A problem that was posed in Dean Oliver’s Basketball on Paper (2003) revolved around the way one can credit assists. In his book, Oliver laments the lack of data available for crediting assists and he begrudgingly uses the one-half rule: credit both the scorer and the assist man with one-half of the assist. However, in the game of basketball, it is not true for all assists to be equal. Let’s take for instance two differing types of cherry-picked assists.

Rondo Assist: Who Deserves More Credit?

Let’s consider a simple play where an uncontested transition turns into an assisted, uncontested transition.

In this play, Rajon Rondo secures the ball on defense after a bad pass is made by the Chicago Bulls. Rondo, takes the ball nearly the length of the court uncontested. Instead of scoring, he drops the ball off to a trailing, also uncontested, Ray Allen. Allen finishes the “pick-six” transition with a dunk.

It’s easy to suggest this moment is a stat-padding situation for Rondo. However, the score would not have been made had Rondo not already initiated the transition. Similarly, Allen would not have scored had he not trailed the transition. Many arguments can be made here about the assist such as “empty” (zero) assist, or “equal” assist, or “scorer” assist. And the arguments, under certain assumptions, are all valid.

But to pull a page out of Dean Oliver’s book: “Two points is two points is two points.” And by definition, since this is an assist, credit needs to be split judiciously. Sorry zero assist guys…

Steve Nash: Who Deserves More Credit?

In this example, we see a standard drive and dump that Steve Nash is well known for.

Here, Nash pulls the defense by driving through the lane. With the double team collapsing on him mid-lane, Amar’e Stoudemire pivots off his helping defender, taking a difficult pass for an uncontested dunk.

This is the style of play that has made Nash such a valuable player to teams and it’s easy to admit that most of the assist credit belongs to Nash. However, Stoudemire deserves some credit too for making himself available to Nash and allowing his defender to pull off him. But the question is… how much credit?

And to answer that, we can propose a probabilistic argument.

Crediting Assists: Probabilistic Argument

For many years running, the argument I have always made when crediting assists (or passing in general) is to credit the passer more whenever they execute a pass that increases the chances of scoring for a team. The argument is as follows: If a passer passes from a lower probability of scoring to a higher probability of scoring, then the passer has found a “more open” scorer. If a passer passes from a higher probability of scoring to a lower probability of scoring, then the passer has either made a bad pass or is looking for their teammate to bail them out of a situation (if an assist occurs). Furthermore, if the probability of scoring does not change, such as a meaningless touch pass between two players on the perimeter, then both players are interacting with equal responsibility.

The argument all comes down to the fundamental concept of blame.

Therefore, the problem of crediting assists merely turns into an exercise of computing the probability of scoring; yet another non-trivial task. Fortunately, there are several resources for developing such a probability model.

So for the sake of argument, we select one such model and jump into analyzing the probabilistic argument of crediting assists.

Probabilistic Attack

For the sake of argument, let us consider a pair of binary variables. The first binary variable is whether a potential assist occurs. The secondary variable is whether the field goal is made. This means that there are four outcomes when it comes to making a decision to pass for potential assist:

• Pass leads to a score.
• Pass leads to a miss.
• Kept leads to a score.
• Kept leads to a miss.

Taking these binary variables, we should be able to form a contingency table outlining potential assists and their relationship to scoring. The difficulty here is that we cannot simply count made and missed FG’s on non-potential assist situations. Fortunately through leveraging a probability model we do get to estimate the probability of scoring from where the pass was made, if such a pass did not exist.

That is, we key off the potential assist locations of the passer and the shooter. Using the probabilistic model, if we say the pass was not made, how likely will the team score? Similarly, if the pass was made (which it is), how likely will the team score? In this setting, we don’t count fictional makes and misses; but rather use the estimated and therefore known probabilistic quantities.

I’ve Got a Fever…

In this setting, we can think of a potential assist as a treatment being applied to a receiver’s itch to shoot. Since the potential assist comes from the passer, the probability of making a field goal becomes the dependent variable.

So let’s go back to that Rondo assist. On the play, Rondo obtains the rebound and has a clear path to the basket. He drives one-on-none and then pulls up, only to dump the pass to a teammate. Given the locations of the players, who the players are, and the state of the game, using a basic logistic regression method for estimating likelihood of scoring, Rondo has a .9342 chance of scoring. After the pass, Ray Allen has a clear path to the basket for an easy dunk with a .9736 chance of scoring. While the pass increases the odds of scoring, the impact of the assist is minimal. In the counting stats it simply reads as Rondo 1 assist, Allen 2 points.

So how much credit does Rondo deserve for this? Let’s reconvene on the binary variables:

In this setting, we can consider the relative risk; which looks at the impact a potential assist has on a made field goal in a particular situation. In this case, the relative risk is given by:

Computing it for the Rondo assist, we have a relative risk of 1.0422. So how do we interpret this value?

Interpretation

If the relative risk is one, then this shows that the “exposure” of a potential assist does not impact the probability of scoring. As the relative risk increases, we start to see that “exposure” to a potential assist actually increases scoring and therefore the passer has accurately identified a passing lane that leads to a better opportunity to score. This indicates that the passer not only identified the improved situation, but also reacted accordingly.

Analogously, as the relative risk decreases, we identify situations where the scorer effectively “bails out” the passer as the pass went from a higher opportunity of scoring to a worse scenario. In this situation, the scorer deserves more credit as they are able to convert a basket despite the incorrect decision made by the passer.

Splitting Credit

Using this construct, we can merely designate credit through the weighting of the relative risk. In the Rondo assist case, the relative risk is 1.0422. Assuming that no impact is one, we baseline the scorer with a value of 1 and therefore assign credit as

In this case, Rondo is credited with 0.5103 of an assist, and Ray Allen is credited with 0.4897 of an assist.

What About the Nash Assist?

In the situation where Steve Nash dished to Stoudemire for a score, the probability that Nash would score under the double team was estimated to be 0.4273. While the uncontested dunk for Stoudemire was estimated to be 0.8842. For this play, the relative risk is 2.0693. Using the splitting credit computation, we suggest that Nash deserves 0.6742 of an assist; while Stoudemire earns 0.3258 of an assist.

As we can see, no longer are we being pushed towards a “fifty-fifty” split of credit and we are finding that credit can be obtained based on the relative degree of difficulty of scoring a basket through a probabilistic argument.

Model Flexibility

Suppose you feel we are being too stingy for giving Nash credit on that assist. That’s not a problem. We can introduce an amount of flexibility through the use of a sigmoid function. Consider the relative risk, r. As this is the variable for determining credit, the larger it is, the more credit is given to the passer. Therefore, the sigmoid function can “push” the value out a little further through the function

So let’s test this out. The value beta is the flexibility factor. If beta is 10. Then Steve Nash’s assist credit goes from 0.6742 to 0.999977. Similarly, if beta is 2, Nash’s assist credit becomes 0.8946.

Similarly in the Rondo setting, for beta = 10 we have .6040 credit. For beta = 2, credit becomes 0.5211. So we see how this sigmoid function stretches higher probability changes to larger crediting of passers. The same is true for scorers if values of beta are below one. Hence the flexibility.

We should note, however, the sigmoid function is merely a transformation on the relative risk. As the assist credit formula above is the exact same computation except using the transformation

Now What…

Now that we are armed with a method for crediting assists, we can leverage any probability model of our choosing and start revamping quantities such as points produced or even adjust assist leaders‘ totals to reflect the difficulty of assists. The former helps us start to understand how much a player actually produces without having to require a 50/50 split. The latter begins to deflate stat-padding situations.

That said, leveraging a data set obtained from a Western Conference team in 2015; we see this subtle change in assist totals:

Using the raw (beta = 1) adjustment, we see a bit of a shuffle. In fact, Deron Williams jumps up to nab the 15th spot after adjustment.

However the adjustment is applied, we can now start developing models to help determine credit for assists. And one such method can leverage the relative risk of a potential assist treatment.