General

Should You Bench Star Players With Lots Of Fouls? Maybe

May 26, 2010, 9:00 PM UTC

â†µMost basketball fans—college especially—have felt the special agony of a second foul on your star player before the first commercial break and the subsequent shots of said star chilling on the bench for the rest of the first half. Armchair quarterback types (guilty as charged) will fume about the strategy here, figuring that a player is going to pick up fouls at a relatively constant rate and that leaving on the bench for extreme amounts of time reduces the overall playing time the star will get in a game, thus lowering the chances of a victory. Not that I've ever expounded on this to friends who find me tiresome dozens of times and can bring this criticism up on demand. â†µ

â†µâ†µAll right, so I have. But at least I’m not the only one. A recent post done by The Leisure Of The Theory Class— a blog run by Northwestern’s Kellogg School of Management that takes its name from an 1899 book by Norwegian-American economist Thorstein Veblen, and is thus high on the list of “least likely places to find a perceptive basketball critique”—asserts as much: â†µ

â†µâ†µâ‡¥â†µâ‡¥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! â†µâ‡¥

â†µâ†µâ†µThis is because the “new distribution of minutes is first-order stochastically dominated, being just a truncation of the alternative.” It went on from there to address a number of caveats, but the upshot is just that: if you have a guy with three fouls and 30 minutes left in the game he’ll probably play more minutes than a guy with three fouls and twenty minutes left in the game. â†µ

â†µâ†µThis promptly got spread across the blogosphere, because any mention of first-order stochastically dominated systems is guaranteed to go viral, and was met by nothing but rapturous approval until Ken Pomeroy stepped in. You may know Pomeroy as the patron saint of tempo-free basketball statistics, the nuclear engineer progenitor of all those slightly fancy basketball statistics that do things like divide. Thus you might expect him to be on board with a mathematically inclined critique of coaching decisions. â†µ

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â†µIn this, you would be thunderously wrong: â†µ

â†µâ†µâ‡¥â†µâ‡¥The problem with the analysis is that it is based on a faulty premise. The object of basketball is not to score as many points as you can and limit your opponent to as few points as possible. You play to win the game. Those two concepts have plenty of overlap, but it’s not difficult to conceive scenarios where they oppose each other. â†µâ‡¥

â†µâ†µâ†µPomeroy takes up the idea that the latter portion of a game is actually more important than what comes before it, which is both the conventional wisdom—to color announcers and anyone who gets all their basketball wisdom from color announcers—and the opposite of conventional wisdom—to internet statty types. His example is from the national championship game, comparing two different three-point attempts by Butler that missed, one of which would have increased the Bulldogs’ chances of victory by 2.5% and the other (the Hayward last-ditch attempt) that would have guaranteed it. Here’s another fascinating related fact: â†µ

â†µâ†µâ‡¥â†µâ‡¥a tie game at halftime barely changes a team’s initial win probability. If a favored team makes it to intermission tied, they haven’t given up anything in terms of their chance of winning the game. In that context, one might see why a coach would err on the side of saving a superstar rather than keeping him in the game to try to build a lead and thereby risk losing him for potentially more important moments later on.â†µâ‡¥

â†µâ†µâ†µSo there you go. This still feels strange to me, like there is a way to hammer this out in a conclusive fashion that will probably say that coaches are too conservative with their decisions in this area—I can’t imagine that having a guy out with under ten minutes left is ever a good idea—but not go as far as the Veblen-inspired laissez-faire folk above. â†µ

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â†µ(Editor's note: Butler's Matt Howard pictured here because, let's face it, he basically starts every game with two fouls.)â†µ

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This post originally appeared on the Sporting Blog. For more, see The Sporting Blog Archives.

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