**Abstract**

The notion of a “good foul” is a prominent concept in basketball; as such, it is virtually a universally applied strategy. The concept suggests that rather than concede what would be a “guaranteed basket”, defenders should foul the shooting player. Ideally, this causes the shooter to miss the field goal attempt and forces said shooter to recoup the lost points by way of the foul line. In isolation, the theory is sound; in the 2018-19 NBA regular season, the average free throw percentage was 76.6%, functionally saving the defense 0.468 points on each foul preventing a “guaranteed basket.” However, the consideration often neglected in this oversimplified calculation is the cost of the foul outside of the immediate consequential free throws. Because the NBA institutes rules disincentivizing the accumulation of fouls for both player and team, fouls can be modeled as a quasi-causal sequence where a single foul negatively affects the fouling team for every subsequent possession throughout the game. Thus, to holistically compute the value of a foul, objective understanding of the circumstances in which the foul was committed is vital. In this paper, through the use of publicly available SportVU tracking data [1], corresponding play-by-play data, and individual player plus-minus metrics [2], we propose a method to characterize the context surrounding a foul and accordingly quantify the profitability of given fouls. We show that the profitability of a foul is very much dependent on the aforementioned context, and as such the range of values for shooting fouls varies wildly, though are almost a universally negative impact as the 25th percentile of fouls provide negative value equivalent to a loss of over one point for the fouling team, whereas the 75th percentile of fouls are equivalent to the loss of less than 1/4^{th} of a point.

**1 Introduction**

In basketball, fouls are often considered a limited resource; expend it in quantity to your advantage whilst refraining from the overindulgence that results in penalties. Under this conventional mindset, fouls are advised primarily with anecdotal premises (exemplified by the 1980’s “Bad Boy” Pistons): foul to “get into the opponent’s head”, to “set a tone of physicality”, to “make them earn their points from the stripe”. While difficult to assess the legitimacy of such implementations, it is however possible to analyze the objective consequences of fouls. That is, the expected impact given fouls have on (1) the fouling player’s rotation minutes, (2) a team’s probability of exceeding the per-quarter foul limit and entering the bonus (and the corresponding opponent added free-throw attempts expected), and (3) for shooting fouls, the differential in points expected on the shot attempt and the free-throw attempts resulting from the foul. The foul impact was then expressed as a point equivalency value through various conversion techniques.

To assess each of these emanations, SportVU tracking data from the 2015-16 season was used in conjunction with corresponding play-by-play data and individual player advanced plus-minus metrics (BPM) from Basketball Reference [1,2]. For this paper, the example used will be a one-game sample, the first game of the season played by the historic 2015-16 Golden State Warriors team. The fouls highlighted will be the seven fouls that overlap in each category (the seven shooting fouls committed by the Warriors during this game). It should be noted that a one game sample was used as an example for the express purpose of disseminating data concisely throughout the paper, but the model can scale up to any amount of games. It is also of note that games where an overtime period was played were omitted from the sample as the expected personal fouls for each player across the game changes, as does the expected personal foul distribution with the time added. These nuances interfere with the foul impact on rotation minutes projections, and thus games entering overtime were excluded.

**2 Rotation Minutes Lost**

In this section, we attempt to project expected rotation minutes lost, individualized per player in a dictated span of games, as a function of a given foul. Fundamental to this calculation is an understanding of each player’s archetypical rotation, thus necessitating a per-player individualized probabilistic model of said rotation, delineated as a time series.

With this expected rotation model, the question then becomes how to estimate a generalized player foul distribution with respect to the aforementioned projected rotation. The rational approach would be to again use a probabilistic model to evaluate individualized expected times of foul occurrences. However, upon implementation of this approach, no discernable trend was identified between player time on court and foul probability (as the dataset of games expanded the correlation coefficient trended towards zero). Thus, dealing with a stochastic observation, the auspicious approach was deemed to be an assumption of an even distribution as it regards to player fouls across rotation minutes (expected player fouls determined by averaging fouls committed by the player over the observed games).

Establishing the baseline expectation for rotation minutes and partitioning of personal fouls across said rotation minutes, we can now evaluate a given foul’s time of offense relative to its expectation, using corresponding play-by-play data to obtain time and type of foul data, and thus evaluate the foul’s likely impact on the liable player. We separate this calculation into three dependent subsets: (1) relative per-foul assessment of time of offense versus expected time of offense; (2) projecting fouls added/subtracted as a function of time of offense; (3) assessing expected minutes lost as a function of projected fouls added/subtracted.

Beginning with the first subset, because a constant difference in time of fouls was assumed, for each foul the difference between the actual time of the foul and the expected time of the foul was added to the expected time of offense for each subsequent foul, thus realigning the expected time of offense for each subsequent foul in order to enable the evaluation of fouls individually. Subsequently, because of the assumed constant difference between times of fouls, a linear fit was formulated, with the independent variable designated as the expected playing time of the given player and the dependent variable designated as the number of fouls expected to be committed across the playing time interval.

Finally, to establish a point equivalency value for the estimate of impacted rotation minutes, a comprehensive, evaluative metric for added value above replacement is necessitated, typically taking the form of an advanced plus-minus metric. For this particular task, Basketball Reference’s Box Plus-Minus (BPM) was chosen primarily because of its availability at all instances throughout a season and because it can very effortlessly be scraped from the web [2]; however, any comprehensive plus-minus metric will do (according to the discretion of the user).

The formula is:

*on the condition that the estimate of impacted rotation minutes is negative. Because it is the value of the given foul attempting to be deduced, it is impossible for committing a foul at any given interval to add expected rotation minutes; rather, it is the minutes gone without fouling that adds expected rotation minutes, no the act of the foul itself. Applying the above formula, we retrieve the following table, where PEV represents the point-equivalency value of each foul:

**3 Accounting for the Bonus Rule**

In this section, we attempt to develop a method by which to assess probability added of entering the bonus per foul, and subsequently identify the point equivalency value of the probability added. In NBA basketball, a team foul penalty, colloquially known as the “bonus”, is imposed upon a team committing its fifth foul of a quarter, or its second foul in the final two minutes of a quarter. The consequences of entering the bonus are that upon committing a team foul while in the bonus, regardless of the type of foul, the fouled player is granted two free throws.

Thus, because the bonus model regards fouls as summative, the task of establishing a point equivalency value as it concerns entering into the bonus is principally centered around determining (1) an individualized added probability of team entering the foul penalty and (2) projecting the amount of non-shooting team fouls expected to be committed while in the bonus.

Similar to the expected rotation minutes lost calculation, team fouls are largely distributed stochastically with a slight uptick near the end of games caused by the common strategy of fouling intentionally when losing in order to prolong the game. As such, the expectation for team fouls is evenly distributed across a given quarter. This then lends the question of how to determine the expectation for team fouls. The two-faceted nature of the bonus rule (in that there are multiple ways of entering the bonus) necessitates two distinct probability calculations, one for fouls occurring before the final two minutes and one for fouls occurring during the final two minutes.

Beginning with the assessment of fouls occurring before the two-minute mark, the expectation should abide by the assumption of a 50% bonus entrance rate. Therefore, because the bonus rule operates as a step function jumping between 4 and 5, the team foul expectation can be assessed as the mean of 4 and 5, a figure equivalent to a 1 foul every 160 seconds distributed across the entire quarter (720 seconds). The probability model can then be expressed as:

Assuming the same distribution rate of fouls with under two minutes left in the quarter, we can ascertain the step function between fouls 1 and 2 and thus the mean, 1.5, would denote a 50% bonus entrance rate. We therefore can use an adaptation of this formula to predict bonus entrance probability while under two minutes, under the assumption of 3 or less previous fouls in the quarter (by equation A.1 in appendix).

From these two probability values, the succeeding calculation is the expected non-shooting fouls to be committed after entering the bonus, and furthermore the points forfeited relative to an average possession. This entire model, demarcated for individual foul value, can be expressed as:

Applying the above formula, we retrieve the following table:

**4 Projecting Shot Value**

In this section, we attempt to determine the value of impeding the shot attempt through a foul (Note: this aspect of the calculation only applies on shooting fouls). For this calculation, there are two principal variables: points expected from shot attempt, and points expected from a replacement level free throw shooter on the attempts resulting from the shooting foul. The first of these factors, points expected from a given shot attempt, is computed by determining the quality of a shot attempt through the methods provided by Second Spectrum [3].

We then subtract from this value the expected points produced from a replacement level free throw shooter on the attempts resulting from the shooting foul, modeled by the equation

Applying the above method, we retrieve the following table:

**5 Final Point Equivalency Value Calculation**

Summing together each of the identified point-equivalency values (PEV), we can thus return:

A range of PEV values between -1.5 and 0 is generally sustained as the sample of shooting fouls expands, allowing us to deduce the anecdotal premise of “fouling to save a basket” as analytically inefficient.

**6 Future Extensions**

- Account for Individual Coaching Preferences
- Observe individual coaching trends as it regards to foul-trouble management and therein individualize minutes lost projection.

- Utilize Individual Player Shooting Splits
- Use individual player shooting metrics to enhance projected shot value accuracy and better capture the context surrounding a shooting foul.

- Account for Defensive Prowess Lost as a Result to Foul Trouble
- Potentially apply Disruption Score developed by Alexander Franks and Andrew Miller [4].

**References**

[1] linouk23. “linouk23/NBA-Player-Movements.” *GitHub*, 19 Sept. 2016, github.com/linouk23/NBA-Player-Movements/tree/master/data/2016.NBA.Raw.SportVU.Game.Logs

[2] *Basketball Reference*, 2016, http://www.basketball-reference.com/.

[3] Chang, Yu-Han, et al. “Quantifying Shot Quality in the NBA.” *Http://Www.sloansportsconference.com/*, MIT Sloan School of Management, 2014, www.sloansportsconference.com/wp-content/uploads/2014/02/2014-SSAC-Quantifying-Shot-Quality-in-the-NBA.pdf.

[4] Franks, Alexander, et al. “Counterpoints: Advanced Defensive Metrics for NBA Basketball.” *Http://Www.sloansportsconference.com/*, MIT Sloan School of Management, 2015, sloansportsconference.com/wp-content/uploads/2015/02/SSAC15-RP-Finalist-Counterpoints2.pdf.

**Appendix**

**A.1: **