By Benedict Brady

It is Tuesday January 7th, and the Minnesota Timberwolves are playing the Utah Jazz in a less than marquee matchup between two small market teams. The Wolves lead for most of the second half, but as time winds down at the end of the game, they find themselves trailing by 2 going into the final possession. With 28 seconds left, Minnesota calls a timeout and draws up a play to get Karl-Anthony Towns the ball, who launches an 11 foot 2-point attempt that bounces out. Luckily, after a tie up for the rebound, the Wolves recover the jump ball in the backcourt with 15 seconds left on the clock. Here Zach LaVine dribbles the ball up slowly and then waits until the final seconds of the game, where he drives in and attempts a contested 19 footer to tie the game. He misses, and the Wolves let another one slip away.

You can hear Wolves fans groan throughout the arena. This type of shot rarely goes in, and it seems like Zach LaVine has made another immature decision. But, upon further inspection, there may be more than one thing wrong with this decision. Following the game, famed basketball gambler Haralabos Voulgaris expresses annoyance with the Wolves game plan.

If you are gonna run the clock out and take the last shot when down 2, at least take a 3. Taking a 2 was absurd. — Haralabos Voulgaris (@haralabob) January 8, 2017

The obvious inefficiency here is the the 20 seconds that Zach LaVine wastes, taking a quick shot allows for time to foul if he misses. But Voulgaris is picking up on a more subtle critique here: the Wolves should be looking for a three, not a two. While a two is more likely to go in and thus extend the game, a simple expected value calculation suggests that a three point attempt may be more favorable. With time expiring in the game, shooting a three at a league average rate of 36% probably yields roughly a 36% chance of winning the game. However, shooting a two at the league average rate of 50% yields only a 25% chance of winning because overtime is probably close to a toss up.

This is a nice theory, but we can put it to the test using actual game data.

To fully understand this scenario, we can scrape all the instances of teams shooting while down two points in the last 20 seconds of a game, going back 10 years, from Basketball Reference. For each shot, we are interested in seconds remaining in the game, shot distance, point value of the shot, whether or not the shot is made, and whether or not that team won the game. There is one thing to be careful of here, the heaves taken in the last few seconds of the game do not fit the criteria we are interested in (as it is the team’s only option). To best combat this, we will limit the analysis to shots within 27 feet of the basket. This gives us a data set of 1536 shots since the 2006-2007 season. This analysis does not include fouls, as Basketball Reference does not record where they take place. Because of this, perhaps shots with a high foul frequency, such as layups, are slightly undervalued (although refs probably call significantly less fouls in the final twenty seconds of a game).

The first hypothesis to put to the test is the idea that holding onto the ball negatively impacts a team’s chance of winning. Below is a graph of two point and three point field goal percentage versus time left in the game. The graphs in this article are made using moving averages along the x axis, since the data set is small and very noisy.

From this graph, we can see the negative effects of holding on to the ball. While this analysis does not consider the time that each team has to set up their play, it seems abundantly clear that field goal percentage plummets as the game comes to a close. There are a few things to note here, though. First, field goal percentage for twos and threes is not significantly different with about 10-20 seconds left on the clock. Since a three seems so much more useful here, there may be some hidden value. Second, three point percentage plummets much quicker than two point percentage between nine seconds and three seconds remaining in the game. Finally, field goal percentage bottoms out with between two and zero seconds left in the game, representing hurried and highly difficult shot attempts. For this reason, we will split the data up into three time frames.

Time Frame 1: The Last Ditch Shot Attempt (0-2 Seconds Remaining in the Game)

Below is a graph representing how shot attempts with less than two seconds on the clock vary by distance. The green line represents the league average here from 2001-2015 for reference (big thanks to Matt Goldberg for help with retrieving this data).

The takeaway here seems to be that unless you are getting a shot at the rim, efficiency is incredibly low. It is interesting to note that teams are hitting threes at about the same rate as most mid range shots, so if there is no way to get to the rim it may be best to settle for a three pointer. This would support the argument that Zach LaVine’s 19 foot jumper was a flawed decision. It seems that NBA teams know this for the most part though. As we can see, most of the shot density is concentrated at the rim and the three point line.

We can also look at the splits between the two point and three point attempts, breaking it down by shooting percentage, winning percentage for the team that attempts the shot, win percentage given that the shot goes in, and number of shots taken since the 2006-2007 season.

Shooting % Win % Win % Given Make Sample Size 3 Pointer Attempted 18.75% 19.23% 94.87% 208 2 Pointer Attempted 30.19% 17.86% 48.38% 308

As seems intuitive, teams almost always win after making a three, and win about fifty percent of the time after making a two. It seems that three pointers are slightly favored, but it is probably advantageous to steer away from this situation if possible.

Time Frame 2: When The Defense is Hedging Against a Three (3-9 Seconds Remaining in the Game)

Here we can run the exact same analysis, switching the time frame. Below is the shot selection graph.

Here, we start to see a little bit of value in the midrange. From the field goal percentage breakdowns, it seems likely that defenses are very scared of a three point attempt in this situation, and as a result are allowing midrange and long twos at a roughly league average rate. Again, we can look at the splits between two and three point attempts.

Shooting % Win % Win % Given Make Sample Size 3 Pointer Attempted 21.97% 18.94% 65.52% 132 2 Pointer Attempted 41.40% 21.87% 37.32% 343

While the three point percentage has not increased much in this time frame, teams are hitting two point attempts over ten percent more frequently. Here we see that the two point attempt is favored by roughly three percentage points in win probability. We can also see that NBA teams are generally aware of this, shooting almost three times as many two point attempts as three point attempts in this time frame.

Time Frame 3: Everything is About Equal (10-20 Seconds Remaining in the Game)

Again, we can look at the graph of shot selection and tease out a few patterns.

Here, we see the three point percentage rebound to normal. Teams are not as scared of a three if they have ten or more seconds to go down on the other end and score a game winner. Breaking it down by two and three point splits yields the following chart:

Shooting % Win % Win % Given Make Sample Size 3 Pointer Attempted 32.10% 27.16% 65.38% 81 2 Pointer Attempted 40.86% 26.86% 37.76% 350

This chart may seem a little odd at first. Teams hit three pointers only a bit less frequently than two pointers, and win almost twice as frequently when they go in. So why do teams only win slightly more off of a three point attempt? The explanation for this is offensive rebounding rates. Teams that miss their two point attempt go on to win the game 28% of the time, while teams that miss a three go on to win 19% of the time. This is a drastic difference, given that most of the shots are misses. The one caveat here is that better shooting teams will see an outsized positive impact of shooting a three versus shooting a two, as rebounding rates become less important.

So, what is the takeaway?

As you probably already knew before reading this article, there is a lot of value in getting to the rim in the last few seconds of the game. Not only do these shots have a much higher field goal percentage, but the teams rebound these shots at a much higher rate. It is also interesting to note how obviously we can see game planning against the three in the second time frame. Still, there is a lot of analysis left to be done on late game strategy, and hopefully the rest of the season and playoffs will add a few more events to the data set.

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