The rise of the three-point jump shot in the NBA has been well documented. Over the past five regular seasons the average team three-point attempt rate has spiked from 24.3% to 33.7%, per basketball-reference. The analytically driven Houston Rockets actually attempted more three-pointers than two-pointers this past season!
In this three-point happy league, analyzing an individual game after removing the effects of “hot” or “cold” three-point shooting can be very informative. It allows us to see how well the teams played in all aspects of the game except three-point accuracy. We can strip aside a particularly unusual shooting performance and observe the “fundamentals” of a game, in a certain sense. This is what Three-Pointer Independent Game Score (abbreviated as 3-PIGS) does.
A team’s 3-PIGS score is the percentage of times the team would win the game if we replayed the game many times* with all three-point attempts randomly simulated. The probability a three-pointer is made in each simulation is equal to the the yearly three-point percentage of the player taking it**.
* I conducted 1000 simulations of 2017-18 regular season games for the purpose of testing the merits of 3-PIGS.
** Players with less than 30 total three-point attempts in the season are given a 30% chance of making each attempt.
The 3-PIGS score answers this narrow question: Given all the information about a particular game except the knowledge of the success or failure of each three-point attempt, what is the probability that a given team won the game?
Computing 3-PIGS is not complicated. I simply subtract all points added from threes from the final game score and then add points back as dictated by the simulation results. I do this a bunch of times (usually 1000) and count the percentage of times each team wins.
There are a couple reasons why 3-PIGS is informative. First, defenses tend to have more influence over opponent three-point volume than opponent three-point accuracy (I examined this to an extent here). 3-PIGS is working at exactly this level. The 3-PIGS score will be influenced by the volume of three-pointers the opponent takes and which players take them, but not whether they are actually made.
Secondly, 3-PIGS tells an interesting part of the story of a game which can be lost by simply looking at the final score. As an example, take games 1 and 2 of the Cavaliers vs. Celtics Eastern Conference Finals, which the Celtics won by 25 and 13 points respectively. By conventional wisdom, the Celtics dominated game 1 even more than they dominated game 2, but 3-PIGS tells a different story. The Celtics’ 3-PIGS scores for the games were 74.2% and 87.7%, indicating that their game 2 performance was even more impressive. The Cavaliers’ game 1 shooting woes had a large effect on the final score. They made only 4 of 26 three-point attempts in game 1, while through chance alone we would expect them to actually make a bit more than 10 on average.
Finally, 3-PIGS goes beyond team three-point percentage and takes into account the actual individuals who took the shots. This is simply because each three-point attempt is simulated with a probability of being made equal to the three-point percentage of the player taking it. The 3-PIGS score gives a bit more information than simply comparing a team’s three-point percentage in a particular game with their season average because it takes into account who shot the threes.
3-PIGS gives the proportion of time each team would win a game (after simulating three pointers), so it makes sense to see how accurate it is as a predictor of the winner. The plot below shows that 3-PIGS is a fairly well-calibrated predictor, meaning that teams win about as often as the percentage given by the 3-PIGS score for the game would suggest. The data used for this plot is all 2017-18 regular season games.
There is a trend for the home team to win more often than their 3-PIGS score would suggest, as depicted by the blue line being above the red line. This is especially true when the home team is given between a 40%-80% chance of winning the game. Interestingly, this trend cannot merely be explained by teams shooting better at home. Teams actually shot 0.2 percentage points higher on threes on the road than at home over the 2017-18 regular season.
There are certainly flaws with 3-PIGS. This statistic is not taking into account the specific difficulty of each three-point attempt. Perhaps in the future I should consider whether each shot was contested vs. wide-open or off-the-dribble vs. catch-and-shoot, and build this knowledge into 3-PIGS. And there is also the fact that missed three-pointers offer the offensive team a chance for a rebound, thus blunting a little of the downside of a missed three. Even with its flaws, I believe 3-PIGS can give us valuable insight into how much three-point shooting variability contributed to the outcome of a game.
For the sake of curiosity, I provide a link to a Google Sheet with the results and 3-PIGS game scores of all 2018 playoff games through Houston vs. Golden State game 6. The column titles ‘Home 3-PIGS’ is the 3-PIGS score of the home team in the given game (so the road team had a 3-PIGS score of 100 – Home 3-PIGS). The columns titles ‘Home Margin’ and ‘Expected Home Margin’ are, respectively, the home score minus the road score and the expected home score minus the expected road score, if all three-pointers were random. In the future, I will probably release 3-PIGS scores for regular season games and other postseason games.
Acknowledgement: Inspiration for this post was provided by Jacob Goldstein’s article for Nylon Calculus, Nylon Calculus: Defining and calculating luck-adjusted ratings for the NBA. He came up with a method for adjusting the net-ratings of individual players to account for three-point shooting (and free throw) luck.