Part 1: Half-Court Offense, An Optimal Stopping Problem.
Welcome to Hoop Theory! For those of you who are familiar with the MIT Sloan Sports Analytics Conference, I am very excited to announce that I have been given the opportunity to present some of my joint research with Justin Rao on Allocative and Dynamic Efficiency In NBA Decision Making at their prestigious venue. I hope that this blog can be a convenient place to discuss the intuition behind the paper and to have a broader conversation about the application of game theoretic models to our understanding of sports.
We are often critical of players who attempt difficult shots early in the shot clock. That player’s team may have had many more opportunities to create a high value shot and settling for such a poor opportunities may have been an especially bad choice. Conversely, as the shot clock ticks toward zero we become progressively more permissive of players settling for difficult shots. Intuitively, we know that every second on the shot clock that passes is one less second left with which our team can try to create another scoring opportunity if we pass this one up. The value of not shooting of continuing the possession is steadily declining with every second that we do not shoot.
If our team’s players are responding to this dynamic optimally, they should be settling for shots more frequently and for progressively lower value opportunities with each passing second. It should come as no surprise that NBA players behave accordingly. The blue line on the graph above indicates how the value of holding onto the ball decreases over the shot clock. Players should shoot when they have opportunities that are more valuable than holding onto the ball. The values of possessions used at any particular period of the shot clock are expressed by the green line.
Finally, the orange line indicates the hazard rate with which NBA teams attempt to use a possession in any given second. A hazard rate expresses a conditional probability that an event will happen at a given instant. Here, the hazard rate expresses the probability that a basketball team that has the ball with t seconds left on the shot clock will use the possession (shoot) before the shot clock gets to t-1. Since you always use the possession with zero seconds on the shot clock (by a shot clock violation, if nothing else) we should not be surprised to see the hazard rate go to 1 on the left end of the graph.
Based on this graph, it is clear that the time remaining on the shot clock is an important determinant of how offensive players should and do behave. In our paper we model player i as being willing to use a possession with t seconds remaining on the shot clock, only if he realizes an opportunity of greater value than some fixed cut-threshold. We posit two intuitive conditions that a player’s chosen cut-off level must satisfy to be optimal for his team.
Dynamic Efficiency: A player should shoot if and only if he realizes a scoring opportunity of larger value than the continuation value of the possession. That is, should be exactly equal to the continuation value of the possession for player its team.
Allocative Efficiency: If two players, i and j, share a court together, player i should not be passing up shots that are better than some of the ones that player j takes. More formally, for all shot clock periods t,
It should come as no surprise that the conditions on optimal shot selection are conditions on the worst shot a player is willing to take. When a player has a very good scoring opportunity (like he is wide open), he has an easy and uninteresting decision to make. Just as in economics, it is his choices on the more difficult –marginal shots that are interesting. Unfortunately there is no simple way to determine the value of the worst shot that a player takes in a given period of the shot clock. We can only observe each player’s average efficiency over all possessions used, and we really have no idea which attempts may have been wide open jumpers and which may have been closely contested by an elite defender.
The main innovation of our research is that we present a novel scheme for identifying and estimating the value of the worst shot a player takes in each period of the shot clock. In doing so we must address the fundamental issue of how basketball players trade off between usage and efficiency.