Game Theory Analysis of Basketball’s Deep 3-Point Shot
In the game of basketball, there are three classes of shots: a 1-point free-throw, a 2-point shot, and a 3-point shot. Free throws can only occur while the ball is not in play. While the ball is in play, a player can choose between attempting a shot outside the 3-point line to earn 3 points for their team, or attempting a closer-range shot for 2 points. If you decide to shoot a 3-pointer, you can either release the ball as close to the line as possible, or release the ball even further out. If you stand close, the shot seems easier to make, but that also makes it easier for defense to stick your shot. If you stand further out, the shot seems more difficult, but you’ll also typically have more space between you and your defender. How far should one stand to the line when releasing a 3-point shot? We can look to game theory for a perspective on that question.
Since real-life basketball is very complex with a large number of strategies and variables, we’ll make some simplifications and assumptions and rely on surveyed statistics to model each player’s strategies. Despite the simplifications, we are still able to conclude with helpful information about shooting strategies in basketball. We can simplify an offensive player’s strategy list to three options. We can consider passing the ball, but someone on offense will have to shoot eventually. “On offense, we will have three choices: take a deep 3, take a regular 3, or go to the rim. On defense, we will also have three choices: press all the way up at half-court (intended to counter the deep 3), stick a balanced defense around the arc (to counter the regular 3), or pack the mid-range and the paint (to counter any drives to the rim)” (Natarajan) A player-strategy matrix can be composed for one offender and one defender. A “Deep 3” is a 3-point shot released considerably far from the line. A “Regular 3” is a 3-point shot released very close to the line. “To the rim” refers to the strategy of an offender driving the ball to the rim for a close-range shot. A “Press” refers to a defensive player staying close to a shooter far beyond the 3-point line. A “Stick” refers to a defensive player staying close to a shooter at the 3-point line. A “Pack” refers to a defensive player staying closer to the rim.
Source: https://fansided.com/2018/07/31/nylon-calculus-game-theory-deep-3/
A red payoff in this matrix refers to the shooter taking a highly contested shot, an orange payoff refers to the shooter taking an average contested shot, and a green payoff refers to the shooter taking an open shot. It is intuitive that the main diagonal is solely red since each red circle corresponds to the defense’s best response to each shot selection.
We can analyze a player’s shooting percentage from each of these ranges if the payoffs of each cell in the matrix the average points earned per shot attempt in each scenario. For example, one can construct the payoff matrix for Stephen Curry, an NBA player well-known for his high 3-point shot rate:
Source: https://fansided.com/2018/07/31/nylon-calculus-game-theory-deep-3/
One can also construct the payoff matrix for James Harden, an NBA player with a considerably lower 3-point shot rate:
Source: https://fansided.com/2018/07/31/nylon-calculus-game-theory-deep-3/
The highest payoff for the shooter in each defensive situation is highlighted in green. Despite the large difference in the two players’ 3-point shot rate, taking a regular 3 is a dominated strategy. Therefore, according to this payoff matrix, regardless of the defensive scenario, one should not take a 3-point shot close to the line.
This may be counterintuitive for some. “Since a close 3-pointer and a deep 3-pointer are worth the same amount of points, the player should always take the closer and easier shot, right?” The payoff matrices above show this is a fallacy. Of course, since various assumptions and simplifications were made to construct these matrices, we cannot simply conclude that basketball players should never take a 3-point shot close to the line. However, it allows us to show that the deep 3-point shot is at least much more valuable than we may initially think.