As a college basketball coach, I’ve always found a strange and symbiotic relationship between statistical analysis and informing what actions are taken on the court. Ben Falk over at Cleaning the Glass, as well as many others, has written extensive and eloquent pieces on the importance of striking a balance between these two polarities. The point of statistical analysis is to find data and trends that can lead to a competitive advantage. The NBA, and sports leagues as a whole, have invested great deals of time, money and effort into seeking such advantages.

One area that constantly sticks out like a sore thumb when watching a basketball game deals with intentionally fouling. No, not the type of fouling done late in contests to elongate a game. This is the Hack-a-Shaq kind of foul—a strategy-based decision by a defensive team to send one specific free-throw shooter to the line. The object is simple: Control which player attempts to score on a given possession, and, by choosing one who is poor at free-throw shooting, the offense won’t perform as well as on a normal possession.

In wrestling with the strategy’s merits, I go back to one of the more specific and, quite frankly, brilliant implementations of it a few years ago as the Houston Rockets hosted the Detroit Pistons. The Pistons were up nine at the half, and interim head coach J.B. Bickerstaff inserted K.J. McDaniels, a seldom-used reserve, into the lineup at the start of the third quarter to guard Andre Drummond.

McDaniels’ sole purpose? To intentionally foul Drummond away from the ball so the Rockets wouldn’t have to wait to reach a certain foul count before utilizing the Hack-a-Drummond strategy:

The strategy isn’t aesthetically pleasing to fans, nor is it able to be employed for long stretches of time (NBA foul-out rules hamper its durability). The league has, in fact, legislated rule changes since this game to alter the frequency and success of such a strategy. Nonetheless, coaches have gone to the well over the years and decided to intentionally foul opposing players, both slowing down the game and playing a mental chess match with their opponents.

In the matchup between the Pistons and Rockets from January 2016, Drummond had 36 free-throw attempts, 26 of which came in the second-half. He only made 13 of them, but the Pistons won the game. From Houston’s perspective, was such a strategy really worth it? The results are wildly hit-or-miss, but the question from a theoretical perspective still holds plenty of merit. Is such a strategy worth employing, and when (or on whom) would it fall into that category?

Understanding the Relevant Numbers

Think about each shot as an investment on the part of the offense. Each time they attempt a field goal, they are looking for as high a return on investment—a make—as they can get. The metric we use to weigh the effectiveness of those investments is points per possession. From an offensive standpoint, teams aim to generate shots that will reliably produce a greater amount of points per possession. Inversely, defenses seek to prevent teams from scoring, and they can measure their success through comparisons to the average across the league. Offenses try to ratchet those numbers higher by selecting shots from the right locations, while defenses discourage them. On a whole-season basis, this type of study can inform the effectiveness of a scheme.

Per Synergy Sports Tech data, the average points per possessions from all teams throughout the league last season measured at 0.976 PPP. In layman’s terms, when all possessions are averaged out throughout the season, teams score approximately 0.976 points each trip down the floor. That number, however, includes transition opportunities, where the NBA is quite clear that intentional fouling will result in free throws and retention of possession for the offensive team. The strategy is accordingly inapplicable, so the number must shift to include only half-court efficiency: 0.949 PPP.

Not all shots are created equal, though. General wisdom conveys the point that the highest-percentage shots are the ones that come closest to the basket; the farther away a shot comes, the lower the odds are that it goes in. However, the three-point line adds a wrinkle into shot selection. At a certain point away from the basket, the shot actually becomes more valuable because it adds an extra point to the scoreboard. This is why points per possession is the right metric to use for understanding best practices associated with intentionally fouling. It naturally includes and already calculates the shot types an offense is highlighting.

It should also be noted that NBA rules further simplify the equation and deter intentional fouling tactics more than in the college game. Every time a foul occurs that results in free throws (with the exception of a one-shot technical and any and-1 situations), the free-throw shooter receives two free-throw attempts. The expected points per possession on free-throw attempts for an individual is simply the shooter’s free-throw percentage multiplied by two, since the percentage is calculated per shot, and the shooter receives two. At collegiate and high school levels, the one-and-one makes intentionally fouling more appealing before the double bonus is in effect, as a miss on the first free throw causes an end to the possession, thus lowering the return for the offense.

Getting Specific, Team By Team

The league-based averages are a good springboard because they dictate a starting point at which we can expect positive results when fouling. Just as not all shot locations are created equal, no teams are either. Taking the league-average offensive output as a data point and expecting to apply it universally would be foolish. The Golden State Warriors and last year’s worst-performing offense in terms of points per possession, the Sacramento Kings, finished the season 0.118 points per possession apart. For every 20 possessions, the Kings would score 17.9 points, and the Warriors would post 20.2.

The best way to account for the difference would be to do the math on every team in the league. The equation would then fluctuate and change throughout the season as the PPP of each opponent changes. For the Warriors, who averaged 1.012 PPP, you would have to be worse than a 50.6 percent free-throw shooter for intentional fouling to lower the expected PPP. For the Kings, however, that number is dramatically lower, as their 0.894 PPP would theoretically save all shooters above 44.7 percent from the strategy. Each team needs a different marker, as determined by their typical output.

Based on last season’s half-court numbers, here is the free-throw percentage a player needs to be beneath in order for fouling them to make sense:

In a vacuum, any free throw shooter would need to be better than 50 percent to be above the lowest expected intentional foul rate on any team in the league. However, offensive rebounding numbers change the metrics and need to be accounted for in the process.

Offensive Rebounding & Weighing the Other Variables

Another factor largely at play deals with offensive rebounding and rebounding rate. Per a study from Kirk Goldsberry, the average team records an offensive rebound on a missed free-throw approximately 12 percent of the time. If that is the case, and teams score approximately 1.11 PPP on offensive boards, that number must be factored into the equation.

Of course, offensive rebounding rates will also change based on who is attempting the free throw. If the best rebounder is the one shooting, the rate might decline. Conversely, if the offensive team is anticipating a miss and ready to crash the glass, the rate might increase. Each shooter would have his own metric, but it is an unreliable measure because it doesn’t indicate who is on the floor. As such, it might be best to deal with the league average of 12 percent as opposed to trying to develop a flawed equation that encompasses all other variables. Assuming that each opportunity occurs at a league-average rate gives coaches a safe, albeit rough, estimate of what to expect.

But if we’re going to factor in the offensive rebounding rates of free-throw opportunities, we must do the same for the half-court metrics as well. While the PPP remains the same at 1.11, the difference comes in rebounding frequency. While 12 percent of available misses are corralled by the offense off a free throw, last year’s league average for live-ball scenarios was 22.3 percent. On its face, that means offensive rebounds are nearly half as likely on free throws. A defense would then, by comparison, need to weight how much they favor that expected rebounding return into the decision of whether they should foul.

So what is the best way to account for offensive rebounding rate given these two factors?

Here’s my math: The coefficient for a missed free throw’s offensive rebounding opportunity is an additional 0.133 PPP (1.11 PPP on exactly the league-average 12 percent rate of rebounding a missed free throw). The coefficient for a missed field goal attempt’s offensive rebounding rate is an additional 0.2475 PPP (1.11 PPP on exactly the league-average 22.3 percent rate of rebounding). I then add the field-goal rebounding variable of 0.2475 PPP to the PPP of a normal possession for each team.  Then I subtract the free throw rebounding adjustment from that expected PPP, and that creates the baseline.

The updated numbers, with the average coefficient added, shift as follows:

Let’s tie this back into intentional fouling strategy. The object of intentional fouling is to lower the PPP—the return on investment of a possession for the offense—by selecting a free-throw shooter whose percentages from the charity stripe are lesser than the team’s general offensive output. If a defense can control which player will be shooting free throws, the equation is fairly simple and becomes a game of odds that, statistically speaking, should tip in the favor of the defense.

The decision to employ the strategy is now even more complex. Such a decision has non-mathematical factors at play to add variables into the equation: the impact of the player committing the foul and their foul count, the time and score of the game, the sample size of the free-throw shooter and the jurisdiction of the referee under intentional foul and excessive force rules that are subject to interpretation. Intentionally fouling also lends itself to fewer transition opportunities for an offense. A team heavily reliant on transition might sour on the strategy if it prefers more fluidity within the game.

Perhaps the largest independent variable to be accounted for comes from flow of the game, where one team can be over or under-performing their expected output on the season. If the Golden State Warriors, for example, are 0-of-20 from three on the game, are the odds that attempt No. 21 goes in any lower than normal?

Some might be skeptical to think the answer is yes, and that doubt might cause the Warriors’ opponent not to foul, rolling the dice when the numbers might suggest they do so simply because Golden State is cold. The same goes for poor free-throw shooters who suddenly make two foul shots when intentionally fouled. Does their confidence and recent success at the line lower the chances they make the next shots, or heighten it? I’ve yet to find compelling data that definitively answer these dilemmas. There has been some intriguing evidence to suggest rhythm matters for shooters, though: They tend to shoot 3 to 5 percent higher on the second attempt than the first.

The Theory in Action

The best way to break down this study is through examples and hypotheticals.

Take Clint Capela and the Houston Rockets, a team frequently subjected to intentional foul tactics over years past. Capela shot 56 percent from the stripe last season, which bears out to 1.12 points per possession. His Rockets averaged 1.253 PPP in the half-court when factoring in offensive rebound success rate. Subtract the coefficient for the difference in offensive rebounding on free-throw attempts, and the safe number to foul for the Rockets is… exactly 56 percent.

In order to determine whether the strategy is safe, given the large amount of variables that are difficult to account for in these calculations, Hack-a-Shaq should only be considered when there’s a clear advantage to doing so. The safest bet is to go by the league’s worst team in terms of offensive output. The Kings only scored 0.894 points per possession. Add in the rebounding coefficients and their number of expected value jumps to 50.4 percent. That would make 50 percent shooting the safe threshold—the number at which point it makes mathematical sense to start fouling a player, even if he plays on the worst offensive team.

Of everyone  in the league to attempt at least one free throw per game and check in during 30 or more of their team’s 2017-18 outings, only six fell cleanly beneath the 50 percent mark. Those six players: Andre Roberson (31.6 percent), Kosta Koufos (44.6), Miles Plumlee (45), Lonzo Ball (45.1), Mason Plumlee (45.8) and Tarik Black (46).

With the coefficient included, below is a look at each team’s individual metrics and the lowest qualified free-throw shooter on their team during the 2017-18 season to see how far away from useful applications of the strategy they really were. Exactly one-third of the teams in the NBA have a player whose free-throw efficiency dictates, in theory, that they should be fouled instead of left to play freely in the half-court:

In summary, myriad variables impact the equation, many of which are too difficult to quantify, if not altogether unquantifiable. The simplistic ways of looking at intentionally fouling as a defensive strategy might provide the most clarity, but they also lower the number of players that should definitively be fouled. Fouling becomes an adequate strategy for all players who shoot below 50 percent from the line, and on better offensive teams that number is heightened because their half-court efficiency is so high.

The league has worked diligently to legislate the issue over the last few years, and players such as DeAndre Jordan and Andre Drummond also deserve credit for improving their efforts at the stripe to exclude them from the criteria. The strategy is far from archaic, and as long as there are free-throw shooters around the league who miss more than they make, coaches should heavily consider utilizing this tactic.

Unless otherwise mentioned, all data is from Synergy Sports Tech, NBA.com or Basketball-Reference and are current as of the conclusion of the 2017-18 NBA season.