In a recent game between the Milwaukee Bucks and the Detroit Pistons, the Bucks employed their typical **switch, slip, and show** match-up zone defense to clog the paint and swat shooters off the perimeter en route to a 115-105 victory. The methodology is fairly straight forward: On the perimeter, try to slips guards and drop screen men. Keep the defensive big on the block if the pain is clear, on the offensive player otherwise. If any cuts occur, switch the big onto the cutter, slip the cutter defender and show on all ball possessions in the paint.

## Example: Bucks Stop Here

Let’s see a typical defensive rotation in action:

In this play, **Blake Griffin** of Detroit runs a **dribble hand-off **(DHO) with **Langston Galloway**. Despite Griffin being a relatively strong perimeter shooter, **DJ Wilson** of Milwaukee **drops** to allow **George Hill** to slip the screen. I use the term slip, because there is no fight as Wilson gave him room to avoid the screen.

While the screen is happening, **Bruce Brown** of Detroit runs a **Deep Cut** from the top of the perimeter to the weak side corner. The aim is to **pull ****Khris Middleton** off the nail and tangle him with **Brook Lopez** in the paint. Middleton merely checks his weak side, sees that the weak side is currently clogged with **Glenn Robinson III **and **Eric Bledsoe** hovering about.

The defensive plan here is to keep drivers out of the paint and chase shooters off the line. The primary option for Detroit’s offense is to find a driving lane, which is now gone thanks to Middleton, Hill, and Wilson. As Wilson had dropped, the only true option for Griffin is to pop to the perimeter, which will require a pass over the shoulder from Galloway; who is not entirely known for hook passes for pick-and-pop three’s. Instead, the Pistons run through their second option.

As Galloway deep cuts to the weak side, we should expect the Bucks to anticipate a 1-on-1 situation between Griffin and Wilson. In the 1990’s this would spell almost certain doom as Griffin is as strong as they come. Instead, the Bucks drop back into their zone-style of play using a **check **from Brook Lopez to allow him to stray deep into the paint with a fresh set of three seconds.

Bledsoe begins to sag back onto the nail in attempt to cover Griffin’s dominant hand should he come crashing into the lane. Thanks to Lopez’s **switch**, Hill is able to **slip** the switch back onto Galloway, allowing Lopez to **show** within his three second window.

As predicted, Griffin turns over to his strong hand, causing Bledsoe to **blitz** Griffin. This minor miscue allows Robinson III to backdoor cut towards the basket. Griffin slips a nice pass to Robinson III into the paint. Despite this, Lopez is in **show **position and contests the field goal attempt with a **block **and Eric Bledsoe **defensive rebound**.

In the annals of play-by-play data, this Detroit possession will be logged as a **Robinson III FGA (3′ Cutting Layup), Lopez BLK, Bledsoe DREB.** It will ultimately be seen as **zero points out of one offensive possession **for Detroit and **one stop out of one defensive possession** for Milwaukee. In this case, the term **stop ** simply refers to a defensive possession where the offense scores zero points.

So the question is, **how do we quantify this stop?**

## Quantifying the Stop: Box Score

Commonly throughout a game we will hear phrases such as, “all we need is one stop” or “[Team A] made this a two possession ball game.” What these statements are referring to is the stop. A stop is simply a defensive possession that results in zero points: the defensive team has stopped an offensive team from scoring in that given possession. Ideally, we would like to assign credit to the defenders. In models such as **adjusted plus-minus **and **RAPM**, there is no credit assigning mechanism other than a **regression-based methodology **that isn’t an actual regression model (IE: short story is, there can never be a 0.14 player in the game; nor is he isolated. These are more akin to poorly managed fractional-factorial designs with heavy aliasing.). Using such a a model will identify some key traits of players, but the numbers themselves are effectively meaningless when relating to true defensive impact. That is, having a defensive RAPM of 4 just means you’re on the court during situations that positively affect the defensive rating more often than someone who has, say a defensive RAPM of 3. It doesn’t mean that player is one more point better per 100 possessions (it’s a biased estimator, remember) and it certainly doesn’t mean that the player contributes to 4 points worth of defensive efficiency (due to aliasing).

We can also use **RPM-style Bayesian models**. While RAPM is a Bayesian process, it’s not a Bayesian process in the eyes of a player. It’s merely a regularizer that controls the variation effect of the parameter space, not the model space where the players exist. In this case, we can apply priors based on **box score stats** that help reduce the effect of the bias of the aforementioned “regression” methods. Using box score type statistics as a prior distribution helps smooth the RAPM estimates to allow for **some** credit of defenders. For instance, the Bucks play above will give more credit to Brook Lopez and Eric Bledsoe, but only because Brook Lopez obtain a block and Eric Bledsoe obtained a defensive rebound. It’s certainly a flawed system, but takes into better account the defensive actions.

Another method is the **Stop Percentage**, as developed by Dean Oliver. In this case, Oliver focuses in on the instances in which a defensive player **terminates ** an offensive possession. And it is broken up into two “orthogonal” parts, which we will liberally call the **personal effect **and the **team effect. ** The result is a cascading equation that breaks down a play from zero points per defensive possession to the box score actions taken over that possession.

Let’s break this all down using Justin Kubatko’s breakdown of Oliver’s stops calculation.

### Partitioning Stops

The first step is partitioning stops into **personal stops** and **team stops**. This reflected into the equation

We define personal stops to be **steals**, **weighted** **blocks**, and **weighted defensive rebounds**. While we know that steals completely terminate the possession. Field goal misses do not. More importantly, defensive rebounds are not entirely attributed to missed field goal attempts. And using box score data, we cannot necessarily separate out free throw and field goal attempt defensive rebounds. Therefore, we need to incorporate a weighting scheme to understand how much a **block** would become a stop and a **defensive rebound **would become a stop as well.To do this, we need to compute three quantities: the **defensive field goal percentage, **the **opponent offensive rebounding** **percentage**, and the **forced miss weight.**

### Defensive Field Goal Percentage (DFG%)

Defensive Field Goal Percentage (DFG%) is simply defined as the field goal percentage of an opponent. It is given by

### Opponent Offensive Rebounding Percentage

Opponent Offensive Rebounding Percentage (DOR%) is also simply defined as the percentage of rebounds obtained by the offense during a defensive possession. It is given by

### Forced Miss Weight

Forced Miss Weight (FMwt) is a slightly more difficult number to compute. It is given by

This quantity appears to be backwards because we think of obtaining defensive rebounds on missed field goal attempts, while this equation coyly places defensive rebounds on made field goals. But that’s not the aim of this equation. The aim here is to **weight the value of a missed FG **versus a **defensive rebound. **In this case, the product is looking at **field goal attempts that either are made or defensive rebounded** versus **missed field goal attempts that are offensive rebounds**. IE: possession ending events on a FGA or possession continuing events.

### Personal Stops

With these three components in hand, we can compute personal stops as

How do we read this equation? Let’s walk through it.A personal stop is when a player obtains a **steal**, **block, **or **defensive rebound**. That’s the three addition components.

However, blocks and defensive rebounds don’t necessarily create stops. Take for instance, a made field goal, and And-1 foul, a missed free throw, and a defensive rebound. In this case, there is no stop on the possession. **This is where that DFG% comes in with FMwt above!**

The value of 1.07, while not in Oliver’s original work, is an adjusted value to account for the number of rebounds off of And-1 (and similar) free throws.In this case, for **blocks**, we have two components, the **blocks ****that results in forced misses** and the **blocks that result in made baskets**. The first part is obvious. The second part is nuanced as these are blocks that go out of bounds and stay in the offense’s possession or are offensive rebounds that result in points. We must subtract these out.

The third component on **defensive rebounds** are simply the remaining component of the personal stop, as we count all defensive rebounds and subtract out the ones that have had points scored on the possession prior to the defensive rebound.

### Team Stops

Now, the second step in computing stops is **team stops**. These are computed rather straightforward, albeit lengthy, using the formula

We call this a **team stop **as these components focus more on the team’s element on gaining a defensive stop. For instance, the first component identifies all opponent non-blocked field goal misses and estimates how many will result in defensive rebounds with no made field goals prior on the possession.

The second component counts the number of non-stolen turnovers committed by the offense and, assuming a uniform distribution over time, estimates the number that should have occurred while a player was on the court.

The third component estimates the number of free throw situations that result in two misses given the personal fouls committed by a player.

Let’s see how these components tie together with the Bucks example from above.

## One Play: One Stop

In the play above between the Bucks and the Pistons, we saw that there was indeed one stop on the play. We’d like to give much of the credit to Brook Lopez, but how much credit does he, and his teammates, deserve? Let’s start naively and suppose the entire game lasts one possession for illustration purposes.

In this case, we compute **DOR%** to be **0** as there are no offensive rebounds and **DFG%** to be **0** as there are no made field goal attempts. This will cause stress in the computation of **FMwt** as the denominator will become **0*1 + 1*0 = 0**. As this is a box score estimate, we should require a complete box score for this play. So let’s go back and leverage the teams’ box score stats.

For this Pistons-Bucks game, the Pistons were 42-89 from the field for **.472**. The Pistons also secured 10 rebounds out of 43 possible rebounds. Note that we are skirting the true rebound total as the actual NBA box score does not list team defensive rebounds. This gives us an estimated **.233 DOR%**. Now, we ascertain **FMwt** to be **.746**.

Since the entire team played this segment without breaks, for this particular play, we will have a factor of **0.2** on the unstolen turnovers. However, there are **zero unblocked FGA’s** and **zero unstolen turnvoers** and **zero personal** fouls. Therefore the team stops for this particular play is zero. This means that all contributions are personal driven.

### Brook Lopez

For Brook Lopez, we recorded one block on the play. This translates to a personal stop value of **.5600**.

### Eric Bledsoe

As Eric Bledsoe obtained the rebound, he also contributes significantly to the stop. In this case, Bledsoe’s personal stop value is **.254.**

### Everyone Else

For Wilson, Hill, and Middleton, they obtained no steals, blocks, or defensive rebounds on the play. In this case, they all come up Milhouse with a value of **.000**. What this ultimately means is that the credit for the stop comes out to be **.814, **slightly shy of the entire **one stop**.

Now, granted, this is a **box score result**. Therefore, the game shall become completed before we make the estimates. Applying this to one play is unfair to the analytic.

### Application to Full Game

By using the box score, we are able to extract out the estimated number of stops in the game. In this case, we have the following:

In case it is too difficult to read, this suggests that **Giannis Antetokounmpo **obtained 5.898 personal stops with 3.726 team stops for a total of 9.624 stops; leading the team for the night. Brook Lopez, on the other hand obtained 3.204 personal stops with 3.592 team stops for a total of 6.796 stops; good for second best on the team.

Continuing in this manner:

- Giannis Antetokounmpo
**9.624** - Brook Lopez
**6.796** - Eric Bledsoe
**6.180** - George Hill
**6.111** - Khris Middleton
**5.046** - Tony Snell
**3.733** - Pat Connaughton
**3.041** - Ersan Ilyasova
**2.997** - DJ Wilson
**2.474** - Christian Wood
**0.000**

This would suggest there were a total of 46.003 stops in the game. But how would we actually verify this?

## Play-By-Play

The easiest way is to crawl through play-by-play. By doing this, we find that there are exactly 12 stops in the first quarter, 9 stops in the second quarter, 11 stops in the third quarter, and 8 stops in the fourth quarter for a total of **40 stops in total**, identifying six over-estimated stops for the game. This means that despite the box score underestimating the number of stops for a single possession, the number of stops are actually higher across the entire game. **This is not always the case**.

This is actually expected as box score analysis is coarser than play-by-play analysis. However, if we shift our focus to play by pay, what are some methods we can use to determine credit for stops?

### A Naive Mathematical Way

One way is to count the number of stops and throughout the course of the game and then fit the number of defensive statistics to each number of stops and perform a “regression” of sorts. This will give answers, but will be quite volatile.

### A Naive Counting Way

The next way is to simply assign credit to each player based on their stats. We can walk through each possession, and if a stop occurs, we can either **blindly set attribution to 0.2** per player (uniform credit) which will drastically undervalue real defensive stoppers, or we can **weight** defensive statistics. For instance, in the example above, Brook Lopez gets the block and Eric Bledsoe gets the rebound. Let’s credit them with **.5 each**.

### Naivete is Not Enough

But in doing this, we drastically underestimate the amount of contribution supplied by Wilson, Hill and Middleton. If we recall, Khris Middleton’s hold at the nail as Brown attempted to pull the defense, along with Wilson’s drop and Hill’s slip on the fight-though stopped the primary option from occurring. If Middleton blindly follows Brown on the deep cut, Detroit sets themselves with a 1-on-2 with Galloway/Drummond on Lopez.

Furthermore, how do we credit Bledsoe’s gaffe that ultimately led to a pass to Robinson III for a layup? Should Lopez get more credit for the stop because of his read of the play and ensuing block? How do we put some of the onus on Robinson III for not taking a floater and instead crazy-braving himself into a 7′ tall shot blocker?

There’s only really one way: measuring the decision making process of defenders.

## Measuring Decision Making Processes

In an effort to understand how stops are created is to really dig in deep onto the X’s and O’s of a defense in response to an offense. Ultimately the game of basketball boils down to an offense making a series of decisions in an effort to force the defense to lose synchronization and open up regions of the court where there is a high probability to score. It’s a chess match where the offense primarily dictates the motion.

The defense, in response, can only implement counter-actions to force an offense to make poor decisions. In this vein we measure defensive contribution through the defender’s ability to move the offense into low probability areas of interest. This, by the way, is an open thread of research:

### Detroit’s Initial Offense

One way to start crediting stops is to look at Detroit’s early offense. Recall this was a DHO between Griffin and Galloway after a reversal and weak-side deep cut from Brown. Middleton’s hold on the nail along with the slip by Hill eliminated the driving lane, which may have been there in the past. Therefore, we can look back at all initializations of this early offense (across all roles) and see all the various directions of play occurred. One way to do this is through **ghosting** to train the **average defensive response**. Then using the ghosting output, we can build a markov model that estimates the decisions made by the offensive team. Using the ghosting/Markov model, we obtain a **probability distribution **on the actions. And we find out in this case, based on this year, the Pistons tend to score an effective 1.02 points off that action.

By the two actions from Hill and Middleton, Detroit’s expected points scored on the play dips to 0.83 points. **That’s a positive 0.19 differential. **If we remove Bledsoe, attach him as a tether to Brown by implementing a **Brown Position – Brown Velocity + Noise** model, and run the Markovian model; Detroit’s expected point value increases to 1.11 points. Therefore that **Middleton action ****may have seemed meaningless, **but it saved the Bucks potentially **0.28 points**.

### Challenge: Making the Pieces Fit

The challenge then becomes “how do we integrate these components on defense?” For instance, Detroit’s expected point value actually increases with the blitz from Eric Bledsoe; from 0.92 points to 1.08 points. Fortunately, Lopez’s show and Robinson III’s extra step drops the value down to 0.98 before the shot, which is ultimately blocked.

If we integrate out the actions, we flatten out most of the work performed. Therefore, some form of localization between defenders needs to be identified. In the end, Lopez, Hill, and Middleton should get most credit for the play as they thwarted the primary option (Hill/Middleton) and then eliminated a gaffe on a back-turning blitz (Lopez).

And on a further note, the next question is whether the **template** **of the defensive scheme **is the real stopper in this situation as Middleton and Lopez play their roles correctly. How do we quantify this effect? How much credit does Budenholzer get for this? Does he deserve credit?

It’s definitely a real challenge. But if you can figure this out, I’ll see you at the next Sloan Conference presenting your work. For now, we rely on carefully thought out work by Dean Oliver, as missing six stops isn’t bad at all. The next game will be -3, the next 1. It’s all an approximate process holding a place for when we figure out how to better quantify the X’s and O’s.

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