Should a Coach Sit a Player in Foul Trouble?

Something that is leftover in my mind from the NCAA Tournament is the issue of whether a basketball team should bench players who are in foul trouble. It's a common practice among coaches, of course, but some folks recommend against it on the rationale that a point is just a point. They argue that a coach's goal should be to maximize a star player's minutes, accounting for necessary rest of course, so therefore sitting a player in foul trouble is bad because it might not maximize his minutes.

Here is a Ken Pomeroy blog post arguing in favor of sitting a player in foul trouble. In it, he links to folks arguing against it. Check out those links for the other side.

Here's what I think I can add to the debate by framing the discussion in a manner that I've not seen done elsewhere. If someone else has already taken this approach (and if they have, it's probably better than this), please let me know in the comments. I'll be talking in terms of college basketball since this is a college sports blog, but it should apply to any level of the sport.

Thinking in terms of points per possession

I think a coach should rest a star player in the event of foul trouble in some circumstances, though perhaps not always. The way I think about this is in terms of points per possession (PPP). It's calculated exactly how it sounds: the number of points a team scores in a particular time span divided by its number of possessions in the same time span.

It's critical to note that teams' PPP rates vary. A team will very likely have a different PPP over a 10-minute span in a game than it has for the whole game, and the PPP rate for a game will likely be different it is for the team over a full season.

I'm going to be talking in a moment about two generic, "normal" teams in a generic, "normal" game to try to make this as broadly applicable as possible. By this, I am referring to teams that are in the middle of the distribution in many ways.

I'm going to base things on a hypothetical game in which both teams have 68 possessions, a figure that's decidedly middle-of-the-pack. I'm going to set 1.3 PPP as the high end of a "normal" PPP rate and 0.7 as the low end of a "normal" PPP rate for any given span of time. The 1.3 PPP figure is just beyond what the best teams were able to put up for the year in both 2011 and 2012, and the 0.7 figure is just below what the worst teams did in the same time. I'm going to consider any PPP rates outside these bounds as unlikely runs and slumps that no "normal" team can can count on intentionally producing.

Visualizing what a team does

I hastily put this chart together to illustrate what a team can do from a PPP perspective. This is for offensive points per possession:

The X-axis is points per possession, while the Y-axis the probability of hitting a particular points per possession threshold. Here I am assuming (charitably, in some cases) that the coach of this team knows what strategy maximizes his team's points per possession, and these curves represent what the team will do following that strategy.

With a star player, the team visualized here hits an average of C points per possession. Without the star player, it hits an average of A points per possession.

Obviously, the team is generally better off with the star player. There is no Ewing Theory at work here. However, having the star player in the game is no guarantee that the team will hit a higher level of performance. The team is equally likely to hit B points per possession with or without the star, and, if the star is having a bad day, there is plenty of overlap that allows for the team to play better without the star in the game than with him.

Even allowing for the possibility of a bad day for the star, the team is able to hit PPP levels with the star in that it cannot hit without him. D points per game is an example of a level that it can hit with the star that it cannot hit without him.

Game time

Let's say that Team A is playing Team B. The teams are pretty close, though Team A is slightly better at full strength:

Again, this represents offensive points per possession for each of the teams in this particular game. Though I am focusing on offensive PPP to keep comparing apples to apples, keep in mind that a given player's presence or lack thereof will affect both offensive and defensive PPP. In an example from this past season, Nerlens Noel made a far bigger impact on Kentucky's defensive PPP than he did on its offensive PPP.

Anyway, that's the matchup with Team A's star in the game. This is what it might look like with him out of the game:

The tables have turned. Team B understandably comes out ahead with Team A's star out of the game.

Even though Team A is favored with the star in and Team B is favored with the star out, there still is plenty of overlap in these curves. Team A can still outplay Team B with the star on the bench, while Team B can outplay Team A with the star on the floor.

Measuring who outplays who is a matter of the PPP differential: one team's PPP minus the other team's PPP.

Uncertainty reigns

Let's say Team A's star picks up two quick fouls in the first two minutes of the game. Uh oh. What should the coach do?

Two minutes aren't enough to determine where on these curves the teams will be playing over the course of the game. With the teams so close together, just about all possibilities are on the table. The bulk of the probability suggests that with or without the star in the game, the teams will be fairly close in PPP rate regardless.

Let's say the coach is thinking about leaving his star on the bench until there are five minutes to go in the half. That would mean the star is on the bench for 13 minutes. Let's also say that the peaks of the curves are 0.1 PPP apart, since we've already granted that the teams are pretty close either way.

The "normal" range I defined above is 0.6 PPP wide at 0.7 to 1.3, so let's say those peaks are 0.95 PPP for Team A and 1.05 PPP for Team B. Let's also imagine that with the star on the bench that Team A's efficiency drops to 0.75 PPP while Team B's efficiency rises to 1.25 PPP. Those figures are still in the "normal" range I defined above, and they give a PPP differential of 0.5 in Team B's favor.

If the 68 possessions of this game are distributed evenly (which in real life they're not, but just go with it for now), each team will have 22 possessions in that span with the star on the bench. Given a PPP differential of 0.5 in Team B's favor, Team B would outscore Team A by 11 points with the star on the bench.

Let's say for simplicity's sake that the teams played to a tie in the first two minutes, so Team B has an 11-point lead with five to go before intermission. How hard would it be for Team A to overcome that in the final 25 minutes of the game with the star on the floor? Not too hard.

Team A would only have to have a PPP differential of 0.26 in its favor the rest of the way, which is just half of the PPP differential that Team B used to build the lead. That's quite doable, especially considering that Team A has an positive expected PPP differential already with the star in the game, and the star will be rested over the final five minutes of the half while the other team's players will be a bit tired. If the PPP differential with the star on the bench ends up being less than 0.5 in Team B's favor, then overcoming a smaller lead will be that much easier.

All minutes (and points) aren't the same

The more time there is left in the game, the easier it is to come back from a deficit.

Again, assuming 68 possessions for each team distributed evenly, here is what the PPP differential must be for a team in order to overcome a 20-point deficit. That's a large lead, but it's not insurmountable in some cases.

The X-axis shows how many minutes there are left in the game.

The amount of PPP differential in your favor required to overcome a 20-point lead spikes later in the game because there are fewer possessions left to go.

Using the definitions I made above, a team can expect a maximum PPP differential of +0.6 (1.3 minus 0.7) without needing unreliably lucky runs. The latest in the game a team can overcome a 20-point deficit with a +0.6 PPP margin is 20 minutes to go. For two roughly even teams, one team can't overcome a 20-point lead anywhere after halftime with its normal strategy without some luck. If you're wondering, it is in fact possible under these same "normal" conditions for a team to run up that large a halftime lead. With a 0.6 PPP differential and 34 possessions for each team, the halftime margin would be an even 20 points.

A 20-point margin is pretty extreme. Think about a more modest 10-point margin:

Obviously, it's more doable to overcome this smaller margin. Still though, the latest in the game a team can go and still overcome a 10-point lead with a +0.6 PPP differential is 10 minutes to go.

Think about those charts above with Team A and Team B's potential PPP rates. The only possible way for Team A to run a +0.6 PPP differential is with the star on the floor. If the star fouls out early, Team A cannot hit that top PPP rate down the stretch. Team B can hit a 10-point lead with 10 minutes to go merely by having a +0.2 PPP differential over the first 30 minutes. That's easily within the given tolerances whether Team A's star sits or plays.

Maybe the coach absolutely needs the star late, and maybe he doesn't. He doesn't know that two minutes into the game. He is acting perfectly rationally by sitting the star down to give him a better chance of being available late. If the lead starts getting out of hand in the first half, he can always put the star back in the game. After all, the larger the lead gets, the closer to the start of the game that the period that requires maximum PPP differential gets. In practice, we see coaches do this all the time. If a deficit grows to quickly, they'll put their foul troubled stars back in sooner.

Not all leads are created equal, as this section shows. That means, by extension, not all points are created equal. Their value depends on what the margin between the teams is and how much time is left.

Hitting a three when the game is tied matters little when it's 30 seconds into the contest because the opponent has 39:30 to overcome it. The shot also had a good chance of going in because the stakes (and therefore the likely defensive pressure) were low, and at that point in the game, only a decent three point shooter would be taking such a shot. Hitting the same triple in a tie game with 3 seconds to go matters a lot because it will be very difficult for the other team to get a good look from behind the arc in that small a time. That opponent will also inherently have a smaller chance of making the shot because the team that hit the three knows it only has to guard shooters who are behind the arc. It also might be able to harass the good three point shooters on the team that's down and force a bad three point shooter to take the shot.

Those are the extreme ends of the spectrum, and it is a spectrum in between. However, it goes to bolster the strategy of sitting a star in foul trouble. A lead of any size is safer later in the game than it is earlier in the game, therefore it is more crucial to have a star player on the floor late than early.

Lead sizes affect strategy

Possessions are not typically distributed evenly throughout a game. There tend to be more at the end of the game because teams that are behind will intentionally foul to stop the clock.

You do not want your team to be in position to have to foul to extend the game. It very rarely actually works. The only reason why teams even do it is because if they didn't foul, the other team could run the clock down so far as to make the game hopeless (or run it out entirely).

The worst free throw shooting team in the country last season was Rutgers. Even that team had two regulars that shoot better than 80% from the line. Even if you manage to foul a 75% free throw shooter in an end-of-game scenario, the expected value of his trip to the line is 1.5 points. A defense playing at the worst end of the normal range has an expected points allowed value of 1.3. Even with a defense that is playing poorly (but not unluckily so), you still want to just play defense.

All throughout this piece I've built a case around why a team would want to have a star around at the end when it's facing a deficit. Here's the flip side, where a team wants its star around when it has a lead. Stars who aren't big men tend to be good free throw shooters. They also tend to be smart and quick. You want your good free throw shooting star available to try to accept an inbounds pass when the other team is forced to foul. He is more likely to get open than a lesser player with a similar (or better) free throw percentage. Having the star on the floor makes it less likely that a poor free throw shooter ends up on the line.

In college, this is even more important than it is in the NBA due to the 1-and-1. Think back to that 75% free throw shooter. His expected point value is 1.5 points when he's guaranteed two shots. When it's a 1-and-1, it falls to 0.94 expected points. The penalty of losing out on the second shot is that high. When a team is only in the bonus (and not the double bonus), being able to hit the front half of the 1-and-1 is critical. That fact makes the good free throw shooting star's presence on the floor that much more valuable.

Layer on the complexity

What I've described so far feels like high school physics problems where you assume no friction or air resistance. Let's add in some more specific elements of the game.

Teams don't usually have equal numbers of possessions thanks to factors like offensive rebounding. You don't have to have as large a PPP differential if you simply have more possessions than the other team does. If a star is particularly good at grabbing offensive rebounds, then he's even more valuable when trying to make a comeback for that reason. A player who fights for offensive rebounds is also more likely to pick up fouls though due to the nature of jockeying for position on the boards, so it's definitely better to sit him with early foul trouble.

Guards tend to pick up fewer fouls than forwards and centers do. An instance of a guard picking up too many fouls too early is more likely to be an anomaly than a frontcourt player doing the same. A guard might not need to sit as much (or at all) in light of foul trouble. Guards also tend to be good free throw shooters, so it may instead be more advantageous to sit him in order to maximize his end-of-game minutes for reasons detailed above. It will depend on the coach's judgment on how close the game is likely to be and just how good a free throw shooter the star is.

Those expected PPP curves above are nice normal distributions. It's very likely that they are not nice normal distributions in real life. Stars tend to be consistent, which is part of what makes them stars. The expected PPP distribution without a star probably has a smaller height at the mean with a greater area to the left of the mean than to the right. What those curves look like will depend on just how good and just how consistent the star is.

I kept talking about the threshold of where a team can still come back with a PPP differential of no higher than +0.6. That part assumed that the team was following the coach's "normal" strategy to maximize efficiency. However if a team gets beyond the point where that strategy can help them come back, then the team will have to employ a higher risk/higher reward strategy. If the star in question is a good three point shooter, then he's that much more valuable in the case of an insurmountable-in-normal-circumstances lead. That will affect how long he could sit early in the game. And even adding this much is still woefully insufficient. Coaches constantly adjust their approaches throughout games; suggesting they only have three strategies (normal, high risk, and foul to extend) is laughably incomplete. What strategies a coach employs, how he prefers to employ them, and how they work together in which quantities will also play into the decision of whether and how much to sit a foul-plagued star.

All along, I've assumed the two teams playing each other are close in quality. If it's a mismatch, the calculations change. A far better team has plenty of options with a star in foul trouble. The team is likely to win with or without him, so the coach could make substitution decisions based on judgments that don't have to do with winning (e.g. sit him to teach him a lesson vs. play him so he can learn how to play with foul trouble). A far worse team wouldn't be able to sit its star for very long, if at all, due to the risk of falling irrevocably behind. In practice, we already see this kind of behavior from coaches. Teams tend to sit their foul-plagued stars longer when favored and shorter when they're a clear underdog.

Finally, while I've tried to exclude them as much as possible, lucky runs and unlucky slumps do happen. It's very important that a coach know his team well enough to recognize when they happen and not to count on them repeating themselves in a regular way. In other words, yes, John Beilein should have benched Spike Albrecht after the plucky guard made 17 first half points in the title game against Louisville. Albrecht was far more likely in the second half to play at his standard benchwarming levels and be a liability against a good Cardinals team than continue to make every shot he took.

In the end

Whether a player in foul trouble should sit and for how long depends on a long list of factors. Those factors include what his particular skills are, the relative qualities of the teams that are playing, how well the teams are playing on the particular day of the game, and the time and score of the game. I do believe, however, that coaches already do consider those factors when they choose to bench stars in foul trouble.

Here, I go back to Pomeroy's take on the matter. Coaches don't abide by hard and fast rules for whether and how long to sit a star who is in foul trouble. They do their best to synthesize what they see happening in front of them and what they expect to see in the future to make a decision. The binary question of, "Should you sit a star in foul trouble, yes or no?" is too simple. The answer is, as with so many things, "it depends".