Dynamic Efficiency, the Nash Equilibrium & Kobe as an undershooter

Shortly after lunch on Friday, Matthew Goldman presented a paper crafted in collaboration with Justin M. Rao called "Allocation & Dynamic Efficiency in NBA Decision Making." The concept was not just how you shoot, but when you shoot -- specifically, when in the shot clock you shoot. Goldman treated the idea as a sort of tradeoff: at every second, whichever player has the ball is making a decision to either shoot (and "use" the possession) or don't (and "continue" the possession).

Thus, we have Goldman's concept of dynamic efficiency. To quote him: "A shot is realized only if its expected value exceeds the continuation value of a possession."

TrueHoop at MIT Sloan Sports Analytics Conference

Some of the ideas Goldman presented were straightforward -- players become more aggressive (and less efficient) with shot selection later in the shot clock, point guards & wings are naturally better than bigs at creating effective opportunities, and teams with either low experience or high salary are ineffective at allocating shots effectively. (Just ask the Heat last night.)

Much more fascinating, however, was his extrapolation of this concept into the idea of "overshooters" and "undershooters." Using his model of dynamic efficiency in the shot clock, Goldman found that since 2006 (not including this season) the best players in the NBA more significantly undershoot than the top gunners overshoot -- and the numbers aren't close.

Within the stipulated time frame, the top seven "undershooters" were Chris Paul, Brandon Roy, LeBron James, Al Jefferson, Joe Johnson, Amare Stoudemire and Vince Carter. Pretty surprising, right? All seven of these guys are known for their relatively high-usage offensive game (although Carter isn't anymore, for a variety of reasons), and yet they're considered the biggest undershooters according to Goldman's data.

The top seven overshooters? Russell Westbrook leads the way, followed by Tyrus Thomas, Lamar Odom, Monta Ellis, Larry Hughes, Drew Gooden, and Tracy McGrady. This list is less surprising -- all seven of these guys are known for shooting more than they should, although Westbrook has turned a corner this year.

What's also notable is that the undershooters "undershot" almost twice as much as the overshooters "overshot." But the most surprising revelation had to be Goldman's figures on Kobe Bryant, which found that he slightly leaned towards the side of undershooting. I don't think anyone in NBA history has accused Kobe of undershooting before (the second half of Game 7 in the 2006 Western Conference first-round exit excluded), but Goldman stood by his formula.

Goldman explains by e-mail that injury concerns could be a factor: "Kobe's undershooting t-statistic is 2.3. This means he undershoots in a very statistically significant way, but not quite on the order of LeBron/CP3/Roy. As a nice counterbalance, Pau Gasol has an undershooting t-stat of 2.83 and as such his undershooting behavior is more statistically significant that Kobe's. ... I think there is a pretty intuitive rationalization for the undershooters we observed. These guys are elite players with very flat skill curves. If LeBron always acted like there were five seconds on the shot clock and crashed hard to the basket to create a shot, his team would certainly be better off in the short run. However, his body would wear out in about a week. I don't think LeBron (and the others) estimated undershooting is at all suboptimal for the larger goal of surviving the season and playing well in the playoffs. It is worth noting that a lot of the guys on the undershooter list have or are currently having knee trouble and may not want to constantly push themselves to create immediate value for their team on the margin."

For one, it was unclear in the presentation how effectively Goldman adjusted for this undershooting phenomenon. If I heard him correctly (and I may not have, as he was speaking quickly), he assumed a mostly linear relationship between usage rate and efficiency. While that works to a degree, I find that as usage increases past a player's effective limit, efficiency decreases exponentially, not linearly. If you assume that LeBron James' efficiency only decreases slightly and consistently, you'd find that he'd be effective using even 50 percent of the Heat's possessions -- which, for obvious pragmatic reasons, isn't the case.

Goldman also didn't expand on how much the undershooters were actually hurting their team by undershooting. I'm reminded of a paper done by Brian Skinner last year on the Nash equilibrium and the price of anarchy in basketball. Skinner used the example of Ray Allen on an imaginary team with four lesser offensive teammates. The hypothetical Allen shot far more than them, which as a result decreased his shooting percentage, thus not increasing the team's overall effectiveness. Skinner found that the team's offense would be optimal with Allen shooting almost exactly as much as his teammates -- even though he's a much better offensive player. The math was striking, and eye-opening.

I don't pretend to have done the math and research that Goldman & Rao have done -- far from it. But with this idea in mind, it's possible that those superstars are, in fact, undershooting -- but that increasing their usage would actually be detrimental to overall team efficiency. If that's true, then guys who are close to average on the scale (like Kobe and Carmelo Anthony) are actually hurting their team's overall offensive efficiency.