Is
there one statistical measure that can determine with finality the value of
each player? Something that can be used to compare Kevin
Garnett to Bob Cousy, or Jason Kidd to Kenyon Martin?
Many attempts have been, usually a linear combination of various officially
reported statistics, sometimes on a per-minute or per-possession basis. I have
looked in depth previously at net team points, and its many variants. Today, I
present the principles of what may be the ultimate foundation for evaluating
player performance: spread options. It’s not just one statistic; it’s a whole new
field of study.

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Is
there one statistical measure that can determine with finality the value of
each player? Something that can be used to compare Kevin
Garnett to Bob Cousy, or Jason Kidd to Kenyon Martin.
Many attempts have been, usually a linear combination of various officially
reported statistics, sometimes on a per-minute or per-possession basis. I have
looked in depth previously at net team points, and various off-shoots. Today, I
present the principles of what will be the ultimate foundation for evaluating
player performance. It’s not just one statistic; it’s a whole new field of
study.

Like
physicists searching for a grand unified theory of everything that would
combine relativity and quantum mechanics, we can first list the properties we
would like our ultimate statistic to have. The ultimate statistical theory of
basketball:

·
Should
reflect the difference between actual performance and expected performance of a
fill-in player. Note that it should not reflect the difference between the actual
performance of a player and the expected performance *of that player*. Otherwise, consistency is punished. Nor should it
reflect the difference between the actual performance of a particular player
and the actual performance of another particular player, because then you are missing
two expectations, and also encouraging teammates to cheer against their
replacements, causing unnecessary team friction.

·
Should
NOT differentiate between teammates who share the exact same minutes. If two
players are on the court at the exact same times, and one of them has a vacuous
stat line while the other has lit up the scoreboard, it could be the case that
the vacuous one drew so much attention, or set such great picks, that without
him, the other guy wouldn’t have accomplished anything. Basketball is by
definition a team game, and though it is often clear by watching a game who is the
greatest player, it is not always, and we are looking for an objective measure
of worth.

·
Should
reward only the net gains of the team, offense minus defense, in evaluating
players. A player who leaks like a sieve on defense but scores consistently on
offense may be a worse player than someone who doesn’t score a lot, but can shut
down the entire opposing team, e.g. Bill Russell.

·
Should
be based on some kind of per-minute or per-possession evaluation to partially
alleviate the potential bias created by coaches’ decisions on playing time.

·
Should
weigh more important shots more heavily. A game-winning bucket in the final
seconds is of far greater importance than one in the middle of the second
quarter, even during the seventh game of the Finals. There is more attention by
both teams on the final possession if it can decide the outcome of a game, and
the best players are, or should be, on the court.

The concept I’m about to
introduce is easier to understand from the background of net team points, which
is just the amount by which a player’s team outscored the opposition during the
time the player was on the court. This is sometimes called plus-minus, but because
plus-minus can also refer to a variant of this in which the net score earned by
the team during the player’s time on the bench is subtracted, I prefer to refer
to it more clearly as net team points. As an aside, the plus-minus variant
where the performance of a player’s replacement is subtracted is terrible for
team morale, as it encourages players to hope for their replacements to play
poorly. Let’s just hope the Dallas Mavericks do not have such a statistic in
the contract for their players; it would destroy the team.

Net team
points solves
virtually all of the bullet points above, but it misses the final two. Net team points per minute or per possession solves all but
one, but it is an important one: net team points (or even net team points per
minute) rewards garbage time equally with important crunch time performance.
The approach I’m going to outline now solves those problems.

The approach I propose as
a new foundation for evaluating players is to calculate the option value they
provide. Two examples will clarify. Suppose you’re a team that consistently allows
your opponent, any opponent, 80 points a game. Your own scoring is more
sporadic, sometimes getting only 60, sometimes as many as 100. Then you are
essentially long a call option on your offense; that is to say, you win games
if you score more than 80, much like the holder of a call option on stock
struck at $80 makes money if the stock rises above $80. Standard Black-Scholes option valuation means that in this situation, you
would like your volatility to increase. In other words, you want to play your
volatile players, your streaky players, the ones that sometimes score 30 and
sometimes score nothing. Why? For the same reason that if you’re long a call
option on a stock, you want that stock to be bouncy: in most cases, higher
volatility increases the probability that it ends up above the strike price.

Alternatively, suppose you
are a team that consistently scores 90 points, regardless who guards you or
what defense they play. On the defensive end, however, you are sometimes an
unstoppable force and sometimes a sieve. Then in this case you are essentially
short a call option on your defense; that is to say, you win games if you keep
your opponents under 90, much like the seller of a call option on stock struck
at $90 makes money if the stock stays below $90. Standard option valuation
means that in this situation, you would like the volatility of your defense to
decrease. In other words, you want to play your safe and consistent defensive
players. Why? For the same reason that if you’re short a call option on a
stock, you want that stock to be less bouncy: it most cases, lower volatility decreases
the probability that it ends up above the strike price.

Putting these two concepts
together is the key. Essentially, the game of basketball is a package of
options, one long and the other short, where they in general are looking to
have risky offense and conservative defense. There is a particular type of
option that describes this specifically, called a spread option.

A spread option pays off
the difference between two risky assets. The initial strike price on the spread
option is zero, because the score starts out zero-zero. You could express the
value of a spread option, if you want to, in terms of the (risk-neutral) probability
of your winning the game, i.e., in terms of the probability of your offense scoring
more points than your defense allows. Then each point scored can be expressed
in terms of the change to this implied probability of winning the game.

This means that a
game-winning shot in the final seconds can change the implied probability of
winning from 0% to 100%, the largest possible increase. Meanwhile, scoring the
first point of the game on a free throw may change the probability of winning
from 50% to 51%. In other words, the crunch time points end up counting more,
which is one of the desired features of a panacea statistic. Finally, we can
look at the change in implied winning probability on a per-minute of playing
time basis for each player.

In the very near future, I
will be presenting the results for the regular season so far, at which players
tend to participate in probability-enhancing runs.