What makes corner 3s in the NBA so efficient, and how can defenders better respond to limit them?
The sport of basketball has come a long way since its founding in 1891, and as of recently the game has taken a huge shift in the form of a 3-point shot revolution, sparked by some of the NBA’s best shooters in Steph Curry and James Harden. One might question the dramatic increase in 3 pointers, but the rapid improvement in 3-point shooting percentage and the increased offensive emphasis on them have allowed them to become an important facet of every game. This paper by Pelechrinis and Goldsberry analyzes a specific type of 3 point shot – the corner 3. It deep dives into the efficiency of the shot and ends with a simplified game in which the Nash Equilibrium is calculated to analyze how a defender should react to “drive and kick” situations, where they can either defend the higher percentage close shot or defend the corner three if a pass is made.
The goal of modern basketball is to create efficient shots. However, efficient shots are not the ones with the highest percentage, but rather ones with the highest payoff. The two most efficient shots in the NBA are surprisingly very different shots. The first as expected is a close high percentage shot from the paint, but the second shot as mentioned above, is the corner 3 pointer, which provides an expected point value of 1.14, much more valuable than a 20 foot jumper that yields an expected value of 0.9 points. The reason for the efficiency of this shot is often (falsely) attributed to the fact that the corner 3 is a much shorter shot than a 3 pointer from above the break in the middle of the court (22 feet vs 23.75 feet). While this is certainly a factor, the main reason described in this paper is that corner 3s are assisted at a very high rate, resulting in them being less contested. Furthermore, defenders do not fully commit to the corner 3 shooter, opting to stand about 10 feet away from them to provide help defense for the short high percentage drive.
The paper then uses game theory and Nash Equilibria to discuss why the strategy above is not optimal. A defender has two options – either commit to the defensive assignment of the shooter or provide help defense towards the penetration of the basket. The offense also has two strategies, either drive to the basket or make the pass to the shooter. Both these strategies are quantized by a variable d which represents the distance in feet the defender keeps from the shooter. The Nash Equilibrium for this game was solved for various values of a, which represents the impact of a double team and thus results in different payoffs. The payoff matrix for a=2 is as follows:
The mixed strategy Nash Equilibrium was calculated for a = 1.3 and a = 1.9 and the results are as follows:
These findings show that the corner defender should commit to either defending the corner 3 or helping the penetrating attacker. Current NBA defenses deviate from this, with the defender typically standing somewhere between the corner 3 and the penetrating attacker. Perhaps more NBA defenses will take this into consideration, or perhaps there is more future study to be done before they commit to this new strategy.
Link: https://arxiv.org/pdf/2105.12785.pdf