorse concepts from game and decision
theory. First, since a shot has to be taken in the 24-second time limit, the team has to solve
an optimal stopping problem, which involves setting thresholds, based on the time remaining
on the shot clock, that determine if a shot opportunity is realized or passed up in favor of
continuing the possession. Second, these thresholds ought to be uniform across players. Or
in other words, the team must share the ball so that marginal productivity at each moment
is equal across players.
Optimizing in basketball thus requires adhering to optimality conditions that are literally
“textbook.” Solving a stopping problem entails both accurately estimating a continuation
value and comparing it to the currently observed outcome, conditional on doing the same
procedure in the next period. This type of reasoning underlies the equilibrium concepts of
nearly all repeated games. Optimal allocation requires distributing a scare resource (shots)
to many factors of production with differing technology (players) in order to maximize output
(points). Sharing the ball is basic producer theory.
For these reasons, we think basketball is an intensely interesting testing ground of fa-
miliar standards of optimality. Moreover, optimizing requires the use of reasoning concepts
that laboratory subjects are notoriously bad at employing, meaning we may have doubts
in people’s ability to solve this type of decision problem in the way predicted by theory. A
further layer of complexity is that appropriately analyzing optimal play in basketball also
presents interesting challenges. NBA players exhibit incredible heterogeneity in both the
ability to score and the chosen technique to accomplish this goal. In order credibly address
shot selection, one must first flexibly model the shot opportunity distribution of each player.
We examine NBA player decision-making using rich data set comprising all shots taken
(over 1.3 million possessions) in the NBA from 2006–2010. Our modeling approach is to
embed allocation within the stopping problem. At each point on the shot clock, the team
must randomize over potential shooters in order to attain equal marginal productivity across
the 5-man lineup currently on the court. We call this allocative efficiency. 1 At each point
1 The vectors of shooting frequency should maximize points per possession. One might think the players
are not consciously randomizing, but instead responding to slight changes in the game. In this case, each
2

in the shot clock, the potential shooter must decide if he should realize the current shooting
opportunity, or pass up the shot. This requires setting the appropriate threshold value for
continuing the possession versus realizing a shot opportunity. If the shot opportunity is
perceived to exceed this value, the shot is taken. We call this dynamic efficiency.
Naturally it is challenging to fit a complex game such as basketball into a tractable
model, but we show that we can establish necessary and sufficient conditions for optimality
with minimal assumptions. Like all stopping problems in the field (see Rust (1984) for a
nice discussion), we have to compute the “marginal counterfactual;” based on the shots we
actually observe a player taking, we have to estimate the shot opportunity distribution he
faces, so we can evaluate if he appears to be passing up shots that he should take (to test
dynamic efficiency) and if he should shoot more or less (to test allocative efficiency). Our
identification scheme, which we describe in detail in Section 3, simultaneously illuminates a
player’s ability, by modeling the shot distribution which they draw shooting opportunities
from, and their chosen rule for realizing an opportunities at each point of the shot clock,
depending on line-up.
Our key finding is that NBA players are superb optimizers. The average NBA player is
shown to adopt reservation shot values almost exactly equal to the continuation value of his
team’s possession throughout the entire range of the shot clock. This “cut threshold” declines
monotonically with time remaining on the shot clock at almost the exact rate implied by
dynamic efficiency. Most teams core line-ups show impressive allocative efficiency, with the
spread of marginal efficiencies within a line-up quite small. Very few players can be shown
to individually overshoot from the dynamic (shoot too soon) or allocative (shoot too much
overall) perspective. Undershooting is more common and seems to occur primarily in players
easily recognized as amongst the NBA elite. 2
The finding that NBA players can reliably solve an optimal stopping problem is the first
field (or lab, for that matter) evidence of its kind that we are aware of, with the excep-
tion of large firm-level decisions such as patent renewal (Pakes, 1986) and tree harvesting
(Provencher, 1997). The evidence on optimal stopping in the field, especially with regards to
individual decision making, is thin because the requirements for reliable estimation are stiff.
One must correctly specify preferences with respect to risk and time, and possess rich enough
data to infer the opportunities an agent has observed, but did not execute. For example,
to understand if job hunters terminate employment search using theoretically implied wage
vector implies a “threshold equilibrium” of play, in the spirit of a purified game (Harsanyi, 1973).
2 Such behavior is suboptimal in our model, but is easily rationalized if we allow for such players to conserve
their energy and health for the long-haul of the season at the slight expense of their team’s immediate
performance.
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thresholds, the econometrician would need financial balances, risk/time preferences, and a
record of all offers. Such data are hard to come by given current collection technology and
privacy legislation. In contrast, we show that risk and time neutrality can be established
for the majority of basketball game-play (with the remainder eliminated) and we can model
unrealized opportunities based on the detailed data we have on taken shots. We believe the
study of basketball, while interesting to some readers in its own right, can also lend insight
to the human capacity to solve this sort of problem more generally.
Our analysis also reveals an additional twist of optimal play. In our main estimation,
we only use possessions in which the team ought to focus only on points, in a risk-neutral
fashion. In contrast, towards the end of games, the trailing team does better by lowering
the threshold function, so that they take a worse shots on average, but save time by doing
so. The trailing team also should induce variance, by taking more 3-pointers. Conversely,
the leading time ought to raise the cut threshold, passing up good shots in order to use time
and reduce variance by taking more 2-pointers. We find that NBA players recognize and
respond to this trade-off. The cut-thresholds significantly change in the correct direction
when time becomes intrinsically valuable and the trailing team reliably induces variance
by taking more 3-pointers. There is evidence, however, that the cut-threshold shift is too
conservative, especially for the trailing team.
Our final piece of analysis seeks to understand which factors of a lineup are associated
with optimal play. We link the sufficient statistics from our tests of optimality to line-up
experience (how often the players share the court), player tenure in the league, player salary
and salary inequality using OLS regression. We find that adherence to allocative efficiency
significantly positively related to line-up output and salary level. Increased variance in
line-up salary leads to more optimal play. On the individual level, star players are more
likely to undershoot, perhaps to conserve energy/health of the course of the season. For
both measures, player tenure in the league does not appear to have a meaningful impact
controlling for these other state measures.
The remainder of the paper proceeds as follows. In the following two subsections we
review the related literature and provide a basketball primer, in order to introduce the key
game elements necessary to understand our model of basketball decision making. Section
2 describes the data, Section 3 presents the model in detail and discusses our identifying
assumptions in detail, the key results on optimal play are presented in Section 4, additional
co-variate analysis and a discussion follow in Section 5 and Section 6 concludes.
4

1.1 Related Work
Our work is most closely related to two different literature. First, since optimal stopping
involves the important game theoretic concept continuation value, there is a deep literature
of laboratory experiments investigating people’s ability to effectively employ such reasoning. 3
Early work in this literature used the “classical secretary problem” (CSP). In the CSP, a
subject observes a series of “applicants” and wins a prize if she “hires” the best one. She
can only hire the current applicant and is given only rank order information on the applicant
(how does the applicant compare to those who have come before). A robust finding is that
subjects “under-search” by hiring a candidate to quickly (Kahan et al., 1967; Rapoport and
Tversky, 1970). Recent work that has relaxed many assumptions of the CSP has also found
early stopping (Seale and Rapoport, 2000; Zwick et al., 2003; Bearden et al., 2005; Bearden
et al., 2006). In the full information CSP payments are simply the chosen draw (like a taken
shot). Just like basketball, the solution requires a monotonically declining cut threshold
with draws remaining; yet the majority of subjects use flat thresholds (Lee et al., 2004),
even when the time horizon is short (Lee, 2006). 4 Overall, across a wide variety of stopping
problems, lab subjects typically stop the process early and use fixed, instead of declining
thresholds. In stark contrast, our results indicate NBA players do not stop the early and
use a declining threshold in line with the theoretical optimum.
The second closely related literature examines the predictive power of game theoretic
equilibrium concepts in professional sports. 5 Mixed-strategy Nash Equilibrium (MSNE) has
been tested in the field using 2×2 simultaneous move games analogous to matching pennies.
In soccer penalty kicks (Chiappori et al., 2002; Palacios-Huerta, 2003) and tennis serves
(Walker and Wooders, 2001; Hsu et al., 2007), players typically randomize across strategies
and over time consistent with minimax play. MSNE is examined in baseball and American
football in Kovash and Levitt (2009), with the authors asserting that baseball pitchers do not
3 Past work using experts has studied firm decisions in dynamic settings such as as harvesting tree stands
(Provencher, 1997), renewing patents (Pakes, 1986), replacing bus engines (Rust, 1987) and continuing drives
in professional football (Romer, 2006). These problems are similar to the stylized lab version of the problem
in one respect — the decision is based on the comparison of expected value of acting versus continuing —
but are dissimilar in that they do not involve random arrivals of opportunities (for instance, one can replace
a bus engine at any time). NBA shooting presents a situation that is strikingly similar to the classic version
of the stopping problem.
4 With a very long time horizon, a high constant threshold is not optimal, but does pretty well since with
high likelihood one will get a good draw with this strategy. The short horizon games are a better analog to
basketball.
5 There is of course also a deep literature of testing game theory in laboratories. Three recent papers
present a nice discussion about the interplay between lab and field in MSNE games, and the reader is
pointed there for a deeper dive (Palacios-Huerta and Volij, 2009; Levitt et al., 2010; Palacios-Huerta and
Volij, 2008).
5

andomize over pitch selection properly and football coaches do not pass enough. However,
the analysis is hampered by the difficulty coming up with the correct dependent variables in
these sports. 6
The only work we are aware of that applies an optimal stopping model to professional
sports is Romer (2006), which examines football coaches’ decisions to play versus punt on 4th
down on the NFL. Romer finds that coaches are too conservative on average—they punt too
often. The key difference between this study and ours is that Romer studies a deliberative
decision made about 10 times per season by the coach, not the players. The low number
of occurrences is in contrast to the roughly 8,000 possessions played each year by an NBA
team. These differences, as well as other obvious features of the decision problem in the
two settings, could explain the lack of consonance in the findings of his paper and ours.
Empirical work on firm-level stopping decisions has typically found that studied firms have
dynamically managed the asset in question in close accordance to theoretical predictions
(Pakes, 1986; Provencher, 1997; Rust, 1987).
2 A Basketball Primer and Key Strategic Elements
In this section we give an overview of the strategic elements in the game of basketball as it
is played professionally the NBA. The purpose is to prevent the relevant background before
jumping into the nitty gritty of the model.
2.1 Basketball basics
In NBA play, two twelve-man teams face each other in a 48-minute game. Each team has
five players on the court. A five player combination is referred to as a “lineup” and the
lineup that typically plays the most consists of the players who begin the game for the team,
the “starters.” The team who scores the most points in the game wins. Points are scored by
shooting the ball in the basket. Long-range shots earn three points, normal shots earn two
points and shots taken after a foul (free throws) earn one point each (a player is awarded
6 For baseball, the authors use a metric of batting performance (slugging percentage + on base percentage)
that does not map directly to the probability of winning (conditional on game conditions). Indeed any bias in
the metric as it maps to winning percentage biases the results, and there is good reason to believe such bias
exists. The metric gives equal weight to slugging percentage and on base percentage, but on base percentage
is a better predictor of runs scored. Since this measure overvalues “big hits” and undervalues walks, it is not
surprising the authors find that pitchers use too many pitches that limit walks but risk big hits (fastballs)
and not enough that limit big hits but risk walks (curve balls). For football, the game is not as clearly a
simultaneous move game, because the offense can change the play (“audible”) after seeing the players the
defense has on the field and it is easier (and more common) to audible to a run.
6

the number of free throws corresponding to the value of the shot he was fouled on). In our
sample period, teams scored on average 92.5 points per game.
2.2 The central limit theorem for basketball
Although basketball game-play is fast-paced and (potentially) continuous, 7 it is effectively
broken up into a series of non-overlapping “offensive possessions.” A possession starts when
one teams gets the ball; at that moment the 24-second “shot clock” starts. The offensive
team must shoot the ball within this 24-second interval, failing to do so results in a turnover
(0 points). 8 Each team has about 100 possessions per game—a critical element that allows
us to model the game as a large number of well-defined pieces, which allows for a tractable,
quantitative analysis of game play. Crucially we show here (and more formally in the Ap-
pendix) that the large number of possessions implies teams are best off behaving like a
risk-neutral point maximizer in each possession. The two key aspects of risk neutrally max-
imizing points are: 1) at a given time in the shot clock the team only cares about expected
points (“risk neutrality”) 2) the team does not value time itself, that is, for two shots of equal
expected value, the team is indifferent between when they are taken (“time-invariance”). In
what follows, we examine these two conditions through a simple model of NBA game play.
The goal of the team is to score more points than the opponent in the entire game.
Consider the two teams, home (h) and away (a). Let Sh,N and Sa,N denote the current
scores for the home and away team with N offensive possessions (for each team) remaining
in the game respectively. Let Ph,i and (Pa,i) denote the number of points scored by the home
(away) possession on the i th possession from the end of the game. The home team wins if
they have more points at the end of the game. This equal to the current score, plus the
points scored in subsequent possessions, as given by:
Sh,0 > Sa,0 ⇐⇒ Sh,N +
N
Ph,i > Sa,N +
i=1
N
Pa,i ⇐⇒
i=1
N
Ph,i − Pa,i > Sa,N − Sh,N (1)
To model how points are generated, let {µh, σ 2 h } and {µa, σ 2 a} represent the mean and
variance of points per possession that each team is able to achieve in the match-up. If the
number of remaining possessions, N, is large, the central limit theorem gives probability of
the home team winning as:
7 The game stops for out of bounds, fouls and time-outs.
8 The clock is prominently displayed right above the basket. Shots must hit the rim to count as a shot
attempt, so you cannot fake a shot.
7
i=1

P (Home Win) = P (Sh,0 > Sa,0) = P
N
= Φ
i=1
(Ph,i − Pa,i) > Sa,N − Sh,N
Sh,N − Sa,N + N(µh − µa)
2 N(σh + σ2
a)
where Φ is the CDF of the standard normal distribution. Examining this expression,
we see that if the score is currently tied and the teams are of equal quality, the game is a
coin toss. Having an ability advantage (µ higher than opponent) matters proportional to
the remaining possessions, which is intuitive. If you are the better team, it is more likely
to be reflected in the game outcome when there are many periods remaining. The marginal
impact on winning for each factor can be easily obtained by differentiating equation (3).
The following expression gives the impact on win probability of a point scored for the home
team:
dP (Home Win)
dSh,N
= φ
(Sh,N − Sa,N) + (N(µh − µa))
2 N(σh + σ2
1
2
a)
N(σh + σ2 a)
2.2 (5)
Expression shows that points become increasingly impactful on the game outcome when
the current score is close (Sh,N − Sa,N small) and few possessions remain, while the impact
becomes exceedingly small when the score margin is high; in the appendix we show that
when a point is worth less than .001 wins, teams tend to give up and pull their starters out
of the game. We have no reason to believe basketball players continue to pursue optimal
play after this point and we exclude the corresponding data from our main analysis.
For the risk-neutral and time-invariance conditions to hold it must be the case that
teams care much more about maximizing the efficiency of their possessions instead of trying
to play fast (slow) or have high (low) variance possessions (ex. shooting more 3-pointers).
By comparing ratios of marginal win effects we can determine the appropriate indifference
ratios. For ease of demonstration, we assume the teams are of approximately equal quality
(µh = µa), the basic intuition is unaffected by this simplification, because the impact of score
difference and ability difference (weighted by possessions remaining), is very similar.
The ratio of the value of possessions (extending or shortening game by one possession for
each team) to the value of a point is given by:
8
(2)
(3)
(4)

dP (Home Win)
dN
dP (Home Win)
dSh,N
= 1
2
(Sa,N − Sh,N)
To understand this equation, let’s examine the case when the home team is trailing with
N possessions remaining. Since Sa,N > Sh,N, the numerator is positive. Differentiating
equation (3) with respect to N, one can show that the team values possessions intrinsically
(it needs to catch-up). The relative values of points to possessions is given by equation 6.
Indeed, as the number of possessions remaining grows, this ratio goes to zero at rate N.
Given that an average possession cycle (one for each team) takes about 30 seconds, the
number of remaining possessions need not greatly exceed the score difference before this
quickly becomes a very unattractive tradeoff. Sacrificing a meaningful amount of points (on
the order of 0.005) to save a few seconds is a losing proposition. In our empirical analysis we
eliminate observations for which the terms of this tradeoff imply that seconds are relatively
valuable as compared points. The threshold we use entails that if one team is trailing by 10
points, we eliminate all observations when less than 33 possessions remain.
The ratio between mean efficiency and variance presents a similar tradeoff:
dP (Home Win)
dσ 2 h
dP (Home Win)
dµh
=
1
σ 2 h + σ2 a
N
(Sa,N − Sh,N)
N
≈ 1
2
(Sa,N − Sh,N)
In terms of variance, the choice is essentially between 2-pointers, which offer a variance
of about 1 and 3-pointers, which give variance about 2. The trailing team would like to
increase variance, 9 what this equation says is that switching into 3-pointers, which would
lead to a drop in efficiency because of defensive response and equilibrium conditions, is only
rational when the number of remaining possessions is quite low. Again in our main empirical
analysis, we eliminate cases when this ratio diverges meaningfully from zero.
In both cases, when the number of possessions remaining is large, the goal of risk neutrally
maximizing points binds more tightly. The intuition is that at any given time, the team
currently expected to lose would like to induce variance in the outcome of the game. However,
with a large number of possessions remaining, the central limit theorem makes the terms
of this tradeoff very unattractive, because per-possession variance is killed at rate root-N,
and extending/shortening the game comes at a large cost (sacrificing efficiency on many
possessions) for a small benefit. Although our main analysis excludes cases in which risk
neutrality and time invariance do not hold in a tight approximation, we also examine the
9 If teams are of unequal ability, the trailing teams incentive to induce variance depends on their relative
ability. If they are much better and a decent number of possessions remains, this incentive goes away, as
they are still favored to win.
9
N
(6)
(7)

excluded cases to see if teams do in fact become risk (or time) loving (or averse) when doing
so increases the chances of winning the game. 10
2.3 Anatomy of a possession
In the previous subsection we showed that a basketball can be broken down into a series
of nodes, possessions, and modeled as a game of risk-neutral point maximization at every
node. To help build intuition for the reader, in this section we explain the basic features
of a possession. We define the offensive output of a possession as the total points scored
before the opponent starts its next possession. For instance, if a shot is taken, but missed,
and the offense rebounds the ball, this does not count as a new possession. We make this
important distinction because the points we assign to a specific shot are not necessarily the
outcome of that shot, rather is a forward looking metric that captures how well the team
does conditional upon the player using the possession versus waiting.
Table 1: Ways Possessions Can Be Used
Outcome Total in Sample Average per-game Points Per Possession
Made 2-pointer 390763 26.3495 2.0579
Made 3-pointer 88009 5.9345 3.0226
Turnover 190443 12.8417 0
Fouled in the Act of Shooting 141737 9.5575 1.5707
Missed 2 pointer 438231 29.5503 0.3547
Missed 3 pointer 134777 9.0881 0.2833
All (possessions) 1,376,893 92.8451 1.0624
This brings up an important concept of “using” the possession. A player is said to use
the possession if he is the first player from his team to shoot or turn the ball over (which
we model as similar to a missed shot with no rebound). Table 2.3 shows the empirical
frequencies for the ways in which a possession can be used and the immediate outcome.
Column (1) gives the total sample N and Column (2) shows the averages on a game level, to
provide an idea of general scoring levels and possession frequency. The table gives the first
usage, which is not necessarily how the possession ends. For instance, a missed 2-pointer
10 In the Appendix we estimate the probability of winning equation, (3), via probit (due to the Guassian
error structure implied by the central limit theorem). This allows us to predict the relevant quantities of
interest per possession, and appropriately eliminate cases when the marginal impact of time and risk exceeds
the tolerance levels we set for our analysis.
10

could be rebounded by the offense, and then lost out of bounds. The key actions are shooting
(either 2 or 3-pointer), turnover and drawing a foul. As a reminder, when a player is fouled
in the act of shooting he is awarded either 2 or 3 unguarded 14-foot shots (depending on
the value of shot he was fouled attempting) worth 1 point each. Free-throws are valuable; a
good player makes them about 80% of the time. In comparison, 50% is considered good for
a 2-pointer taken in the course of the game and 33% for a 3-pointer, roughly speaking.
2.4 Basketball as a stopping problem
We are now in a position to describe our model of a possession. Given that a team must
use a possession before the 24-second shot clock expires (or face an automatic turnover), it
is natural to model a possession as a stopping problem: which shots should be taken versus
continuing the possession? The added twist is that team must also ensure everyone gets
the ball so as to maximize production across the lineup. We formally present the model in
Section 3, but give the basic intuition here to build intuition, just as theory papers often
present a toy model before jumping into abstract formalism.
We follow the standard tools for modeling a stopping problem: opportunities are stochas-
tically generated by some process and the decision maker must decide whether to realize an
opportunity or wait for a (potentially better) opportunity down the line. Here the relevant
quantity is a ‘shot opportunity.” We model shot opportunities as draws from a player-lineup-
defense specific distribution. That is, our most general specification allows for not only player
ability heterogeneity, but also lineup composition effects and opponent effects. In each sec-
ond of the shot clock (which we observe in discrete increments), we assume that the player
with the ball observes a shot opportunity with mean-zero noise (bias in observation is per-
fectly co-linear decision error, so this assumption is necessary). The player must quickly
decide whether to shoot or not. While it might not be literally true that a player observes
an opportunity every second, we easily incorporate these “non-opportunities” as draws that
have very low value.
To complete the model, we assume that the defense must allocate a finite amount of defen-
sive attention to minimize offensive output. We solve for equilibrium using familiar dynamic
programming techniques. Equilibrium requires that offensive players adopt monotonically
declining (with time remaining on the shot clock) “cut-thresholds.” A cut-threshold deter-
mines which opportunities should be realized and which should be passed up—if a realization
is observed to be above the threshold, the shot is taken. We call this condition dynamic effi-
ciency. It ensures that points are maximized given the time-constraint. Allocative efficiency
is our concept of optimally sharing the ball. It requires that all players in a given lineup
11

Figure 1: Shooting hazard by how the possession originates as a function of time remaining
on the shot clock.
have the same marginal efficiencies each point of the shot clock. If this condition fails, one
player should get passed the ball more (or less), because output would increase by doing
so. If both conditions hold, all players in the line-up use the same cut-thresholds—these
thresholds maximize total output, conditional on the opportunity generating function. 11
Like all stopping problems, the key challenge is modeling shots observed by the player,
but not the econometrician. Since our equilibrium concept says the marginal shot should
exactly equal the continuation value we must answer the question, “What would happen if
player X shot more with t seconds remaining on the shot clock?” We describe our approach
in Section 4. The intuition is that we use the fact that later in the shot clock, usage rates
(probability of shooting conditional on reaching that period, “shooting hazard”) increase for
all players. This gives us an instrument to identify marginal shooting opportunities, and thus
the distribution of shot opportunities facing the player. Figure 1 shows the combined hazard
for 5-man lineups presented by how the possession starts. Note that the hazard increases as
time winds down. This is intuitive, as the end node approaches, the cut-thresholds drop, and
thus shots are taken at higher frequency (conditional on reaching that point). The theoretical
optimum line is explained in detail later—it is derived through the assumption of player-
specific uniformly distributed shot opportunities. The figure also shows that possessions
originating as steals tend to have a “surprising” number of shots taken early in the shot
clock. These are what are known as “fast-breaks,” in which the team rushes down the court
to take an unguarded shot. Possessions that start with the ball being in-bounded from under
11 It is of course possible that the team could run a better overall offense, that is one that generates better
opportunities for all players, what this paper studies is a question of constrained optimization, where the
constraint is the offense, the coach, player ability, etc.
12

the opponent’s basket (made field goal, made free throw, dead ball turnover) tend to “start
later,” in that the hazard is very low until about 17 seconds remaining, indicating the ball
had not been brought into the offensive range yet. The hazards converge at 12 seconds. Since
this gives us an equal playing field (we are confident the team has the ball in the offensive
half court and are not getting easy shots based on steals, etc), we only include the final 12
seconds in our main analysis.
In the next section we present the general model in full detail. In Section 4 we present
our identification strategy. The purpose of the basketball primer is to make these sections
parsable by readers heretofore unfamiliar with basketball.
3 Data
This analysis is entirely based on play-by-play data for all NBA games from 2006-2010 (four
seasons). The game logs detail all the players on the court the outcome of every play. All
variables discussed herein are constructed (via some fairly extensive coding) from the raw
game logs. Approximately 100 games (out of the 4,920 played during this time period) are
missing from this data set. We have no reason to believe their omission is anything other
than random.
Table 2: Data Overview and Description
Event/Action Description
Offensive/defensive line-up Players on court at given time
Game-time Minutes and second of each event
Game day Date of game
Shooter Player/time of the action
Rebound/assist Player/time of the action
Foul Shooting, non-shooting, flagrant, illegal defense
x,y coordinate of shot Physical location of shot
Turnover Broken down by bad pass, dribbling error, charge, lost ball
We performed data cleaning to eliminate cases where the assumptions of our model are
clearly violated. First, we remove possessions where risk-neutrality is violated, such as end of
quarter and end of game situations. Since our model applies only to half-court sets and not
to “fast-breaks” (actions taken immediately after steals or offensive rebounds, which involve
fairy trivial decision making). We define a fast break as any possession used within the first
seven seconds of the offense and have purged all such possessions from the data. Finally,
13

Table 3: Overview of Possession Data
Method of Possession Origin N Ē ˆσ 2 E
SC ¯
¯
SCerror ˆσ 2 SCerror
All 1376893 1.0624 1.2255 9.8089 -1.5575 13.1961
Opponent Made Field Goal 537811 1.0316 1.2277 7.0405 -2.3012 4.3343
Defensive Rebound 477576 1.0727 1.2285 12.2681 0.3059 4.1053
Steal (game play does not stop) 106397 1.2555 1.1550 15.3404 -0.1231 0.7034
Opponent Turnover (game play does stop) 104394 1.0227 1.2225 9.0007 -0.4148 0.5411
Opponent Made Free Throw 150715 1.0308 1.2284 8.5500 -0.4940 0.3936
we remove situations when one team is leading by a point margin so wide that the game
outcome is no longer in doubt as our model only applies to situations where the outcome of
the game can reasonably be effected by chosen offensive strategy.
4 Model
In this section, we first motivate our general model of basketball as one of point maximization
per possession. Importantly this requires an assumption of risk-neutrality, which we defend
with data and simulations in the Appendix. The basic intuition is that a game involves a
large number of possessions, so sacrificing mean efficiency to increase or decrease variance
has a first order effect on the mean but only a second order impact on the variance, so it
is almost surely a losing strategy. It is trivial to rule out the rare end of quarter situations
where this logic collapses. The general model incorporates the intertemporal trade-off of
realizing an opportunity versus continuing the possession and the interpersonal trade-off
of how frequently each member of the team should shoot at a given time interval. The
latter will be interpreted here as cut-points for each player’s worst selected shot in a pure
strategy equilibrium. But, could also be interpreted as a vector of shooting frequencies in
the corresponding Harsanyi purified mixed strategy equilibrium. To fit an abstract model
to observed play we must make assumptions that sufficiently simplify the complex game of
basketball. The more restrictive the assumptions, the stricter the test of optimality, but the
greater chance of misspecification. As such, we adopt a telescoping approach in which we
start with a minimal set of assumptions that provide a necessary condition of optimality. We
then add assumptions to increase the power our optimality tests and use robustness checks
(overidentification tests) to ensure their validity.
14

4.1 Definition of Variables
There are 820 distinct players in our six year sample, denote each one by i ∈ {1, ..., 820}.
These players are observed to play 1376893 different possessions - all equilibrium concepts
will be possession specific. Each possession will be indexed by p ∈ {1, ..., 1376893}. In any
possession p there are exactly five offensive and defensive players.
Op ≡ {i: player i is on offense on possession t}
Dp ≡ {i: player i is on defense on possession t}
and the following indicator variables.
The vectors Op and Dp are sorted by position such that Op,1 denotes the current Point
Guard for the offensive team and Dp,5 denotes the Center for the defensive team. Additionally
there are 30 different teams in our data playing over 6 seasons. We will consider each team-
season a separate unit and denote their presence on offense (defense) during possession p by
the notation T Op ∈ {1, ..., 180} (T Dp ∈ {1, ..., 180}).
The game of half-court offense is defined to occur only when the current value of the shot
clock is weakly below 12 (t ∈ T 12 ≡ {0, ..., 12}). More generally, convex sets of shot clock
periods may be denoted:
T t = {s ∈ N : 0 ≤ s ≤ t}
Tt = {s ∈ N : t ≤ s ≤ 24}
T t′
t = {s ∈ N : t ≤ s ≤ t ′ }.
In every second of half-court offense, it is possible that an individual offensive player might
use the possession by attempting a shot, drawing a shooting foul, or committing a turnover.
Once a possession is used, half-court offense ends and the offensive team is not allowed to
’use’ another possession until the other team has had a turn with the ball. Possession use
events are denoted as below:
15

Up,i,t = 1{player i used possession p in shot clock period t}
Up,i =
Up,i,t = 1{player i used possession p}
t∈T
Ui,t =
Up,i,t = # of possessions used by player i in shot clock period t
p
Ui =
Ui,t =
Up,i = # of possessions used by player i
t
p
Similarly, we may be interested in the total number of opportunities an offensive player
has to use a possession.
Np,i,t = 1{i ∈ Op ∩ {∩i∈Op ∩s∈t∗ (1 − Up,i,s)}}
The above expression should be understood to mean that Np,i,t = 1 if and only if player i
is in the offensive lineup for possession p and his team has not yet used the possession with
t seconds remaining on the shot clock.
Finally, any time Up,i,t = 1 we track the total number of points scored before the other
team begins its possession and store it as Pp,i,t 12 . It is not necessary that player i score all
(or any) of these points, merely that they are an end result preceded by his initial decision
to ’use’ the possession.
{P, N, T }p,i,t will comprise the observed features of our model, but we will often refer to
two intuitive, reduced form expressions instead. Usage rates are defined as the probability
that player i will use a particular possession conditional on having the opportunity to do
so. Traditional usage rates 13 refer to the aggregate probability of player i using a possession
conditional on being on the court.
up,i ≡ P (Up,i = 1|i ∈ Op)
In this paper we will be more interested in hazard rates of possession use at a given value
of the shot clock. Informally, we want the probability that player i uses a possession with
exactly t seconds remaining on the shot clock conditional on the possession being unused up
12 Up,i,t = 0 =⇒ Pp,i,t = 0
13 www.basketball-reference.com presents one version
16

to that point.
up,i,t ≡ P (Up,i,t = 1|Np,i,t = 1, p ∈ Op)
Also, conditional on player i selecting to use a possession, we are interested in the average
return of points to that possession. Informally, we are be interested in the efficiency of player
i’s possession use.
ep,i,t ≡ E(Pp,i,t|Up,i,t = 1)
For purposes of estimation, we will be interested in empirical analogs of these quantities.
Even though our equilibrium concepts will be possession-specific, we will have to average over
subsets of possessions in order for this to be meaningful. We will denote this by replacing
the p subscript with notation indicating a given subset of possessions. As one example, we
may wish to evaluate player i’s performance only against opposing team x. Then:
A ≡ {p : T Dp = x}
uA,i,t = P (Up,i,t = 1|Np,i,t = 1 ∩ p ∈ A)
ûA,i,t = UA,i,t
NA,i,t
eA,i,t = E(Pp,i,t|Up,i,t = 1 ∩ p ∈ A)
êA,i,t = PA,i,t
UA,i,t
.
The empirical analogs (denoted by hats) are simple sample averages and thus obey stan-
dard Central Limit Theorems.
4.2 General Model
In this section we will write down a general model of a half-court offense game between an
offensive and a defensive player. This game will give economics significance to the observable
quantities introduced in the previous section.
On each possession the Offensive and Defensive lineups compete in a zero-sum game
over the number of points resulting on the possession. Offenses do this by running plays to
17

create scoring opportunities and then making quick decisions about when an opportunity is
valuable enough to warrant its exploitation. Defenses respond by allocating scarce defensive
attention to interfere with the Offense’s ability to realize efficient opportunities.
Formally, defensive strategy on each possession is summarized by the static selection
of five choice variables dp ≡ {dp,j}j∈{1,...,5} which represent the average level of attention
devoted to each offensive player during the possession. These variables are constrained by
the abilities of the defenders according to (8).
j∈{1,...,5}
αT Dp,j dp,j ≤ 1 (8)
Here the team specific parameters (αx ≡ {αx,j}j∈{1,...,5}) allow arbitrary heterogeneity in
the budget constraints faced by different defensive teams. Intuitively, if team x has a very
good defensive Shooting Guard, it might face a relatively low cost toward defending opposing
Shooting Guards (αx,2 relatively small). This is not equivalent to, but will generally result in,
a higher level of equilibrium defensive pressure exerted against opposing Shooting Guards.
In real life, basketball defenses have a far higher dimensional choice set. There are
many different ways in which a defense might allocate more effort to contain a particularly
proficient scorer. The above expression views the selection and implementation of such
specific strategies as a black box inside each team’s budget constraint.
Offensive strategy is more richly modeled and allows for dynamic decisions by each player
over the course of the shot clock. It is variation in these decisions as the shot clock ticks
toward zero that motivates our comparison to the optimal stopping literature. At every one
second long interval of the shot clock, each offensive player has the opportunity either to use
the possession or to wait until the next period of the shot clock. Specifically, with t seconds
remaining on the shot clock, every player i ∈ Op draws an unbiased measure of the expected
number of points his team would get from his immediate use of possession p. Formally ∀t,
denote a probability space by {Ωt, Ft, Pt}. Then Φ will denote a family of random variables
on this space for the realization of scoring opportunities.
Φi,t,dp,i : Ωt → R
ηp,i,t = Φi,t,dp,i (ω)
The measure Pt is assumed continuous so that ∀i ∈ Op we may write the corresponding
distribution of ηp,i,t with measure and density below.
18

Fi,t,dp,i (x) = P (ηp,i,t ≤ x)
dFi,t,dp,i (x)
fi,t,dp,i (x) =
dx
ηp,i,t represents not just expected points on an immediate shot, but also whatever value
his team is likely to get (including points scored after offensive rebounds or from foul shots)
before ending their possessions. ηp,i,t is also effected by the likelihood that player i should
turn the ball over in his attempt to use of possession, thus immediately netting his team
zero points.
on:
The various subscripts indicate that the opportunity distribution may depend arbitrarily
1. The player shooting the ball: i.
2. The period of the shot clock we are in: t.
3. The level of defensive attention focused on player i: dp,i
Restrictions to the above dependencies will be discussed as necessary for identification
and convenience of estimation. For now we only impose a little bit of reasonableness on
the structure of defensive impact in order to insure an equilibrium defensive allocation in
our model.
Assumption RD: Every single quantile of a players opportunity distribution is strictly
decreasing and strictly convex in the level of defensive attention received by that player.
This can be interpreted to mean that defense is effective at all margins, but has diminishing
returns to scale. Finally, if any player is completely ignored on defense, than a small amount
of defensive attention is extremely effective.
∀x ∈ [0, 1] :
∂F −1
i,t,dp,i (x)
∂dp,i
∂2F −1
i,t,dp,i (x)
∂d2 p,i
∂F
lim
dp,i→0
−1
i,t,dp,i (x)
∂dp,i
< 0 (9)
> 0 (10)
= −∞ (11)
Also we make no restriction on the precision with which a player observes the quality of
potential shots, but do require that he has unbiased expectations.
19

Assumption UE: E(Pp,i,t|Up,i,t = 1, ηp,i,t) = ηp,i,t
This restriction is necessary because we have no means of differentiating players who
choose to shoot too much from player’s who are just deluded into thinking all of their shots
are good shots. Otherwise, this is not a meaningful restriction.
If no player chooses to exercise the possession in time period t, then that period of the
shot clock is allowed to pass and each player will realize new scoring opportunities in period
t − 1. Note that we are not assuming a viable scoring opportunity for every player at every
interval of the shot clock, it is entirely possible that during many seconds of the shot clock
a player will realize scoring opportunities of arbitrarily small expected value. Since these
opportunities are very rarely exercised, it is not of paramount importance how we model
them.
In period 0, the shot clock is about to expire. If no player uses the possession during this
period, then play is stopped and the ball is awarded to the other team with no points for
the Offense.
4.2.1 General Solution: Offense
Conditional on defensive strategy dp, the offense seeks to maximize the expected value of
their possession with t second left. They do so by adopting monotone cut-thresholds
(cp,i,t) for each player to use the possession in each period of the shot clock. If (and only
if) player i realizes a scoring opportunity of larger value than his cut threshold than he will
choose to use the possession.
Assumption MC: ∃cp,i,t, ηp,i,t > cp,i,t ⇐⇒ Up,i,t = 1
This restricts all shooting decisions to be monotone in perceived shot quality. However,
we place no assumptions (beyond being unbiased) on the abilities of player’s to accurately
recognize the quality of their shot so this is not restrictive nor incompatible with perceived
’inconsistent’ shot selection in basketball games. Additionally, for a player with a given
opportunity distribution, selection of a cut-threshold (cp,i,t) is equivalent to selection of a
usage hazard rate (up,i,t) and a level of aggregate efficiency for each player in each time
period (ep,i,t).
20

∞
up,i,t = P (ηp,i,t > cp,i,t) =
cp,i,t
ep,i,t = E(ηp,i,t|ηp,i,t > cp,i,t) =
fi,t,dp,i (x)dx
∞
cp,i,t
xfi,t,dp,i (x)dx
The cut threshold carries greater economic significance, but for technical game theory
reasons we will regard the implied selection of the usage hazard rates as the ”action” of the
”offensive player”.
The offense’s problem is now entirely reduced to the selection of cut-thresholds for each
player and in each shot clock period. Optimal choices are derived in the value-function
approach below 14 . In writing down the value function below we must formally impose our
previously discussed assumption of risk neutrality.
Assumption RN: The Offense’s value function on possession p with t seconds remaining
(Vp,t) is exactly equal to their expected point return on the possession. The Defense’s value
function will not be seperately written down, but their objective is the minimization of Vp,t.
up,i,t
Vp,t = E(Pp|UOp,Tt+1 = 0) = eOp,T t (12)
We may now formally write the offense’s problem in (13) below.
Vp,t = max Vp,t−1 +
{cp,j,t}i∈Op
∞
(x − Vp,t−1)dFi,t,dp,i
cp,j,t
j∈Op
Vp,−1 = 0
Optimal cut thresholds are derived from a first order condition, which gives a familiar
stopping result.
Dynamic Efficiency: ∀i ∈ Op, cp,i,t = Vp,t−1
A player should only choose to shoot if he realizes a scoring opportunity more valuable
than his outside option of continuing the possession. Dynamic Efficiency is a direct test
14 To preserve tractability we ’rule out’ cases where two players are induced to shoot in the same shot clock
period by assuming that each player already knows none of his teammates might also use the possession in
this period.
21
(13)

of whether or not basketball players understand and correctly solve their optimal stopping
problem. Dynamic Efficiency by all five players is a sufficient condition for optimal offensive
strategy.
Under knowledge of the opportunity distribution, the value function, and optimal cut-
thresholds could then be solved by recursively applying the criterion of Dynamic Efficiency.
Thus,as seen in the following lemma, the offensive player always has a unique best response
to any allocation of interference chosen by the Defense.
Lemma 1. For a given set of offensive players there exists a mapping BRO : R5 T ×5 → R
that denotes the Offense’s best response cut-thresholds for every possible defensive allocation
dp. The best response is always unique and has nice properties. Further the selection of cut
thresholds, cp, is equivalent to the selection of conditional usage rates, up, or unconditional
usage rates, qp.
Proof.
cp,Op,0 = Vp,−1 = 0
cp,Op,1 = Vp,0 =
i∈Op
...
cp,Op,t+1 = Vp,t =
∞
i∈Op
∞
η · fi,0,di (η)dη
0
Vp,t−1
(η − Vp,t−1) · fi,t,di (η)dη
For the case of a uniform opportunity distribution and some T periods remaining on
the shot clock, it is straightforward to write down such a ’shooter’s sequence 15 . While our
identification results will not rely on such strict assumptions, it is interesting to compare
observed usage behavior to the implications of such a model.
As shown in Figure 1, prior to t = 12, opportunities seem to differ by how the possession
starts. However, from t = 12 on it looks like the origin of the possession is no longer important
and NBA offenses are doing something that closely resembles optimally solving a ’half-court’
offense game while facing a uniform opportunity distribution. However, our results do not
rely on dynamic programming to solve for the value function. Rather the assumption of risk
neutrality allows us to compute possession values directly based on observed data, as in (12).
15 See Skinner 2011 for an independent deriviation of this condition
22

We also have a priori interest in the hypothesis that players anchor their shooting deci-
sions to the shooting decisions of their teammates, rather than to the solution of a dynamic
programming problem. Thus we also formulate a strictly weaker hypothesis.
Allocative efficiency: ∀i, j ∈ Op, cp,i,t = cp,j,t
Allocative Efficiency requires that all five players on the court have the same reservation
shot-quality in mind. Conditional on a given team-wide shooting rate, Allocative Efficiency
generates the best set of shot opportunities.
4.2.2 General Solution: Defense
For a given matrix of usage hazard rates {up,i,t} i,t∈{Op×T 12 }, we now derive the defense’s best
response selection of dp in order to minimize average points given up. For convenience we
iterate (13) in order to get a more convenient expression of the value function in (14).
Vp,t =
T
(
(1 − up,Op,s))
t=0
s∈T 12
t
j∈Op
up,j,t ep,j,t =
T
t=0 j∈Op
qp,j,t ep,j,t
Where qp,j,t may be considered to be the unconditional probability of player i using
possession p with t seconds remaining on the shot clock.
Lemma 2. For a given set of offensive players, a defensive team with budget constraint (8),
there exists a mapping BRD : R T ×5 → R 5 that denotes the Defense’s best response allocation
of interference to every possible level of offensive unconditional usage probabilities qp. The
best response is always unique and has first derivative given in the proof.
Proof.
ep,k,t = 1
dep,k,t
ddp,k
d 2 ep,k,t
dd 2 p,k
up,k,t
= 1
up,k,t
1
= 1
up,k,t
1−up,k,t
1
1−up,k,t
1
1−up,k,t
F −1
n (p)dp
dF −1
k,t,dp,k (p)
dp 1
< 0
ddp,k
d 2 F −1
k,t,dp,k (p)
dd 2 p,k
dp 2
> 0
Where inequalities (1) and (2) above follow from the assumptions on defensive interference
in equations (9) and (10). Thus (14) is concave in the defensive allocation. D ≡ { d : α ′ T D d ≤
1 ∩ d ≥ 0} is a compact set and thus a unique best response level of defensive allocation
exists.
23
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Under (11) we can be sure that all players face some equilibrium defense and write down
the following first order condition.
∀j = k ∈ Op,
T t=0 qp,k,t
∂ep,k,t
∂dp,k
T ∂ep,j,t
t=0
qp,j,t ∂dp,j
= αT Dp,k
αT Dp,j
The numerator and denominator of the left side can be viewed as the marginal interference
created by additional defensive attention to offensive players k and j respectively. Intuitively,
(15) just states that their ratio should be equal to the ratio of the marginal costs of such
defensive attention as specified in (8). We may then apply the Implicit Function Theorem
to establish derivatives of our best response correspondence:
dBR i D
dqp,i,t
=
i = j =⇒ dBRi D
dqp,j,t
dep,i,t
− ddp,i + qi,t d2
ep,i,t
ddp,idqp,i,t
t
t=0 qp,i,t d2 ep,i,t
dd 2 p,i
= 0
= − dF −1
i,t,d
(1−up,i,t)
p,i
ddp,i
t t=0 qp,i,t d2ep,i,t dd2 p,i
4.2.3 Existence, Uniqueness, and Continuity of Equilibria
Here we outline a series of results that demonstrate the existence of a unique equilibrium
in the half-court offense game between the Offensive and Defensive player. This equilibrium
will be dependent on the capabilities of the five offensive players and the interference budget
constraint faced by the opposing defense. Additionally, we will demonstrate that small
changes in the abilities of the offensive players will result in small changes in equilibrium
play. Taken together, these results allow us to credibly average over separate possessions in
order to learn about the decision making of individual players.
Theorem 1. In the game of half-court offense, with preferences defined by the value function
in (14) and defensive budget constraint outlined in (8) and under the previously maintained
assumptions, there exists an equilibrium selection of cut thresholds and allocation of defensive
interference
Proof. We regard the offensive player as selecting the unconditional shooting probabilities
q p,Op,T 12 ∈ [0, 1] 5×12 subject only to the constraint that they sum to exactly 1. The defensive
24
> 0
(15)

player is still regarded as selecting {dp,j}j∈Op subject to (8). These are both closed, compact
subspaces of R n . In appendix A, we demonstrate that the offensive value function (14) is
concave in q and convex in the levels of defensive attention. This is a standard sufficient
condition for existence of a Nash equilibrium. See Osborne and Rubinstein, 1994, Proposition
20.3 for one demonstration.
Theorem 2. In the game of half-court offense, with preferences defined by the value function
in (14) and defensive budget constraint outlined in (8) and under the previously maintained
assumptions, there may not exist multiple equilibria.
Proof.
5 Identifying Violations of Optimality
In the next section, we will consider what observable restrictions Allocative and Dynamic
Efficiency impose on our data and lay the groundwork for our ultimate tests. We will
begin by only considering the direct implications of the model as presented. In order to
increase power we will consider an additional assumption that will allow us to take variation
in shot selection across the shot clock as exogenous. This assumption enables (almost)
complete identification of our structural model and will generally provide narrow regions of
set identification for cut-thresholds. We will briefly consider estimators of this form, before
ultimately settling on a parsimonious parametric specification. Finally, we will address the
implications of error in our measurement of the shot clock for our identification results and
show how the availability of the auxiliary data set of shot clock violations can resolve this
problem.
5.1 A Necessary Condition for Dynamic Efficiency
In this section we develop a minimalistic necessary condition for dynamic efficiency. We
address the question “do players take shots too frequently and/or too soon in the shot
clock.”
On possession p, player i only shoots at t if ηp,i,t ≥ cp,i,t. Thus:
ep,i,t = E(ηp,i,t|ηp,i,t ≥ cp,i,t) ≥ cp,i,t
1
= Vp,t−1
2
= ep,Op,T t (16)
Equality (1) is Dynamic Efficiency and equality (2) is the appropriate valuation of a
possession under risk-neutrality. Further we can integrate this condition over any set of
possessions and time periods yielding a very general null hypothesis.
25

eA,i,T s
t ≥ eA,O·,T t−1 (17)
Testing of this hypothesis is straightforward from Central Limit Theory.
êA,i,T s
t
∼ N(eA,T s
t , σ2 A,i,T s
t
UA,i,T s
t
êA,T t−1 ∼ N(eA,T t−1, σ2
A,T t−1
)
UA,T t−1
T s
t ∩ T t−1 = ∅ =⇒ êA,i,T s
t ⊥ ê A,T t−1
êA,i,T s
t − êA,T t−1 ∼ N(eA,i,T s
t − eA,T We ran this test for all players in our sample...
RUN THIS TEST
)
t−1, σ2
A,T t−1
U A,T t−1
+ σ2 A,i,T s
t
UA,i,T s
t
Our results show that this optimality condition is rarely violated in NBA play. However,
this analysis makes no statement about the possibility that NBA players ’undershoot’. Fur-
ther it cannot even be taken to fully rule out ’overshooting’ in the NBA. Typically there
is a wide divergence between a player’s “worst shot” ((cp,i,t)) and average shot (ep,i,t), with
the average shot necessarily offering higher efficiency. If a player’s average is consistently
below their team’s continuation value, then the team would be better off if he did not shoot
at all. We can make this statement because we know the counterfactual of a player taking
zero shots (his team would get zero points and they would be able to continue all those
possessions). However, we are also interested in situations in which players are not actively
hurting their team, but are perhaps still using the wrong cut threshold.
In order to make such statements we will need an appropriate counterfactual that gives
us information about the value of a player’s marginal shots. Approaches in this paper will
be based off the exploitation of variation in shooting hazard rates observed over the course
of the shot clock.
As an example, we may observe a hypothetical player i with the following reduced form
statistics.
up,i,t = .04 ep,i,t = 1 (18)
up,i,t−1 = .05 ep,i,t−1 = .95 (19)
26
)

We suppose there is not important difference between periods t and t − 1 and that if our
player increased his aggression he could achieve the same levels of usage and efficiency in
period t. Then by choosing his allocation in period t (and not the one he chose in period
t − 1) he is passing up opportunities that can be approximately valued at.
cp,i,t ≈ ep,i,t−1up,i,t−1 − ep,i,tup,i,t
up,i,t − up,i,t−1
= .75 (20)
If Vp,i,t−1 < .75 then we can conclude that player i should in fact alter his period t
behavior toward what is observed to be capable of in period t − 1.
In the next section we will build general identification results based on this logic. We
will rely critically on the assumption that variation in the shot clock provides a valid source
of exogenous variation in shot selection.
5.2 Structural Identification
We now consider the identification of opportunity distributions and cut-thresholds for a
fixed offensive lineup (Op) and opposing defense (T Dp). Given uniqueness of equilibrium in
the overall game, we need not consider variation in defensive intensity from possession to
possession. As such we will omit notation for defensive intensity and the impact of teammates
on the notation below. We do observe variation in player shooting decisions across the shot
clock and we will proceed by formally taking this variation as exogenous.
Assumption ID: A player’s opportunity distribution is invariant to the particular
period of the shot clock.
Fi,t,Op/i,dp,i
L
= Fi,s,Op/i,dp,i
L
≡ Fi,Op/i,dp,i
Under this assumption, the reduced form quantities {up,i,t, ep,i,t} contain structural infor-
mation about our time-invariant opportunity distribution.
ep,i,t =
∞
F −1
η p,i (1−up,i,t) ηp,idFηp,i
up,i,t
(21)
= E(ηp,i|ηp,i > F −1
ηp,i (1 − up,i,t)) (22)
If for some player i and ∀u ∈ [0, 1], we observed pairs {e(u), u} then we would achieve
full, non-parametric identification of the opportunity distribution and chosen cut-thresholds.
27

de
du
−1 (Fηp,i (1 − u) − e(u))
=
u
c(u) = F −1
de
ηp,i (1 − u) = e(u) + u
du
Given observation in a finite number (T ) of discrete shot clock values, we are only able
to identify sub-blocks of the opportunity distribution and provide set identification of cut-
thresholds.
UL(t) ≡ {s ∈ T T : up,i,s < up,i,t}
UH(t) ≡ {s ∈ T T : up,i,s > up,i,t}
Then ∀l ∈ UL(t), h ∈ UH(t), observables place the following restrictions on cut-thresholds.
ep,i,tup,i,t − ep,i,lup,i,l
up,i,t − up,i,l
ep,i,hup,i,h − ep,i,tup,i,t
up,i,h − up,i,t
=
=
F −1
η p,i (1−up,i,l)
F −1
η p,i (1−up,i,t) ηp,idFηp,i
up,i,t − up,i,l
F −1
η (1−up,i,t)
p,i
F −1
η p,i (1−up,i,h) ηp,idFηp,i
up,i,h − up,i,t
≤ F −1
ηp,i (1 − up,i,t) = cp,i,t
≥ F −1
ηp,i (1 − up,i,t) = cp,i,t
It is important to notice that the failure of full identification here has two different
sources. The first is that no NBA basketball player is ever observed to shoot with a hazzard
rate in excess of 35% or below .2%. Thus we have no information whatsoever about what
player’s opportunity distributions look like outside these confines. This is not a meaning-
ful failure. The second problem is that we do not observe continuous variation in player
shooting behavior, but rather discrete jumps corresponding to the lumpiness of our shot
clock observation. Thus we cannot pin down exactly what cut-threshold corresponds to an
observed hazard rate, because we do not have local knowledge about the behavior of the
opportunity distribution. In practice, 13 different periods of half-court offense is still plenty
of information about the underlying structure. One way to think about this is that as long
as players adopt different hazard rates in each of the 13 periods 16 , then we could identify a
parametric model of the opportunity distribution that had up to 13 parameters.
Alternatively, we will actually compute the bounds by (??) and (??) for an example
16 As we will address in our Results section, they almost exclusively do.
28
(23)
(24)
(25)
(26)

player and show that identified sets are generally quite narrow. But that variability in
the estimation of the upper and lower bounds of these sets tends to make the fully non-
parametric approach uninformative. Below, we consider the aggregation of all NBA centers
as one composite player in an attempt to minimize the impact of sample variance on our
estimator. We will combine the information from various bounds in (26) and (25) into a
single MSE minimizing estimator of a composite lower and upper bound 17 .
Figure 2: Diagram of the Aggregate NBA Center’s preformance in our data set. Blue dots
indicate observed combinations of efficiency and usage in each period of the shot clock. The
other lines indicates estimates of the area of set identification via (26) and (25). Even for this
aggregate player, nonparamtric estimators produce very weak pointwise confidence intervals
on cut-thresholds.
5.3 A Parsimonious Parametric Model
The conditions of the previous section are quite permissive of potentially suboptimal play
for two reasons: 1) they provide no tests for undershooting 2) they give no insight toward
allocative efficiency. Furthermore, they offer no convenient way to characterize the overall
distribution from which a player realizes scoring opportunities. In this section we address
these difficulties. In order to do so w must make a somewhat restrictive assumption. Namely
that ∀t ∈ T player i draws his opportunities from a common distribution. This will allow
us to compare player i’s performance at the beginning of the shot clock (when he will rarely
shoot) to the end of the shot clock (when he will have to shoot a much higher fraction of
17 In the selection of an optimal estimator, the increased bias of looser bounds is approximated by imagining
the shot opportunity distribution is uniform with density .25.
29

the time). Intuitively, we are using the shot clock as an instrument to identify player i’s
opportunity distribution and implied usage curve.
We model player i’s scoring opportunities as drawn from a uniform distribution along
the interval [Bi, Ai]. Additionally, the player selects a shot in period t if his draw lies in the
interval of [ci,t, Ai]. For the vast majority of players, Bi will take a negative value. This does
not mean that players have opportunities to lose points, but merely reflects the fact that
the vast majority of players do not realize a good scoring opportunity in most periods of the
shot clock. As long as ci,t > 0 the part of our theoretical uniform opportunity distribution
that lies below zero is irrelevant.
Conditional on the parameters θi = {Ai, Bi, {ci,t}t∈T }, it is straightforward that:
ei,t = Ai + ci,t
2
P rob({Ei,t, Ui,t}t∈T |θi, Ni,t) =
φ(
t∈T
, ui,t = Ai − ci,t
,
Ai − Bi
dei,t
dui,t
Ui,t − Ai−ci,t
Ai−Bi
Ni,t Ai−ci,t
Ai−Bi
(1 − Ai−ci,t
Ai−Bi )
= Ai − Bi
)φ( Ei,t − Ai+ci,t
2
σi √
Ni,t
We achieve parametric identification because the likelihood matrix is non-singular for all
players who choose at least some variation in cutoff levels across periods of the shot clock.
This should not be a surprise, if a player did the same thing in every period of the shot clock,
our shot clock instrument is irrelevant and we have on hope of learning about his tradeoff
between usage and efficiency. It is doubtful that any NBA player’s shooting behavior is
truly invariant to pressure from the shot clock. However, lack of identification at this point
has the potential to create distortions in standard Wald statistics if the truth is in a small
enough neighborhood of the unidentified region. We do not believe this difficulty is relevant
to the vast majority of players in our data and in this version we will not explicitly address
it. But we do note that recent advances in the econometric literature (Dufour et al. 2010;
Andrews et al. 2010) may enable us to better understand and be robust to any nonstandard
properties of our estimators.
5.4 Measurement Error in the Shot Clock
Our identification argument in the previous section relies crucially on our ability to infer the
correct value of the shot clock that a player observed when making his shooting decisions.
There are precise guidelines for when the shot clock should be reset in the NBA(?) and we
were able to estimate the true shot clock time by tracking the value of the game clock at
which all shot clock relevant events occur. However, it is a typical occurence that the shot
30
)

clock operator and the person who tracks the play by play data might slightly disagree on
exactly when a possessions starts. Thus our shot clock measurement may be off by as much
as 5 or 6 seconds on some possessions. We were originally unaware of these complications
and would like to thank participants at the Sloan Sport’s Conference 18 for making us aware
of this obstacle.
An easy way to demonstrate the imprecision of our shot clock estimator is to consider
the distribution of our estimates of the value of the shot clock when the offensive team
experiences a shot clock violation. A shot clock violation only occurs when the true value
of the shot clock is exactly zero, as you can see our estimates are generally centered around
zero, but somewhat imprecise.
Figure 3: The distribution of error for shot clock violations varies greatly across different
types of possessions.
As seen above, the distribution of shot clock error varies greatly depending on how the
possession starts. When the possession starts off a dead ball (the other team throws it out
of bounds or makes the second of two free throws) their is very little room for error by
the shot clock operator. Similarly when the possession starts off a steal, their is typically
only a small lag before the shot clock operator notices this event and resets the shot clock,
leading to only slightly more measurement error. A third way a possession can start is with
a defensive rebound following a missed field goal by the other team. Battles for rebounds
are often harshly contested and it is not always clear when one team has secured the ball. It
is entirely plausible that the shot clock operator and the person keeping the game log could
come to very different conclusions here and thus we experience significantly more shot clock
measurement error on these possessions. Finally, it is also possible (and very common) for
18 Allocative and Dynamic Efficiency in NBA Decision Making was the version of our paper presented there,
accounting for measurement error in the shot clock has slightly changed the overall flavor of our results in
that we generally find more overshooting and less undershooting now.
31

the possession to start after a made basket by the other team. On these possessions, the NBA
game clock continues to run after a basket is scored and before the next possession starts.
However, the offense’s shot clock does not begin to run until they inbound the basketball.
Players often lolly-gag or exchange brief communication before choosing to inbound the ball
leading to our perception that the shot clock should start significantly before it actually
does. As seen in above graph, average measurement error on these possessions is strongly
negative (we think player’s are out of time when they are not) and has significant variability.
Now that we understand the source of our measurement error, we need to consider how
we can overcome its bedevilling effect on our identification. For purposes of notation, let mp
be a discrete indicator for the mechanism by which the possessions starts and let zp,t denote
the observed value of the shot clock on possession p when the ’true’ value of the shot clock
is t.
zp,t = t + ɛp,t
In order to restore identification of our model, we will need to know the distribution of
’true’ shot clock values conditional on all observables. That is we want, P (t|z, m) which by
application of Bayes’ rule we may write this down as:
P (t|m)
P (t|z, m) = P (z|t, m)
P (z|m)
We will do this in pieces. The denominator (P (z|m)) is the unconditional distribution
of observed shot clock values when possessions are used, it is directly observable from the
data. P (t|m) is the unconditional distribution of possession use times and P (z|t, m) is the
conditional distriubtion of shot clock error.
Learning about these two terms requires the following assumption on (27), which is
justified by the above discussion.
Assumption CME: Shot clock measurement error is classical. It is entirely caused by
mismatches between the game and shot clock that occur at the beginning of the possession.
The distribution of this error may vary across the four different ways possessions originate,
but is invariant to the true value of the shot clock. This is succinctly expressed in the
equations below.
∀t, p, ɛp,t = ɛp
ɛp ∼ Γmp
32
(27)
(28)
(29)
(30)

This assumption yields constructive identification of our desired terms.
P(z|t, m) = Γm(z − t) = P (z − t|t = 0, m) (31)
P(t|m) × P (z − t|m) = P (z|m) (32)
t|m ⊥ (z − t)|m (33)
Where the × in (32) indicates the convolution operator. Then given (33), P (t|m) is the
solution to a straightforward deconvolution equation.
Combining all these pieces of information we identify our target density: P (t|z, m). Com-
bining this information, with the model in Section 5.3 we achieve a likelihood model that is
robust to measurement error.
6 Estimation
L ME (Up, Pp|z, m, θ) =
P (t|z, m)L(Up, Pp|t, m, θ) (34)
t
For purposes of tractability we impose a little bit of structure on Φ allowing for a seperable
penalty function of defensive interference.
Φi,Op/i,dp,i,t = Φi,Op/i,t − hilog(dp,i) (35)
6.1 Parametric Uniform Shot Distributions
The conditions of the previous section are quite permissive of potentially suboptimal play
for two reasons: 1) they provide no tests for undershooting 2) they give no insight toward
allocative efficiency. Furthermore, they offer no convenient way to characterize the overall
distribution from which a player realizes scoring opportunities. In this section we address
these difficulties. In order to do so w must make a somewhat restrictive assumption. Namely
that ∀t ∈ T player i draws his opportunities from a common distribution. This will allow
us to compare player i’s performance at the beginning of the shot clock (when he will rarely
shoot) to the end of the shot clock (when he will have to shoot a much higher fraction of
the time). Intuitively, we are using the shot clock as an instrument to identify player i’s
opportunity distribution and implied usage curve.
We model player i’s scoring opportunities as drawn from a uniform distribution along
the interval [Bi, Ai]. Additionally, the player selects a shot in period t if his draw lies in the
33

interval of [ci,t, Ai]. For the vast majority of players, Bi will take a negative value. This does
not mean that players have opportunities to lose points, but merely reflects the fact that
the vast majority of players do not realize a good scoring opportunity in most periods of the
shot clock. As long as ci,t > 0 the part of our theoretical uniform opportunity distribution
that lies below zero is irrelevant.
Conditional on the parameters θi = {Ai, Bi, {ci,t}t∈T }, it is straightforward that:
ei,t = Ai + ci,t
2
P rob({Ei,t, Ui,t}t∈T |θi, Ni,t) =
φ(
t∈T
, ui,t = Ai − ci,t
,
Ai − Bi
dei,t
dui,t
Ui,t − Ai−ci,t
Ai−Bi
Ni,t Ai−ci,t
Ai−Bi
(1 − Ai−ci,t
Ai−Bi )
= Ai − Bi
)φ( Ei,t − Ai+ci,t
2
σi √
Ni,t
We achieve parametric identification because the likelihood matrix is non-singular for all
players who choose at least some variation in cutoff levels across periods of the shot clock.
This should not be a surprise, if a player did the same thing in every period of the shot clock,
our shot clock instrument is irrelevant and we have on hope of learning about his tradeoff
between usage and efficiency. It is doubtful that any NBA player’s shooting behavior is
truly invariant to pressure from the shot clock. However, lack of identification at this point
has the potential to create distortions in standard Wald statistics if the truth is in a small
enough neighborhood of the unidentified region. We do not believe this difficulty is relevant
to the vast majority of players in our data and in this version we will not explicitly address
it. But we do note that recent advances in the econometric literature (Dufour et al. 2010;
Andrews et al. 2010) may enable us to better understand and be robust to any nonstandard
properties of our estimators.
Additionally, our likelihood equation is dependent upon the normality of the distribution
of points from an individual shot. In small samples, this is trivially incorrect as the actual
distribution of points is discrete and slightly skew. Additionally, we have no reason to
believe that our uniform specification is exactly correct. As such we shall not assume that
the true data generating process falls within our model and apply consistently estimated
covariance matrices from White (1982). Further we shall only perform Maximum Likelihood
Estimation (MLE) for each player with at least 15 used possessions in every relevant period
of the shot clock. Despite the fact that we are fitting a uniform distribution, the support
of observed efficiency and usage rates over any finite sample does not change so we do not
have a regularity problem and are able to calculate standard asymptotic variances from the
likelihood matrix.
34
)

Figure 1 provides the aggregate results of this estimation procedure. We see that indeed
NBA players use a monotonically declining cut threshold consistent with the predictions of
an optimal stopping problem with finite periods. More impressive is the fact that the cut-
thresholds are nearly identical to the continuation values of the possession and the functions
have the same shape. We present the results in two panels to enhance the contrast of the
slope, while still showing it for all periods of the shot clock. Overall, Figure 1 is strong
evidence in favor of near optimal play. NBA players appear to be well-tuned to the contin-
uation value of the possession and adjust their shot choice to reflect it. This is precisely the
mechanics of optimal stopping. Not only do they get the mechanics right, but the rate at
which the players lower their cut threshold matches the continuation value nearly exactly!
We do note, however, that in Figure 1 the cut-threshold does lie slightly above the
continuation value, which is evidence that undershooting is more common that overshooting.
To extend the analysis, we now examine which player’s tend to overshoot or undershoot on
average. We take as a null hypothesis that each player is dynamically efficient on average.
Namely that for each player i,
t∈{2,..,17} ci,t =
t∈{2,..,17} eO,t∗. To maximize power to detect
deviation, we test this hypothesis with a weights inverse to the variance of our estimated
cutoffs. Figure 2 provides a histogram of the resulting t-statistics. A negative t-statistic
indicates overshooting, a positive t-statistic indicates undershooting.
The results again indicate that undershooting is much more common than overshooting.
In fact, only 5 players are found to be significant overshooters — less than we would expect
to find by chance alone (although if we take the mean to be 1, not 0, the evidence these
players overshoot strengthens considerably. 19 In line with this reasoning, the distribution
appears standard normal but shifted over about 1 (the mean 0.98) Most players appear to be
optimizing and mistakes tend to come in the form of undershooting early in the shot clock.
In this sense, some players wait too long to shoot or do not expend maximum effort on each
possession. In contrast, lab subjects tend to pull the trigger too early, typically through
the use of a fixed threshold (Lee et al., 2004). We also compute the loss in surplus due to
sub-optimal shooting decisions (intuitively integrating between the two lines in Figure 1).
The median value of DWL across players is 4%, consistent with nearly optimal play.
In the discussion section we examine how much of a role effort conservation plays in this
finding. While we cannot observe effort directly, we can observe which type of players tend
to undershoot. We find that under-shooters are frequently highly paid “star” players. Our
intuition is that some of undershooting is driven by effort conservation by star players who
play relatively more minutes-per-game.
19 Lamar Odom, Monta Ellis, Rafer Alston, Russell Westbrook and Tyrus Thomas are the guilty parties.
35

6.2 Allocative Efficiency
Allocative Efficiency is the hypothesis that the players on the court, (I) use the same cut-
point in all periods of the shot clock. In our estimation we allow the the set I to be an
entire five man lineup or a three or four man ”core” that more frequently shares the court.
The concept of cores is convenient to improve power in estimation Because our estimation
is more precise for these periods, we will focus on cut-points in the first ten seconds of the
shot clock (T = {9...18}). Let c be the T I × 1 vector of relevant cut points, sorted first
by period of the shot clock. Based on our parametric procedure we have:
ĉ ∼ N(c, Vc
N )
We define the ”true” deviation from Allocative Efficiency as spread.
SI,T =
t∈T i∈I (ci,t − ¯ct) 2 = (c − ¯c) ′ I(c − ¯c) = z ′ Iz
Where ci,t is the cutoff chosen by player i in period t and ¯ct is the average over the five
teammates of the cutoff chosen in period t and ¯c is the appropriate corresponding T I×1
vector of averages.
Note that, if we define M = IT ⊗ (I5 − J5 ), we may also simply write z = Mc and
5
S = c ′ M ′ Mc = c ′ Mc.
The ”seemingly” natural empirical analog to S is:
ˆS = 5 t∈T i=1 ( ˆ ci,t − ˆ¯ct) 2 = ˆz ′ I ˆz
However, a little bit of calculation reveals that this measure is biased upwards, especially
for small samples.
E(ˆz) = c − ¯c
V (ˆz) ≡ Vz = MVcM
N
Thus: E( ˆ S) = E(z2 i,t) = E( zi,t) ˆ 2 + V ( zi,t) ˆ = S + diagVz = S + diag(MVcM)
N
36

turns out to be upward biased and especially so for small n. In order to correct for this,
we essentially subtract off the bias and define:
ˆS ∗ = ˆ S − diag(M ˆ VcM)
N
We use this as our measure of dispersion in the marginal output of players sharing the
court and refer to simply as spread. It turns out that the same 5-man line-ups share the
court relatively in frequently. We thus need to define the concept of “4-man core.” A 4-man
core includes all line-ups that share the same 4 players and and the fifth player is occurs less
frequently than the core-4. An example is if 4 starters typically share the court with 1 of 3
back-ups (the fifth player). In this case, each back-up plays less than the starters with the
line-up, so is not in the 4-core. We then compute spread as defined above.
Figure 3 presents the distribution of spread for the 246 most common 4-man cores in
our data. The median is exceedingly small; it comes in at 0.036 (the mean is slightly larger
at 0.06). We also see a large spike at 0 in the distribution. In most line-ups, the players
show near equal marginal efficiencies at each period of the shot clock. Again, this is strong
evidence of near optimal play, but again play is not perfect. Players seems to be sharing the
ball incredibly well, but still about 10% of line-ups are estimate to have spread in excess of
0.25. These line-ups show clear room for improvement in terms of ball allocation.
In the Appendix we present the distribution of spread for 3-man cores. The results
confirm the results presented in Figure 3 and in fact the distribution is even more closely
centered around 0. This makes sense, in a 3-man core we compute spread over 3 players,
instead of 4, that are more familiar with each other and have more similar characteristics.
In the discussion section we delve into how spread correlates with features of the line-up.
7 Deeper analysis and discussion
The analysis in the preceding section gave sufficient statistics dynamic and allocative effi-
ciency. We now link these measures to features of the line-up such as aggregate efficiency,
experience and features of the players.
7.1 Line-up performance, line-up attributes and allocative effi-
ciency
In this section we examine how line-up performance and other attributes of the line-up cor-
relate with adherence to allocative efficiency. We view these regressions and figures as at the
37

very least informative correlations. Clearly we, as academic researchers, cannot exogenously
improve adherence to optimality in order to gage the impact on performance. Similarly we
cannot exogenously impose an a more experienced line-up and measure the impact on shot
allocation. Indeed our estimates themselves provide answers to the question “how much meat
is left on the table” (our estimates of DWL are small). One potential endogeneity problem
is that a coach might continue to play a high output line-up (driven by talent), despite
relatively poor decision making, because the results are still better than experimenting with
something new. This would tend to dampen the impact of offensive output on adherence to
optimality. 20
Figure 4 presents a scatter plot of 4-man core spread on average output per possession.
There is a detectable, but not overwhelming, pattern of more productive line-ups showing
lower spread. Table 2 demonstrates that this relationship is significant at the 0.10 level
using a two-tailed test. Given that our hypothesis is in fact one-tailed, this significance
level can be adjusted to 0.05. Table 2 also gives estimates of other correlates of spread. N
gives the number of possessions the core shared in the sample. The impact of N is weak
and positive. This is not necessarily evidence of a negative impact of repetition because of
numerous endogeneity issues based on coaches’ decisions. Column (1) includes the average
salary of the line-up in logs. Column (2) includes the simple arithmetic mean. For both
cases, higher paid line-ups perform significantly better in terms of spread. Higher salary
inequality, as measured through the standard deviation, is associated with larger values
of spread. This result is potentially of interest to labor economists concerned with salary
equality and production. “Experience” gives the number of years played in the NBA at the
start of 2006. The mean of playing experience across the line-up does not appear to impact
adherence to allocative efficiency. The standard deviation of experience seems to have a
slight negative impact on spread.
7.2 Aggregate player characteristics and efficiency
In Table 3, we regress the t-stat from adherence to dynamic optimality on individual player
characteristics. Consistent with the reasoning offered in Section 3.3, salary is positively
related to the t-stat. Recall higher t-stats indicate under-shooting. Indeed the league’s star
players such as Chris Paul, Lebron James and Kobe Bryant have high t’s. Table 3 shows
that this the case generally, salary is highly significant in the both the linear specification
(1) and linear-log specification (2).
20 One might expect salary to be a nice control for line-up output, but unfortunately the correlation is
not strong (regressions available from the authors). Essentially there are just too many highly paid lousy
38

Table 4: Impact of line-up characteristics on allocative efficiency (spread).
(1) (2)
Output -0.214* -0.205
(0.124) (0.124)
N 4.41e-05*** 4.88e-05***
(1.61e-05) (1.69e-05)
ln(Mean Salary) -0.0862**
(0.0425)
Mean Salary -1.16e-08*
(6.23e-09)
s.d. Salary 1.38e-08* 1.30e-08*
(7.56e-09) (7.38e-09)
Mean Experience 0.00714 0.00658
(0.00598) (0.00625)
s.d. Experience -0.0171 -0.0182*
(0.0103) (0.0104)
Line-ups 238 238
Robust standard errors in parentheses
*** p

Table 3 also shows that over-shooters tend to offensive rebound specialists; defensive
rebounding does not appear to have a reliable impact, although the estimate is noisy. The
player position dummies are not significant and are thus suppressed for brevity. The fact that
higher paid players are more likely to under shoot is perhaps surprising at first. For instance,
some readers might have the intuition that NBA players interests diverge from team interests
in that they have the incentive to raise their point average through suboptimal play. Under
this view, the labor market rewards the wrong attributes (points per game as opposed to
efficiency, for example). Our results are inconsistent with the view of labor market. Boosting
individual production at a cost to the team is not a strategy employed by NBA players. We
think this is interesting in its own right. Teams still have a principle-agent problem in that
long-term contracts create a moral hazard for effort, but it is interesting that very few players
exhibit “selfish play.” Our belief is that the prevalence of under-shooting among the higher
paid players is evidence that the better players conserve energy at times due to their high
playing time and long season (over-shooting would be impossible to rationalize this way and
we say far less over-shooting).
7.3 MSNE, purified games and our contribution
Past work has studied relatively simple games. We study a much more difficult game and
in this way really put the theory to the test. Importantly, while this game is difficult, it
is far easier than equilibria implied by many models that have complicated MSNE, such as
bidding in common value auctions. If we take our theory seriously, we should apply these
“hard” tests.
The results are supportive that experts can reach equilibrium in complex games. NBA
players appear to be superb optimizers. Furthermore this work incorporate the concept of
purified games by modeling MSNE in the context of dynamic thresholds. Players adherence
to these dynamic thresholds is striking. Exceedingly few are found to over-shoot and within
line-ups spread is quite low. We do find that some (especially star) players under-shoot,
potentially to conserve energy. Overall all the hallmarks of optimal stopping and MSNE are
present.
7.4 Dynamic stopping problems
Past work has also studied stopping problems similar to this one in the lab. It is hard to know
how to generalize these studies though, because these are difficult problems that my require
line-ups that it weakens the relationship considerably.
40

experience and training. Perhaps, however, people just do typically have the capacity to
solve these complex problems, even with experience. This would be a very important results
for models of labor search. There have been papers that have looked at optimal stopping
problems using highly trained professionals. These studies use deliberative decisions made
by firms harvesting trees (Provencher, 1997), renewing patents (Pakes, 1986) and replacing
bus engines (Rust, 1987). In the field of sports, Romer (2006) is most similar to this paper
in the study of coaches decisions to “go for it” on 4th down on the NFL. Again this is a
deliberative decision made perhaps 10 times per season by the coach (in this sense it is one
that he may not have a lot of experience in making).
This study uses a huge volume of quick decision made players observing a random arrival
of shooting opportunities. The findings indicate the players do this quite well, far better
than lab subjects. Trained individuals appear quite capable of solving problems with the
level of complexity of classical optimal stopping problems.
7.5 Players are not perfect
While NBA players do show striking adherence to optimality overall, they are not perfect
optimizers. We find that a minority of players over-shoot and some line-ups show significant
deviations from allocative efficiency. Furthermore this can persist even for line-ups that play
together quite frequently. Other work has established that minor mistakes are made in NBA
player. For instance Rao (2010) finds that a minority of players behaviorally respond to past
shot success by taking more difficult shots, despite the fact they perform no better on these
shots (hot hand fallacy). Our findings here are consistent with the idea that play is near
optimal, but that some players and coaches have room for improvement. The better teams
adhere more tightly, which is natural.
8 Conclusion
Past work has studied experts playing relatively simple games. We study a much more
difficult game and in this way really put the theory to the test. Importantly, while this game
is difficult, it is far easier than equilibria implied by many models that have complicated
MSNE, such as bidding in common value auctions. If we take our theory seriously, we
should apply these “hard” tests. The unique decision environment we study allows us to
extend a stylized stopping problem from the lab to a field setting. Furthermore, we extend
the analysis of allocative efficiency across pure strategies to a markedly more difficult game.
The trade off is that is more challenging to get hard-and-fast optimality conditions, but our
41

modeling approach telescopes in the strength of assumptions and still provides meaningful
conclusions about the adherence to the theoretical standards employed.
The paper uses a huge volume of quick decisions made by players observing a random
arrival of shooting opportunities. The decision to shoot is modeled as a dynamic allocative
stopping problem. By using a realistic modeling approach we are able to derive strict tests
of optimality. We find that players overall adhere quite closely to the theoretical predic-
tions; overall they are suburb optimizers, although mistakes are made. In the context of
dynamic efficiency, the shot threshold has precisely the correct slope and nearly overlaps the
continuation value of the possession. The mistakes that are made tend to be undershooting,
in that the continuation value is lower than the marginal shot; these “mistakes” could be
rationalized by the conservation of energy across possessions. In allocative decision making,
most teams show a very low variance of marginal efficiencies across players on the court for
each interval of the shot clock, consistent with near optimal sharing of the ball. Overall our
results extend the realm and difficulty level of games that humans can play according to
game theoretic equilbrium.
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9 Appendix
9.1 Proof of Theorem 1
Consider two arbitrary, feasible combinations of ”actions” by the offensive and defensive
player: {
d N p , q N p }N=0,1. Additionally, pick an arbitrary λ ∈ [0, 1] and define the appropriate
convex combination of the previous two actions
{ d λ p, q λ p } = λ{ d 1 p, q 1 p} + (1 − λ){ d 0 p, q 0 p}
We first fix the defenders action at d 1 p and demonstrate that better than sets are convex
in hazard rates.
Vp,t( d 1 p, q λ p ) =
=
T
T
q λ p,j,t ep,j,t(d 1 p,j , qλ p,j,t )
t=0 j∈Op
∞
(λ + (1 − λ))
F −1
η (1−qλ p,j,t )
ηp,j,tdFη
t=0 j∈Op
T
∞
=
λ
t=0 j∈Op
F −1
η (1−q1 p,j,t )
∞
ηp,j,tdFη + (1 − λ)
F −1
η (1−q0 p,j,t )
−1
Fη ηp,j,tdFη + λ
(1−qλ p,j,t )
F −1
η (1−q1 p,j,t )
−1
F
ηp,j,tdFη + (1 − λ)
= λVp,t( d1 p , q1 p ) + (1 − λ)Vp,t( d1 p , q0 T
−1
Fη p ) +
λ
t=0 j∈Op
(1−qλ p,j,t )
F −1
η (1−q1 p,j,t )
−1
Fη ηp,j,tdFη + (1 − λ)
(1−qλ p,j,t )
F −1
η (1−q0 p,j,t )
ηp,j,tdFη
≥ λVp,t( d 1 p, q 1 p) + (1 − λ)Vp,t( d 1 p, q 0 p)
η (1−qλ p,j,t )
F −1
η (1−q0 p,j,t )
ηp,j,tdFη
Where the final inequality follows because the term in parenthesis is non-negative for ev-
ery player and time period. We demonstrate below, by assuming (without loss of generality)
that q 1 p,j,t ≥ q λ p,j,t ≥ q 0 p,j,t.
45

λ
= λ
F −1
η (1−q λ p,j,t )
F −1
η (1−q 1 p,j,t )
F −1
η (1−q λ p,j,t )
F −1
η (1−q 1 p,j,t )
−1
Fη (1−q
ηp,j,tdFη + (1 − λ)
λ p,j,t )
F −1
η (1−q 0 p,j,t )
−1
Fη (1−q
ηp,j,tdFη − (1 − λ)
0 p,j,t )
F −1
η (1−q λ p,j,t )
ηp,j,tdFη
ηp,j,tdFη
≥ λ(q 1 p,j,t − q λ p,j,t)F −1
η (1 − q λ p,j,t) − (1 − λ)(q λ p,j,t − q 0 p,j,t)F −1
η (1 − q λ p,j,t)
= 0.
Concavity of the value function in the defenders action is more straightforward and follows
directly from the previously established fact that d2 ep,j,t
dd 2 p,j
Vp,t( d λ p, q 1 p) =
≤
T
t=0 j∈Op
T
t=0 j∈Op
9.2 Proof of Theorem 2
q 1 p,j,t ep,j,t(d λ p,j, q 1 p,j,t)
< 0.
q 1 p,j,t (λep,j,t(d 1 p,j, q 1 p,j,t) + (1 − λ)ep,j,t(d 0 p,j, q 1 p,j,t))
≤ λVp,t( d 1 p, q 1 p) + (1 − λ)Vp,t( d 0 p, q 1 p)
Thus we have demonstrated the existence of a Nash Equilibrium in our game. That is we
have a fixed point solution to the problem.
Or equivalently:
d ∗ p = BRd(u ∗ p)
u ∗ p = BRu( d ∗ p)
d ∗ p = BRd(BRu( d ∗ p)) = BRd,u( d ∗ p) (36)
u ∗ p = BRu( d ∗ p) (37)
Suppose, for sake of contradiction, that half-court offense had a multiplicity of equilibria:
46

∃ d ∗ p, d ∗∗
p , such that:
BRd,u( d ∗ p) − d ∗ p = BRd,u( d ∗∗
p ) − d ∗∗
p = 0
( d ∗∗
p − d ∗ p) = ∆p = 0
In particular we know ∃j ∈ Op such that ∆p,j = d ∗∗
p,j − d ∗ p,j > 0.
α ′ T Dp ∆d = 0
Now that we have demonstrated the existence of a unique equilibrium in our game, we
can proceed to estimate our model without concern for the possibility of sudden regime
changes that would change our interpretation of player shooting decisions.
9.3 Parametric Model Specification Tests
Our model is identified by assuming invariance of a player’s ability to realize scoring oppor-
tunities across different values of the shot clock. However, because we make this assumption
for 16 different periods of the shot clock, our model is overidentified and our assumptions
can be tested. Suppose for example that defenses became progressively more tenacious over
the course of the shot clock and that player’s opportunity distributions generally declined
across the shot clock. Then, we would find that a single distribution could not accurately
reflect a players ability to score in both the beginning and end of the shot clock. Players
would end up shooting less and less efficiently towards the end of the shot clock than our
model would predict. In future versions of this paper, we hope to present formal tests of
our specification. For now, we present plots showing how our model preforms across the
shot clock as an average across all players in our sample. We take solace in the lack of any
obvious trends in our residuals.
9.4 Additional support for Risk Neutrality
The following plot shows the simulated winning percentage for an underdog with baseline
mean expected point value of 1.38 playing a team who averages 1.4 points per possession
with standard deviation 0.45. Each “game” was simulated 10,000 times. As evidenced by
the figure, although the underdog wants to increase the standard deviation of the expected
value of shot attempts it does not want to trade off any meaningful amount of mean to do
so.
47

Appendix Figure 1: Underdog winning percentage as a function of standard deviation and mean
9.5 When Does Half-Court Offense Begin?
We decided that half-court offense began with 17 seconds on the shot clock. Our reason for
doing so, is that prior to 17 seconds the average value of exercising a possession is found to
be strongly correlated with the mechanism by which the possession originated (steal, dead
ball, or defensive rebound). However, from 17 seconds on team’s are in a half court set and
the average value of possession use is now independent of how the possession originated.
48

9.6 3-man Cores
Distribution of spread for 3-man cores. Higher values indicate larger deviations from
optimality.
49

Figure 5: Player-by-player t-statistic for deviations from dynamic optimality. Positive values
indicate “under-shooting.”
51

Figure 6: Distribution of spread for 4-man cores. Higher values indicate larger deviations
from optimality.
52