An analysis of the New York Knicks' most used five-man lineup. The article attempts to solve for the optimal shot allocation among the five players.

1. Does Carmelo Anthony Shoot Too Much?
Vincent La
Anirudh Jayanti
May 29, 2013
Introduction
One of the chief responsibilities of a basketball coach is determining how to most efficiently
utilize his players. At any given time, there are five players for each team on the court.
On offense, each of these players can shoot the ball with a certain efficiency. When a coach
diagrams a play, he is essentially listing a set of actions leading to a player taking a shot.
Although the coach cannot control every shot that each player takes on a possession-by-
possession basis, he can, in general, plan to allocate a certain proportion of shots to each
player. How should the coach allocate shots between his players? Is it optimal to always give
the ball to the most efficient player? Is there a way to arrive at this answer mathematically?
These questions are the subject of this article.
Common sense tells us that even though a certain player may be more efficient than his
teammates, it probably would not be optimal if he took 100% of the shots. For a variety
of reasons, such as fatigue or the defense adjusting, one can imagine that if a player takes
too many shots, his efficiency starts to drop. This intuition suggests there is (in general) a
negative relationship between shooting efficiency and the proportion of shots players take.
Dean Oliver terms this relationship a player’s ”skill curve.” If we can determine skill curves
for each player in a five-man lineup, it is possible to mathematically determine the optimal
allocation of shots between them, as we demonstrate below.
Much of this work is motivated by Professor Brian Skinner’s article, “The Price of An-
archy in Basketball.”1
The basic idea of his paper is that if one player is more efficient than
the others, the optimal strategy is to let that player shoot only until his efficiency is at its
maximum. He also discusses skill curves. To measure shooting efficiency, he uses a statistic
known as the “True Shooting Percentage” (TS%), which is essentially a player’s field goal
percentage adjusted for free throws and three-point shots. True Shooting Percentage is given
1
The full paper can be found here: http://arxiv.org/pdf/0908.1801v4.pdf
1

2. by:
TS% =
1
2
(points scored)
(field goal attempts) + 0.44 × (free throw attempts)
.
Professor Skinner also calculates the fraction of the team’s shots that a player takes while
he is on the court. He denotes this as x where
x =
player shots/game
team shots/game
×
48 minutes/game
player minutes/game
.
Professor Skinner denotes the player’s efficiency function by fi(xi), which is simply player
i’s TS% as a function of x, the fraction of his team’s shots that the player takes. This is
what he calls the “skill curve.”
In “The Price of Anarchy in Basketball,” Professor Skinner goes through a hypothetical
example to illustrate his thesis. He makes strong assumptions about the five-man lineup
(treating four of the five players as identical and shooting with constant efficiency) that we
do not make. We apply Professor Skinner’s analysis to an actual NBA team: the 2012-13
New York Knicks. The Knicks’ star player, Carmelo Anthony, is often questioned for the
number of shots he takes and the low efficiency at which he shoots. We ask whether An-
thony really does “shoot too much.” We empirically determine the skill curves of the Knicks’
most-used five-man lineup and, using the method of Lagrange for constrained optimization,
find the optimal shot allocation for that lineup.
Methodology
We first establish the New York Knicks’ most used five-man lineup by number of minutes
played during the 2012-13 NBA season. The lineup is: Carmelo Anthony, Tyson Chan-
dler, Raymond Felton, Jason Kidd, and JR Smith. All data comes from www.basketball-
reference.com. For our analysis, we use seven seasons of data from the 2006-07 season to the
2012-13 season.2
For each season, we calculate the average True Shooting Percentage (TS%)
and the fraction of the team’s shots taken (x) for each player (results are shown in Tables 1
and 2 below). We then regress TS% on x using an Ordinary Least Squares regression to get
linear skill curves for each player. These are shown in Figures 1 and 2 below.
2
2011-12 was a shortened season with each team playing only 66 games.
2

3. 2006-2007 2007-2008 2008-2009 2009-2010 2010-2011 2011-2012 2012-2013
Anthony’s TS% 0.55 0.57 0.53 0.55 0.56 0.52 0.56
Chandler’s TS% 0.62 0.63 0.58 0.64 0.70 0.71 0.67
Felton’s TS% 0.48 0.50 0.48 0.52 0.52 0.49 0.50
Smith’s TS% 0.58 0.60 0.58 0.52 0.55 0.51 0.52
Kidd’s TS% 0.52 0.50 0.55 0.58 0.50 0.52 0.53
Table 1: Historical Shooting Efficiencies (TS%) by New York Knicks’ Lineup
2006-2007 2007-2008 2008-2009 2009-2010 2010-2011 2011-2012 2012-2013
Anthony’s x 0.34 0.30 0.32 0.34 0.33 0.33 0.36
Chandler’s x 0.12 0.14 0.13 0.12 0.14 0.13 0.13
Felton’s x 0.21 0.20 0.21 0.19 0.21 0.19 0.21
Smith’s x 0.24 0.26 0.25 0.27 0.23 0.23 0.27
Kidd’s x 0.18 0.16 0.12 0.13 0.13 0.11 0.11
Table 2: Historical Fraction of Shots Taken (x) by New York Knicks’ Lineup
0.30 0.32 0.34 0.36
0.530.55
CarmeloAnthony_x
CarmeloAnthony_TS
Figure 1: Estimate of Carmelo Anthony’s Skill Curve. TS% = 0.55603 − 0.02158x
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4. 0.115 0.125 0.135
0.580.640.70
TysonChandler_x
TysonChandler_TS
(a) TS% = 0.5102 + 1.0957x
0.190 0.200 0.210
0.480.500.52
RaymondFelton_x
RaymondFelton_TS
(b) TS% = 0.6389 − 0.6758x
0.23 0.24 0.25 0.26 0.27
0.520.560.60
JRSmith_x
JRSmith_TS
(c) TS% = 0.6658 − 0.4586x
0.12 0.14 0.16 0.18
0.500.540.58
JasonKidd_x
JasonKidd_TS
(d) TS% = 0.57883 − 0.37649x
Figure 2: Estimated Skill Curves for (a) Chandler, (b) Felton, (c) Smith, (d) Kidd.
From the above figures, we see that four out of the five players show a negative relationship
between TS% and fraction of team’s shots taken, x, as we suggested before. The only player
who shows a positive relationship is Tyson Chandler. This isn’t surprising, though; Chandler
typically attempts only high-percentage shots such as dunks and put-backs. If Chandler is
getting more shots, it is likely that these shots are high-percentage shots. We must be careful
here, though; if we attempt to use the method of Lagrange with these skill curves, Chandler
will get a high shot allocation because, according to our estimated skill curve, he “becomes”
more efficient the more shots he takes whereas everyone else becomes less efficient. But
Chandler is not a skilled enough offensive player to create his own shots; many of his shots
are assisted. If he were asked to take more shots, we believe he wouldn’t actually be more
efficient despite what his skill curve says. To address this, we bound the allocation of shots
given to Chandler by the highest fraction of shots he took in the seven-year sample (0.141).
With the estimated skill curves for each of the five players, we can write down an equa-
tion for team efficiency and maximize it using the method of Lagrange to determine optimal
shot allocation (the constraints are that all five “fraction of shots taken” variables have to
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5. add up to 1 and that Tyson Chandler’s fraction of shots taken has to be less than 0.141).
For readers not familiar with this method, it is essentially a technique that is used to maxi-
mize a function subject to some constraint(s); it gives back values for “choice variables” that
maximize the objective function. In our case, the objective function is team efficiency and
the choice variables are each player’s fraction of shots taken. The function and constraints
are:
max
x1,...,x5
F(x1, . . . , x5) = x1f1(x1) + · · · + x5f5(x5)
subject to x1 + x2 + x3 + x4 + x5 = 1,
x2 ≤ 0.141(Chandler constraint)
where F is the team efficiency function, xi is fraction of shots taken by player i and fi(xi) is
player i’s skill curve (with i = 1 corresponding to Anthony, i = 2 corresponding to Chandler,
i = 3 corresponding to Felton, i = 4 corresponding to Smith, and i = 5 corresponding to
Kidd). Finally, we can compare the “optimal” shot allocation to the Knicks’ actual shot
allocation during the 2012-2013 season and determine whether or not Carmelo Anthony
actually did “shoot too much.” Of course, this methodology is not perfect (the five Knicks
players we consider were all on different teams over the time period we study), but we believe
the results will be interesting enough to compensate for the lack of rigor.
Results
The (rounded) results from the Lagrangian are shown in the table below:
Player Optimal Fraction of Shots Actual Fraction of Shots (2012-13)
Anthony 0.57 0.36
Chandler 0.14 0.13
Felton 0.08 0.21
Smith 0.15 0.27
Kidd 0.06 0.11
Team Efficiency 0.57 0.59
Table 3: Optimal vs. Actual Fraction of Shots Taken for Knicks’ Most-Used Lineup
Surprisingly, the results from our Lagrangian tell us that Carmelo Anthony should shoot a
lot more than he usually does. This can be explained by the fact that the slope of Anthony’s
skill curve is not too negative; as Anthony shoots more, his efficiency doesn’t decrease that
much. We also see that Chandler’s shot allocation is right at the constraint we set. This
is to be expected; as we discussed before, Chandler is the only player who becomes “more
efficient” the more he shoots. Felton, Smith and Kidd should all shoot less than they did this
past season. It is interesting that J.R. Smith, another player known for inefficient volume
shooting, is allocated fewer shots while Anthony is allocated more relative to the fraction of
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6. shots they took in 2012-13. This comes, again, from the slope of the skill curves; Smith’s
skill curve has a far more negative slope than Anthony’s. We see that the Lagrangian is able
to take this into account and allocate shots accordingly.
Another thing to note in this table is that the Knicks’ actual team efficiency in 2012-13
was greater than what our calculations said was the “optimal” team efficiency. This could
indicate that the Knicks outperformed expectations in 2012-13 and shouldn’t expect to shoot
that well going forward. If we go back to all five players’ historic shooting efficiencies, many
of them were more efficient in 2012-2013 than they were in the past few seasons.
As we mentioned before though, this is obviously a crude method of estimating optimal
shot allocation. There are many factors affecting player performance that we did not ac-
count for (quality of defenses they face, coaching, offensive system, etc.). A more accurate
skill curve might include some of these variables. Furthermore, in estimating the skill curve,
we are inherently assuming that data points in previous seasons are still relevant predictors
of how players will perform today. This may not be entirely accurate; players can change
their games to become more efficient.
Ultimately, we still get the interesting result that Anthony should perhaps shoot more de-
spite his alleged inefficiency, and that the Knicks may not be able to shoot as well as they
did in 2012-13 in future seasons. If it were possible to more accurately determine a player’s
skill curve, then we could get a better estimate of the optimal shot allocation among players.
Further refinements to the method we present here could change how basketball coaches use
their players and develop offensive game-plans.
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