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# The fallacy of foul subs explained

Should coaches leave their star players in if those star players are in foul trouble? The conventional wisdom says no -- you want your star player involved in the final moments of a game, because that's when star players do the things that make them stars, and if you keep a player in with four fouls and he picks up his fifth early in the second half, you're in trouble. But conventional wisdom in sports is a tricky thing. It hardens, codifies and becomes entrenched. That doesn't necessarily make it right.

Enter the big economic brains at The Leisure of the Theory Class, a blog typically devoted to economics and game theory, which decided to spend some time looking at whether coaches are being smart when they sit players in foul trouble. The answer? Not exactly, no (warning, long blockquote ahead):

Is the rule of thumb reasonable? No! First let’s consider a simple baseline model: Suppose I simply want to maximize the number of minutes my star player is in the game. When should I risk putting him back in the game after his nth foul? It’s a trick question: I shouldn’t bench him at all! Those of you who haven’t been brainwashed by the conventional wisdom on “foul trouble” probably find this obvious. The proof is simple: if he sits, the only thing that has changed when he gets back in is that there is less time left in the game, so his expected minutes have clearly gone down. In fact, the new distribution on minutes is first-order stochastically dominated, being just a truncation of the alternative. This assumes only that his likelihood of getting a foul is time-invariant, which seems reasonable. [...]

Conventional wisdom seems to regard foul management as a risk vs. safety decision. You will constantly hear something like, “a big decision here, whether to risk putting Duncan back in with 4 fouls.” This is completely the wrong lens for the problem, since the “risky”* strategy is, with the caveats mentioned, all upside! Coaches dramatically underrate the “risk” of falling behind, or losing a lead, by sitting a star for too long. To make it as stark as possible, observe that the coach is voluntarily imposing the penalty that he is trying to avoid, namely his player being taken out of the game!

There are plenty of caveats to include here, of course. Coaches want their star players in at the end of the game, and whether that psychological effect is overstated, it still registers. There's also the idea that a player needs rest anyway, so extending that rest via foul trouble is marginally less drastic than the idea that players should stay in the game with fouls no matter what. And maybe a player just needs time to chill out. Or maybe a certain lineup isn't working. Or maybe that player in particular is bad at avoiding fouls while in foul trouble, and maybe that player's defense when he does avoid fouls is so bad that it makes sense to let a sub play hard, physical defense in the meantime.

But that bolded point is the key one, and it's the one worth remembering. In all levels of basketball coaches bench their foul-prone stars. It happens everywhere; every coach does it. It's one of those entrenched pieces of wisdom that goes by unnoticed when followed, but causes a riot when it's ignored. "Why didn't you rest X player there? Weren't you worried about his fouls? Did you want him in for the end of the game?" Fans, media and sportswriters expect coaches to do this one thing, and so coaches do this one thing because there's no real incentive not to. It's the safe play. The "smart" play. Even if it's not really smart at all.

(Here, the resemblance to baseball's entrenched managerial wisdom -- the sacrifice bunt, for example -- is keen. The sac bunt often actively hurts a team's chances of winning; it trades a base for an out, which is the game's most precious resource. But don't tell managers that. And don't tell the wizened scribes who write about them.)

Asking coaches to ignore foul trouble is silly, of course, because that's not going to happen. But the smarter coaches out there should take a look at this sort of thing and challenge their own beliefs just enough to make their own decisions more fluid. Not all foul-outs are created equal. Sure, sometimes the guy should sit. But when the benefit is greater than the cost, and it often is, coaches should think about -- just think about! -- keeping their foul-riddled stars in the game. It's not really that complicated.

Maybe this will never become conventional wisdom. But it would be nice to see a coach or two, if only occasionally, steer away from the unblinking herd.

(Hat tip: Rock Chalk Talk)