### By: Arnav Saxena

In November of 1967, the Dallas Chaparrals led the Indiana Pacers 118-116 in a midseason American Basketball Association matchup. With only one second left of regulation, Indiana inbounded the ball to Jerry Harkness, who launched a desperation hook shot at the buzzer. From 92 feet away, he banked the shot in off the backboard, and it is still the longest shot ever made in professional basketball. The crowd erupted and the Pacers huddled up, readying themselves for overtime. However, 1967 was the first year of the three-point line in many major basketball leagues; Harkness’s shot was Indiana’s only three-point attempt of the game. Only one official realized that Harkness’s miracle heave was behind the new three-point line, and that the shot didn’t force overtime, but instead gave the Pacers the win in regulation. The crowd celebrated once more, realizing that the Pacers had won the game 119-118.

Soon after, the NBA wanted to bring a new level of excitement to its games. George Mikan, the ABA commissioner, said the three-pointer "would give the smaller player a chance to score and open up the defense to make the game more enjoyable for the fans". NBA coaches had to adjust their gameplans quickly and defenders had to guard players farther out from the basket. Even then, the three-point shot was used sparingly- most coaches only used it when their team was losing late and needed points. In 1979, the first year of the three-point line, players only made 28.0% of their three-point attempts. But as time passed, players like Larry Bird popularized the shot and its value began to be realized.

Fast forward to now, and NBA teams are taking more than ten times the number of three pointers per 100 possessions than the 1979-1980 season when the shot was first introduced. The three-pointer has become the favored shot of basketball analysts, and for good reason: in the most recent NBA season, the expected value of a three-point attempt was 1.056, compared to the 0.983, the expected value of a two-point attempt. The three point shot remains exciting, but because it’s become so common, some of the initial excitement has died down. The NBA’s shift to preferring the three-point shot has ushered in a new generation of basketball players capable of draining shots from almost anywhere on the court. Damian Lillard’s 36-foot series-winning buzzer beater over Paul George, Steph Curry’s game winner against the Thunder from 38 feet out, and, as painful as it is, Jalen Suggs’s halfcourt buzzer beater to lift Gonzaga over UCLA have become some of the most iconic, exciting, and memorable shots in recent history. With this shift in the playstyle of the NBA along with the newfound excitement of long distance three-pointers, it begs the question if the NBA should reward these types of shots for their higher degree of difficulty. One potential solution is introducing a four-point line. In this article, I will explore a possible introduction of a four-point line in the NBA, using data and history to analyze where it should be placed, and if it would really give us more iconic finishes to games.

Examining the history and introduction of the three-point line can serve as a useful starting point for the introduction of the four-point line. When the NBA first introduced the three-point line in the 1979-1980 season, NBA players only hit 28.0% of their three-pointers, giving the three-point attempt a PPS (Points per Shot), or expected value, of 0.84. Two-pointers dominated the game as the game plan for most teams at the time was getting the ball to the paint and taking short-distance shots with taller, bigger players. As such, the value of the two-pointer was higher at the time, as the field goal percentage for two-point shots in the 1978-1979 season (the season before the three-point line was introduced) was 48.5%, giving the two point attempt a PPS of 0.97, considerably higher than the initial PPS of the three-point shot. Clearly, the two-point shot was more valuable, and the league even tried shortening the distance of the three-point line in the 1990s. Over time, as players practiced and got better at the three-point shot, its expected value rose dramatically. Today, NBA players convert at a similar rate on two-point attempts, scoring 49.2% of their attempts, giving the two-pointer a PPS of 0.9834. However, the three-pointer is being scored at a much more efficient rate with 35.2% of these attempts hitting nylon for a PPS of 1.0566, supporting the claim that the three-pointer is the more valuable shot in the modern day NBA.

Taking the ratio of the two-point and three-point PPS of the 1979 NBA seasons gives us a scalable, historical factor that is usable for finding a desirable initial four-point expected value. The ratio calculated is 0.84:0.97, or 0.866, meaning that the three-pointer generated points less effectively (86.6% as effective) than the two-pointer when it was originally introduced in 1979. It is clear that the NBA was happy with the development of the three-pointer after its inauguration, as it remained untouched for fifteen years and was allowed to develop naturally. If we assume that the four-point shot will follow a similar rate of development, then we can use the naturally occurring ratio to scale the current NBA PPS’s to find bounds for a potential 4-point shot PPS. Multiplying the current PPS’s by the ratio would give us an initial expected value for the 4-point shot that provides a balance between the two-pointer and three-pointer. This balance would match up with the balance between the initial expected value of the 1979 three-point shot and the 1979 two-pointer since we multiply by the historic ratio, which we know the NBA was satisfied with. Using the two-point PPS from the NBA 2020 season, we get 0.866 (0.9834) = 0.8516, and using the three-point PPS, we get 0.866 (1.0566) = 0.9150. In the context of a four-point line, the NBA could introduce it with an expected value between these two calculated values to have the four-point shot be a balanced shot compared to the current two and three pointers. However, it is more advisable to introduce it with an expected value similar or slightly below the scaled two-point value of 0.8516, as players have already been practicing long distance shots and the development rate is likely faster. Even over the last decade, with no points incentive to take longer shots, the field goal percentage for 30+ foot shots has increased from 6% in 2010 to just over 22% in 2019. Introducing the four-point line with an expected value less than the calculated lower bound using the historic ratio prevents the shot from becoming overused and gimmicky, and instead a skill reserved for only the best shooters.

The current NBA three-point line is 23.75 feet away from the basket but only 22 feet away from the corner wings. The league has shown that it is willing to change the game for more desirable results, evidenced by the shortening of the three-point line to a universal 22 feet during the 1994-1995 season in an effort to increase scoring and counter the highly physical nature of the game. Now, as the league faces a problem of not rewarding longer, more exciting shots, the four-point line will require the NBA to be flexible once more. Knowing that the NBA should introduce the four-point shot with an expected value around 0.8, we need to find an appropriate distance to place a potential four-point line. To do so, we will use shot log data from the 2017-2018 NBA season in which James Harden won the Most Valuable Player award while leading the league in three-pointers made with 265, again signifying the importance of the shot. The NBA also stopped publishing shot distance data in 2019, making the 2017-2018 season data the most relevant to the conversation about the current day NBA. Over 150,000 shots were taken in the season from varying distances on the court, but the data used was on shots from over 23.8 feet away, just farther than the current NBA three-point line. This data was then binned by the foot, and the field goal percentage at each distance was calculated. Naturally, as players attempted shots from farther away, their field goal percentages dropped. At 24 feet, players made 38.4% of their shots, but 10 feet farther at 34 feet, players only made 14.3% of their shots. Obviously, the four-point line would be introduced farther away from the three-point line, so only the percentages from 24 feet onwards were relevant. To find the expected value of the four-pointer at each distance, we take the field goal percentage of the shot at that distance and multiply it by the value of the four-pointer, 4. This would give the four-pointer a PPS of 0.384 (4) = 1.536 at 24 feet and a PPS of 0.143 (4) = 0.571 at 34 feet. The plot below represents the expected value of the four-pointer at various distances on the court.

As can be observed, there is a general decreasing trend in the chart with the PPS decreasing as shot distance increases. There are some interesting trends that can be identified. At 29 feet, there is a drastic decrease in expected value until it levels out around 32 feet. The drastic slope indicates that this distance is where most players stop practicing three-pointers, meaning that there is room for improvement at these distances. We also notice that past 32 feet, the curve begins to level out again, staying slightly below 0.6 PPS even as the distance increases. Because these shots past 32 feet have a similar PPS to halfcourt shots, or “lucky” shots, we know that there isn’t much skill involved in these shots. We want our shot value to be around 0.8, so any distance past 32 feet seems to be too far and the shot would become luck based. Observing the intercept of 0.8 PPS and the curve, we find that 30 feet yields a PPS of 0.8, and would serve as an appropriate distance to place the four-point line. 30 feet is a distance that isn’t luck based and has room for improvement, as indicated by the steep downwards slope of the curve at this point.

Like the many changes the NBA has made to the game of basketball, the four-point line would be introduced to increase excitement, particularly towards the ends of games. We can group our dataset once again into quarters to look at the success rate of long-distance shots at different points in the game. The graph below represents the field goal percentage of shot attempts from 29 feet or farther.

Immediately, the field goal percentage is visibly higher on these types of shots in the fourth quarter, and even in overtime, both reaching just over 20%, while the average of the other three quarters hovers just below 15%. Another interesting trend is the slight decrease in field goal percentage as the game progresses until the fourth quarter and overtime starts. This can likely be attributed to fatigue, as the force required for long distance shots is much greater than the average three-point attempt. Additionally, better players getting more playing time in the fourth quarter and overtime, as well the “clutch” factor, can be used to explain the increase in the field goal percentage towards the end of games. To see if there is a significant increase in field goal percentage of shots over 29 feet from the early game (first, second, and third quarter) to the end of games (fourth quarter, overtime), we can perform a 2-sample t-test. The average field goal percentage in the early game is 14.806%, while players make 20.76% of these shots in the late game. Using a one-tailed hypothesis, we get a z-score of 1.655 and a p-value of 0.04846, which is statistically significant, so we reject the null hypothesis that the field goal percentage on 29ft+ shots is the same across the early game and late game and accept the hypothesis that the field goal percentage on 29ft+ shots is higher in the late game than in the early game. This supports the claim that the four-point shot would add excitement to the ends of games since players are converting at a higher rate.

The NBA could feasibly introduce a four-point line to the game. With the three-point shot already dominating the game, the four-point shot could add another dimension to the game that rewards the best shooters. After comparing the expected value of the four-point shot at various distances, it is clear that the NBA could introduce the shot without it becoming a luck-based gimmick shot. The 30 foot distance would also allow for development and improvement of the shot based on the expected value curve as well as historical evidence from the introduction of the three-point shot. Similarly, there is evidence that the four-point shot would be more useful in late-game scenarios, which is the primary reason the NBA would want to introduce the shot. Of course, some basketball purists would argue that the shot is a gimmick, similar to the criticisms of the three-point shot in 1979. But for a league whose slogan is “Where Amazing Happens”, why not introduce a new shot to let even more amazing happen?

*Sources: *__Basketball Reference__*, *__Kaggle NBA Shot Logs__

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