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Layups: The Game Theory of Foul Subs

Posted by Neil Paine on July 6, 2010

I didn't see this back in April, but an econ blog called The Leisure of the Theory Class had several posts about the underlying philosophy behind benching players who are in "foul trouble":

Foul Trouble

Foul Follow-Up

ESPN's Eamonn Brennan also had a reaction here. The basic premise (one which I happen to strongly agree with) is that by benching a player on pace to foul out before the end of the game, a coach has voluntarily exacted the very same penalty -- not being able to use the player -- that he's afraid of having happen if the player fouls out. In other words, the coach is so afraid of something that might happen at the end of the game (Player X getting his 6th foul), he's willing to guarantee that the player doesn't play a certain amount of time in the middle of the game -- often more time than the player would have missed if he had not been subbed out!

Obviously, there are more complexities to the argument than that, so you should read both posts (and the comments). But the core idea remains that coaches: 1) overestimate the "risk" of leaving a star in, 2) overvalue the final minutes of the game at the expense of minutes in quarters 1-3, and consequently 3) give themselves a harsher penalty in the middle of the game than the one they're afraid of the referees giving them in the 4th quarter.

26 Responses to “Layups: The Game Theory of Foul Subs”

  1. Shareef Says:

    Sitting the player is not due to fear he might foul out. It is to quell opposing coaches from their pursuit of accumulating fouls on him. You try to "steal" those minutes. A problem arises only when coaches roboticly continue to sit the foul troubled player through obvious momentum shifts.

  2. Neil Paine Says:

    So you're claiming that the coach is correct to pursue fouls against a player at a greater rate once he gets several fouls? If that worked, why didn't he pursue fouls against the other team's star player at that increased rate from the opening tip?

  3. Neil Paine Says:

    I should clarify that I'm not disputing whether or not the coach actually emphasizes "going after" a player in foul trouble (you hear that kind of talk all the time during timeouts), I'm disputing whether it actually works -- both in whether they would be able to draw even more fouls by taking that strategy, and in whether that focus wouldn't cause them to play worse. In essence, I'm asking why that's necessarily a good strategy -- if the consequence of leaving your foul-troubled star in is that your opponent gets inordinately caught up in trying to get him more fouls, isn't it possible that this strategy actually hurts your opponent (and, by definition, helps you)?

  4. Spree Says:

    I've always thought this myself!

    The question is if the player in foul trouble will play up to optimal ability/aggressiveness if left on the floor.

    In the Finals we saw LA leave Kobe out with 5 fouls against the Celtics and his defensive aggressiveness seemed to wane and he was reluctant to attack the basket for fear of drawing a charge.

    It is a complex idea because I agree that coaches overestimate the fear of fouling out and actually punish themselves by sitting guys, but if they feel a guy will lose effectiveness worrying about fouls it might be worth sitting them.

  5. Neil Paine Says:

    Also, how is "quelling the opposing coach from their pursuit of accumulating fouls on him" not all about the fear of your star fouling out? If you sit the player because you think the opponent will go after him and draw more fouls, that's still coming from a position of fear -- that down the line, they will be able to either a) draw 6 fouls and disqualify him from the game; or b) use his own fear of fouling out and/or being benched against him, turning him into a passive defender.

  6. Neil Paine Says:

    This is a classic game theory problem, in fact. Once in foul trouble, Kobe (for instance) has two options: he can continue to play with his normal aggression on D, or he can be timid and try to avoid fouling. Also, his coach has two options: bench Kobe until n/6 of the game has passed (where n = # of Kobe fouls), or leave him in as though there was no foul trouble. So you can set up a payoff matrix for each of these options, with efficiency differential as the variable outcome.

    Let's say Kobe at normal efficiency vs. an average team (with average teammates) is worth +7 points to the bottom line. He plays 39 minutes, fouls 3 times, is worth +9 when on the floor, and the team gets +0 when he sits. That's Kobe at normal aggression. Now, say he's more timid when in foul trouble, his defense goes from being +2.5 to zero, he's now an average, timid defender. Also, maybe his offense gets slightly worse, so we'll say his on-court impact is now only +5, all because he's worried about fouls and is not his usual aggressive self.

    So the coach can either bench him or keep him in the game and risk him fouling out. Kobe fouls 0.066 times per minute normally, but let's say the other team's coach goes after him really aggressively and raises his expected rate...We can just look at the expected time he'll foul out with various pf/mp rates.

    If Kobe grabs 2 fouls in the first 10 minutes of the game, and he plays 31 of the remaining minutes, he'd have to average less than 0.13 fouls per minute to avoid fouling out. This is not asking too much (that # is even more than I predicted his "aggressive pf/rate" to be), but say that Kobe can only average less than 0.13 by being playing at a timid +5 rate instead of a +9 rate when fully aggressive. So let's simplify: if the coach chooses to play Kobe and he's aggressive, you get +9 but he fouls out before the game is over. If he chooses to play him and he's timid, he plays until the final buzzer but is +5. If he benches an aggressive Kobe, you get +9 but he has to sit a certain # of minutes in the 2nd quarter. And if you bench a timid Kobe, he gets his normal rest and plays at +5. What should the coach do?

    The equilibrium obviously depends on the minutes sitting in the 2nd because of foul trouble vs. the minutes lost at the end due to fouling out. In our hypothetical situation, the coach would probably play Kobe at least 5 minutes fewer than usual if he cared about foul trouble. In order to foul out with 5 minutes left in the game, Kobe would have to commit 0.15 pf/min; if aggressive Kobe does that, the team is +6.6 for the game. Note that sitting an aggressive Kobe for 5 minutes has the same effect, so the coach would effectively have to predict how many minutes would be left in the game when Kobe fouls out and sit him for 48 minus that amount in the 2nd in order for the benching strategy to be effective.

    Meanwhile, Kobe playing aggressive for 10 and timid for 29 minutes would be worth +4.9 to the bottom line, which is way below his expected contribution when aggressive even if he fouls out with 5 minutes left in the game. In order for playing timid to be worth it, you would have to expect a timid Kobe to be worth +8.25 when on the floor, even while trying to avoid fouling, or you would have to expect an aggressive Kobe to foul out with 16 minutes left in the game!

    The bottom line: unless you think you can still play at ~ 90% of your normal output while trying to avoid fouls, you shouldn't play with less than normal aggression, foul trouble or not. And you should never sit a player more than the amount of time you expect to be left in the game when he would foul out. A player with 2 fouls in 10 minutes would have to foul at 0.105 pf/min the rest of the game to foul out, which is Theo Ratliff/Andres Nocioni levels. And that's if he played all 38 remaining minutes.

  7. Sean Forman Says:

    It's the exact same thing as going for it/not going for it on fourth down. The cost/benefit says you should (and I believe you should) be more aggessive for a long-term payoff, but woe to the coach whose star fouls out with five minutes left to play in the third.

  8. Neil Paine Says:

    I guess these are great case studies for those who say our "gut instinct" is to be risk-averse above all else. As it turns out, humans have very poor intuition when it comes to uncertainty -- which not only explains coaching behavior, but also perhaps the factors that drove the housing crisis and subsequent economic meltdown.

  9. Gil Meriken Says:


    Kobe is seldom in foul trouble. I think for your example you should substitute the name "Darryl Dawkins" for "Kobe".

  10. Neil Paine Says:

    Ha, well, to be fair #4 Spree brought up Kobe as the example, I just went with it.

  11. Jason J Says:

    Well when LeBron is in foul trouble... wait... LeBron is never in foul trouble. Nevermind.

  12. Spree Says:

    I want to quote a part of the first article linked that I find objectionable:

    "By the way, my subjective sense is that the last possession is more similar to any other than conventional wisdom suggests: a wide-open John Paxson or Steve Kerr is a better bet than a double-teamed Michael Jordan any time in the game. On a couple of major occasions, Jordan agreed."

    This is where Game Theory and really theory in general crashes with practice. I don't believe the last possession is in any way like any other in a practical sense. In theory it shouldn't be any different, but in practice it is very different due to pressure felt by players and heightened awareness by everyone.

    It seems to me that in some sense practitioners always clash with theorists on these matters because what seems sensible in theory is not always working in practice. Michael Jordan in the clutch was able to stay calm and peform as well if not better than normal for his talent. Karl Malone...not so much.

    While I'm not defending the "foul trouble" fallacy, I'm willing to give some leeway to it in the sense that having great players available late in games is valuable if and only if they have proven to be capable of handling those pressure situations better.

    In this case I'd have the example of someone like Robert Horry, who is sometimes absent for entire seasons until all is on the line in the waning moments of a playoff game.

  13. Jason J Says:

    Spree I think another factor to consider that agrees with your assessment is that Kerr or Paxson can't get a shot unless a play is run through Jordan first. While they may have wound up with the open shots and hit them in big moments in the playoffs and finals, those open shots were created because the defense reacted to Jordan first. If Jordan is benched with 4 minutes to go, the spot shooting specialists probably won't be viable scoring options on the final possession.

    On the other hand, I don't like the way teams abandon running plays on the last possession of a quarter, half, or game. Why does Boston have to isolate Pierce or Rondo to finish a quarter. If there's time to run a normal play to try to get someone an open shot, why not do so?

  14. Neil Paine Says:

    Right, and I don't think the idea of a clutch player (about which I'm agnostic, btw) is necessarily at odds with the theory about benching stars in foul trouble. When your guy has too many fouls in the 1st half, it's basically a tradeoff between sitting him now by your own hand, or later by the referee's hand. If you sit him now, the only efficiency-maximizing choice is to sit him less time than he would miss at the end of the game if he were to play and foul out (the trouble is, you can never know that exact amount of time). Recognizing that points in the 1st half are worth just as much as points in the final minutes, they're arguing that it's more rational to choose against the benching option you can control, rather than hurting your team by preparing for an eventuality that may never come.

    If you knew exactly when "Darryl Dawkins" would foul out at the end of the game, the agnostic would be indifferent as to whether you sit him now or later, and the clutch disciple would consider it a no-brainer to have your money guy available later in the game. The problem is that you know (and even control) how much you are benching him now, whereas you have only a foggy sense of how long he'd be available at the end of the game if you left him in.

  15. Neil Paine Says:

    Also, when he says "a wide-open John Paxson or Steve Kerr is a better bet than a double-teamed Michael Jordan any time in the game," I'm not sure we can argue with that. Obviously, Paxson and Kerr were open because Jordan was doubled, but the best option is sometimes for your "finisher" to pass for the good of the team. The "alpha dog" ethic of the league might tempt a player to shoot over the double-team despite an open teammate, which is a case where all of this nonsense about "killer instinct" and "wanting to take the last shot" clearly hurts the team. We have to make sure we don't glorify the "alpha dog" so much that players are incentivized to not make the highest-percentage play.

  16. Anon x 2 Says:

    As someone mentioned, I think aggressiveness is a big factor.

    While you're correct that a player should play nearly as aggressive regardless, the psychology part kicks in. If you are a low post defender with 4 fouls and your man just got an iso on you on the block, as he makes his move you're going to challenge with far far less aggression even if this isn't the optimal strategy.

    Nash Equilibrium in a game with imperfect information can be interesting and horrifying. I do know from watching Lamar Odom for years now that Lamar with foul trouble is no better than Adam Morrison and Adam Morrison is god awful. He just mentally shuts down.

    Sometimes I wonder if players believe the refs are out to get them or that if they try anything they'll get the next foul, etc etc. All psychology.

    It's like you said about being agnostic about being "clutch." I pretty much agree. There's no such thing as a clutch player other than stating a clutch player is someone largely unaffected by the moment (doesn't do better or significantly worse). However, there are certainly un-clutch players.

  17. UB Says:

    A few responses, in inverse order.

    @15 - "Obviously, Paxson and Kerr were open because Jordan was doubled, but the best option is sometimes for your "finisher" to pass for the good of the team."

    I don't think that's the point some readers are trying to make. I think the point is, if the 'finisher' isn't on the court, Paxson/Kerr/etc aren't open at all. That star player necessitates the double-teams. Now, Paxson/Kerr/etc have to HIT the shot, and 'finisher' X has to have the willingness to pass the ball, but the finisher still NEEDS to be on the court to create that good look.

    @13 - I think it's because there's still a high value in depriving the opponent of a possession. Let's imagine the following:

    1) Team A scores 1.2 PPP when running up-tempo, usually scoring within 10 seconds. Team A scores .7 PPP when isolating its best 1-on-1 player(so we've got a HUGE delta here between offensive options).
    2) Team B scores 1 PPP when running up-tempo, and .8 when isolating its best player.
    3) Let's say a half-court heave type play (plays run with 2 or less seconds) returns .1 PPP.

    If Team A has the ball with 22 seconds left in the quarter, it has 2 options:
    It can try to run its uptempo offense, which results in an average of 1.2 points and leaving 12 seconds on the clock (PLENTY of time for a regular possession). Or, it can run an isolation set and effectively run out the clock for a shot.

    BUT, if Team A plays up-tempo, Team B gets a final possession in the quarter with a reasonable amount of time to shoot - let's say Team B plays iso on final possessions basically every time they have enough time to do so. Their PPP is then .8 on this possession.

    So we set up the payoffs for each team based on Team A's chosen strategy:

    If Team A runs, they'll earn 1.2 points on average, and Team B will earn .8 points on average - Team A increases its lead an average amount of .4 points.
    If Team A isolates, they'll earn just .7 points on average. But Team B earns only .1 points on average (flings at the basket and full-court passes, etc) - Team A increases its lead an average amount of .6 points.

    Possessions are valuable, and by playing isolation at the end of quarters, a team can effectively guarantee itself an extra one relative to its opponent for that quarter.

    @Neil's original article - I actually got into a brief email correspondence with that article's author, Prof. Jonathan Weinstein. Unfortunately, he didn't have enough time to respond to all of my questions. One that he didn't really address, I'll reprint here:

    If I may, I'd like to refer to (Berri's models have their flaws, but this post is a guest post by Brian Burke; admittedly, his data does come from college basketball). To quote:

    "One thing I’ve already noticed that’s interesting about basketball is that the win probability equation is the same for nearly the entire game. In other words, a 6-point lead for the home team in the first 10 minutes of the game yields the same WP of 0.86 as a 6-point lead with 10 minutes to go in the 2nd half. This surprised me. I would have expected any certain lead to be more decisive as the game went on, gradually becoming more and more insurmountable. In the graph I cited above, the “slope” of the curve would theoretically get steeper and steeper as the game goes on. When I went to make a graph of selected times in the game to show how the curve steepens, I could only see a single curve. I thought I had made some kind of error in Excel, but the curves were just superimposed. Not until the final couple minutes do the curves become very steep, when ultimately a 1-point lead with zero seconds remaining is as decisive as a 10-point lead."

    ... Coaches, with their years of experience, can be extremely well-tuned to the proper response to a lead or deficit for their respective teams, and adjust their in-game strategies accordingly. As such, they prefer to have all of their resources available for as long as possible. And, in evidence, as the game begins to get more 'out-of-hand' (i.e. as the deficit increases) you're much more likely to see coaches put back in players who are in 'foul trouble.' That's because they ARE aware that being blown out in the 2nd quarter makes a game hard to win in the 4th... But if their reserves can keep the game close in the 2nd (which sometimes happens, sometimes doesn't) they much prefer to have their star players available to improve win probability late in games.

  18. Walter Says:

    Neil, Let me add one more interesting point while we are discussing game theory...

    One of the assumptions of the argument is that 5 minutes in the 2nd quarter is the same as 5 miutes in the 4th. Obviously people will debate clutchness until the end of time, but lets ignore that for the time being and focus on something else... playing from behind (or ahead).

    A study was done by Johan Berger and Devin Pope of Wharton University in which it found that college teams were more likely to win a game when down a point at halftime than up only a point. The reason (linked to Prospect Theory) is that they team that is losing can see victory being down only a little and will thus play harder to win while the team winnin will likely not play any harder and thus play less hard relative to the opposition.

    So in this situation, if the coach decides to bench the foul plagued player then he is "betting" on his team keeping the game close enough that when the player returns the whole teams effort will improve as they can see the victory and will try harder to reach it. On the other hand, if he plays the player and that player fouls out and the game is still within reach by the opposition, the impact of losing a player and the other team being close to victory could lead the other team to "play harder" and win the game.

    The basic puzzle using this logic is which is more likely to occur, The player continues to play and if he fouls out the team will have generated a large enough lead to overcome a final push by the opposition or by benching him will the remainder of the team keep the game close enough so that he can return and lead them to a final push to win the game in the end.

    The study would suggest that the last 5 minutes are more important in a close game than 5 minutes in the second quarter (even ignoring clutchness).

  19. Jason J Says:

    @17 "Possessions are valuable, and by playing isolation at the end of quarters, a team can effectively guarantee itself an extra one relative to its opponent for that quarter."

    That makes a lot of sense, though I think in a possession where you have 22 seconds, there's no special reason to think you couldn't run a play that required most of a shot clock. It doesn't necessarily have to be up-tempo to generate a better look than a predictable isolation. Going back to the Celtics example, Doc Rivers ran a lot of good motion to get open looks for Garnett and Allen coming out of time outs, where the play was by definition executed in the half court.

    I understand that handing the ball to Pierce and telling him to dribble out the clock and shoot over Artest and Odom at the last second guarantees that LA doesn't get another shot, but is running a down screen to free up KG for an elbow jumper with a couple seconds left, a couple seconds in which the other team will have to force up a contested buzzer beater, really a worse option? I guess, as you numerically posit, it depends on your estimation of Pierce's ability to finish the isolation play.

  20. Neil Paine Says:

    This is a good discussion, great work, guys...

    #17 - Good point about the fact that Kerr/Paxson wouldn't be open w/o Jordan on the floor. My feel for what Prof. Weinstein was saying in the original post was that the points created by having the star on the floor in the final minutes aren't worth any more than those same points earlier in the game. I was simply adding that sometimes the alpha dog mentality might actually make those points harder to come by in the final minutes, because the ego of the alpha dog could cause him to make the incorrect play in the final minutes due to an obsessions with "the last shot".

    Jordan didn't have that problem when he passed to Kerr/Paxson, but then again, they made the shots -- remember how the press killed LeBron when he did the same thing with Donyell Marshall (who missed)? Sometimes in the 4th quarter, there's needless pressure to make an incorrect basketball decision because of the egos involved, while that pressure doesn't exist for the same play in quarters 1-3.

    Also, very interesting point about the win probability being different only in the final couple minutes... This does suggest that you should put a premium on saving him for the final sequences of the game, which is another layer of complexity you'd have to take into account in the payoff matrix.

    #18 - I remember that study, and thinking about the Bill James aphorism about leaders and trailers merging at .500 because of not only RTM, but also because the team in the lead has a tendency to change to a conservative strategy while the losing team will be more aggressive. I also wonder if the seeming paradox of the team down 1 after a half winning more has anything to do with getting more possessions in the 2nd half -- maybe the leading team is up by one b/c they won the tip and had 1 more possession than the opponent, bur in the second half that advantage is perhaps erased.

  21. Neil Paine Says:

    And for the nitpickers out there -- on the Paxson shot, Jordan actually passed to Pippen, who passed to Grant, who passed to Paxson. But the important thing is that Jordan didn't force ownership of the possession, instead making the right play, which led to the 3-pointer and the win.

  22. Jason J Says:

    #21 And the whole play started with the Suns doubling Jordan in the back court - which sounded smart at the time since he had all of Chicago's points in the 4th quarter to that point.

  23. UB Says:

    @19 - "That makes a lot of sense, though I think in a possession where you have 22 seconds, there's no special reason to think you couldn't run a play that required most of a shot clock."

    Very true, very true. Upon a reread, I misinterpreted your initial post ("On the other hand, I don't like the way teams abandon running plays on the last possession of a quarter, half, or game)". I read it as you believed teams should 'run' the ball (e.g. uptempo looks) - which on reflection is, I think, not what you meant at all.

    For teams running a 'play,' it may be that certain ones fear the inability to execute the play. At that point, it probably depends on how good of a shooter your point guard is. A team like New Orleans or Phoenix probably is much more likely to run a play at the end of a quarter (and, anecdotally, I feel like they do), because they have excellent shooting PGs. Even if David West doesn't come open off a screen, Chris Paul is still a high-percentage shooter likely to score on a jump shot.

    Taking the example of the Garnett down screen, let's look at the Celtics. A team like Boston, with a PG who is extremely poor on jumpers, may feel less comfortable running their usual offense with the ball in his hands. If Garnett doesn't get free for the screen, that leaves a 20-foot Rondo jumper. I actually feel like this happened once or twice at the end of quarters in the Finals, where they attempted to free up Allen or Garnett but the play broke down.

    This is all speculation and hearsay - it'd be interesting, though very labor-intensive, to put together a database of how often teams run isolation plays for 1-on-1 players at the end of quarters, vs running offensive sets, and then comparing that against jump-shooting percentage for each team's primary distributor.

  24. Neil Paine Says:

    #23 - "It'd be interesting, though very labor-intensive, to put together a database of how often teams run isolation plays for 1-on-1 players at the end of quarters, vs running offensive sets, and then comparing that against jump-shooting percentage for each team's primary distributor."

    I could be mistaken, but I think Synergy can do that.

  25. Jason J Says:

    #23 Good point re: the point guard's shooting ability / confidence.

  26. BSK Says:

    Holy crap!!! I've thought about this for a long time and never, EVER seen anything on it. I'm glad I'm not alone, thinking up crackpot theories that somehow miss the obvious.

    I think there are occasional good reasons to sub a guy out, such as calming down a player who is clearly not in control of his game on the court, or if you were going to give a guy a breather anyway starting a minute early might not do any harmful. You also have to take into account an individual's likelihood of continuing to accumulate fouls. Some guys are far more likely to foul than others. If Kobe picks up two fouls right off the bat, that is unlikely (and not just because he's Kobe; wing players generally pick up less). He can probably make only the most minor of adjustments and remain out on the court at 99% effectiveness without real risk of fouling out. Greg Oden, on the other hand, who is in there primarily for defense and who hasn't met a foul he didn't love yet, might need to be benched if you think you'll need him for specific match-up purposes in the 4th. So, it's not optimal to have a blanket "never bench a guy with 'foul trouble'" dogma. But voluntarily giving something up to avoid the RISK of potentially giving it up just is obviously an ineffective tactic.

    Most importantly, though, I'm just glad I'm not crazy!