Balgebaskestry
(some algebra, with a flavor of chemistry, applied to basketball)
To help assess a player's contribution, it would be useful to know the
value of various things, like field goals, missed field goals, rebounds,
and turnovers. By working with balanced equations, that express
seemingly undeniable truths, the values of the items previously listed,
can be solved for, and expressed in terms of points (as in points on the
scoreboard) and an "unknown" constant, b.
b will denote the value of possessing the basketball (and so it
makes sense to refer to b as ball). Surely, a team having
the ball, as opposed to the other team having it, is worth something;
otherwise one should not get at all upset when their team commits a
turnover and the opponent takes possession of the ball. So for now,
think of b as some positive value.
The theory of balgebasketry makes use of axiom and a simplfying
assumption. (Really, if I wanted to view this as pure mathematics, I
suppose that both of the statements below, as well as a few of the
initial expressions that follow, should be taken to be axioms.
But while I may add a few theorems with proofs later, I don't want to
spend too much time worrying about the most correct way to present my
results. Rather, I want to spend my time worrying about whether my
results are useful, and so, I may be informal below.)
- Axiom
- An offensive rebound and a defensive rebound have the same
value. (One might think that an offensive rebound should have extra
value, since an offensive rebound may lead to an easy shot. But it
needs to be noted that a defensive rebound corresponds to denying the
opponent an offensive rebound, and denying the opponent something of
value is also valuable --- just as valuable. It seems that to make
things balance, the two types of rebounds need to be considered to have
the same value. As for the added value of an easy shot following an
offensive rebound, that should show up in the rebounding player's shooting and
scoring numbers (and we need to be careful not to double count things).)
- Assumption
- A missed three point field goal attempt has the same value as a
missed two point field goal attempt. (In both cases, the ball has
not gone in the basket, and is up for grabs. One might think that a
missed three point attempt may bounce farther and be more easily
rebounded by the shooting team, but the bouncing of the ball off of the
rim or backboard depends on more than if the shot was a two or three
point attempt: a long two point miss typically shouldn't bounce much
differently than a short three point miss, and the shooting style
of the shooter could influence the bounce also. While it should not be
thought that all missed shots bounce equally, in order to have a relatively
simple mathematical model to work with, it's convenient to make the
simplifying assumption that a missed two point attempt has the same
value (a negative value) as a missed three point attempt.)
Below, I will develop my results by considering two teams playing one
another. Team
A's
actions and results will be shown in
red,
and team
B's
actions and results will be shown in
blue.
I'll let
- FG denote the value of a two point field goal;
- TR denote the value of a three point field goal;
- MFG denote the value of
a missed field goal attempt (whether it be a 2
point attempt, or a 3 point attempt);
- RB denote the value of a rebound;
- TO denote the value of a turnover.
(Recall that b denotes the value of possession of the ball.)
The five rows in the table below correspond to five sequences of events
that could occur in a game.
In words, we have the following for the five sequences.
- sequence 1
- A starts with the ball.
- A shoots and misses.
- A rebounds the missed field goal attempt.
- A ends with the ball.
- sequence 2
- A starts with the ball.
- A shoots and misses.
- B rebounds the missed field goal attempt.
- B ends with the ball.
- sequence 3
- A starts with the ball.
- A commits a turnover.
- B ends with the ball.
- sequence 4
- A starts with the ball.
- A shoots and makes a 2 point field goal.
- A is awarded 2 points.
- B is given the ball after the made shot,
and ends with the ball.
- sequence 5
- A starts with the ball.
- A shoots and makes a 3 point field goal.
- A is awarded 3 points.
- B is given the ball after the made shot,
and ends with the ball.
| |
what they
start with |
what they do |
what they
end with |
| sequence 1 |
b
|
MFG RB
|
b
|
| sequence 2 |
b
|
MFG
RB
|
b
|
| sequence 3 |
b
|
TO
|
b
|
| sequence 4 |
b
|
FG
|
2
b
|
| sequence 5 |
b
|
TR
|
3
b
|
A few things can be noted about these sequences.
- sequence 1 indicates that a missed shot followed by a
rebound results in no net change (and so a rebound can "cancel out" a
player's missed shot).
- sequence 2 and
sequence 3 together indicate that a missed shot followed by a
rebound by the opponent has the same net effect as a turnover.
- sequence 3 and
sequence 4 together indicate that a two point field goal has
the same net effect as a turnover, except that the shooting team
is awarded 2 points. (The fact that something (the ball) is lost when a
field goal is made should not be overlooked. The net difference before
the made shot and after the made shot is not 2 points --- the
ball also changed hands from one team to another.)
- sequence 3 and
sequence 5 together indicate that a three point field goal has
the same net effect as a turnover, except that the shooting team
is awarded 3 points. (The fact that something (the ball) is lost when a
field goal is made should not be overlooked. The net difference before
the made shot and after the made shot is not 3 points --- the
ball also changed hands from one team to another.)
The equations below relate to the net change in the difference in the
value of
A
and the value of
B.
On the left side of each equation are put the starting values, and the
values corresponding to the actions that take place. The ending values
are on the right side of each equation. On both sides of the equations,
the values for
B
are subtracted from the values for
A,
and so both sides give the difference in the value of
A
and the value of
B.
The equation corresponding to sequence 1 is
b
+ MFG
+ RB
= b,
which implies
MFG
= -RB.
(1)
The equation corresponding to sequence 2 is
b
+ MFG
- RB
= -b,
which implies
MFG
= RB - 2b.
(2)
The equation corresponding to sequence 3 is
b
+ TO
= -b,
which implies
TO
= -2b.
(3)
The equation corresponding to sequence 4 is
b
+ FG
= 2
- b,
which implies
FG
= 2 - 2b.
(4)
The equation corresponding to sequence 5 is
b
+ TR
= 3
- b,
which implies
TR
= 3 - 2b.
(5)
The -2b term in
(4) and
(5)
correspond to a loss in scoring potential from the change of possession
of the ball (sort of like in physics how potential energy is lost when
energy is used to do work).
To recap, we have the following equations.
MFG
= -RB.
(1)
MFG
= RB - 2b.
(2)
TO
= -2b.
(3)
FG
= 2 - 2b.
(4)
TR
= 3 - 2b.
(5)
These five equations can be used to obtain other relationships.
(1)
and
(2)
together imply that
-RB
= RB - 2b,
which implies
RB
= b.
(6)
(1)
and
(6)
together imply that
MFG
= -b.
(7)
Note that now
TR,
FG,
MFG,
RB, and
TO can be expressed in terms of points and b.
FG
= 2 - 2b.
(4)
TR
= 3 - 2b.
(5)
MFG
= -b.
(7)
RB
= b.
(6)
TO
= -2b.
(3)
(Before continuing on, perhaps an example to check what's been done so
far will be instructive. Suppose that a team has the ball, makes a 2
point field goal, their opponent inbounds the ball after the made
basket, and then turns it over to the team that just scored. The net
effect of all of this is that the team that originally had the ball now
has it back, and they also have 2 points. So before the
FG and
TO sequence of events, the difference in the value of the two
teams is
b, and after the
sequence of events, the difference in the value of the two
teams is
2 + b. So as a check, if
one starts with b, adds
FG,
and subtracts TO
(the subtraction being because the
difference in the values of the two teams is being considered,
and the opponent of the shooting team had the turnover),
the result should be
2 + b. This is indeed the case since it follows from
(3)
and
(4)
that
b
+ FG
- TO
= b
+ (2 - 2b)
- ( -2b)
= b
+ 2.
One can consider more complicated sequences of events invloving
TRs,
FGs,
MFGs,
RBs, and
TOs, and the exercise of writing an appropriate equation, and
simplifying it using
(3),
(4),
(5),
(6),
and (7), will always yield a result that
is in agreement with the reality of what would happen in an actual basketball
game.)
Sutton Value System
a system which gives a partial measure of a player's overall value to the team based on
data obtained from box scores, and so not including defensive contributions
other than steals and rebounds
Above, I developed the following:
FG
= 2 - 2b.
TR
= 3 - 2b.
MFG
= -b.
RB
= b.
TO
= -2b.
b denotes the value of possessing the basketball, thus making a change
in possession (e.g., a turnover) worth -2b. After a lot of
thought, I have decided that a value of 0.5 (points (as in points on
the scoreboard)) is a reasonable choice for b. This choice
results in the values shown below.
| |
value |
| TR |
2
|
| FG |
1
|
| MFG |
-0.5
|
| RB |
0.5
|
| TO |
-1
|
At first, it may appear odd that a 2 point field goal is worth 1, and
a 3 point field goal is worth 2, but a way to perhaps get comfortable
with the concept is to consider that it can reasonably be expected that
a team score about 1 point, on average, per possession. In light of
this, the value of 2 for a 3 point field goal represents the number of
points, exceeding the expectation of 1, that went onto the scoreboard
when the shot was made. Similarly, the 1 for a 2 point field goal is
the extra point, in excess of the benchmark value of 1, that the
player's successful shot resulted in. It can be noted that, with this
way of assessing things, -1 is a
sensible value for a turnover, since when a turnover occurs, the
possession ends with the team having scored 1 point less than the
benchmark value of 1. A missed field goal, followed by the shooting
team giving up a rebound to the opponent, is equivalent to a turnover,
and likewise results in a change in value of -1 using my system.
To make things easier to implement, for my value system, I'll
double the above values above to arrive at the easier-to-use values
below. (Note: These values are typical of those used in fantasy
leagues, except that I weigh turnovers more heavily than is typical,
and there is the perhaps seemingly strange value of 4 for a 3 point field goal.
Although, with these new values, the value of a 3 point field goal is
harder to explain, it should be kept in mind that its relative value,
compared to the others, does make sense when one relates all of the
values to deviations from the expectation of averaging 1 point per
possession.)
| |
value |
| TR |
4
|
| FG |
2
|
| MFG |
-1
|
| RB |
1
|
| TO |
-2
|
Although I won't bother to add explanation at this time, after giving a
lot of thought to the matter, I've chosen to augment the above list of
values in order to include more aspects of the game into a player's
overall value. While the extra values added below aren't as clearly
defensible as the original values arrived at, I believe that it is much
better to make perhaps imperfect adjustments for each
block (BLK),
assist (A),
foul (F), free throw (FT), and
missed free throw (MFT),
than to not include them at all. (Giving a steal
(S) the value of 2, seems perfectly defensible, since a player
should be able to cancel out a turnover with a steal.)
| |
value |
|
TR |
4
|
|
FG |
2
|
|
MFG |
-1
|
|
RB |
1
|
|
TO |
-2
|
|
S |
2
|
|
FT |
1
|
|
MFT |
-0.5
|
|
F |
-1
|
|
A |
1
|
|
BLK |
0.5
|