Basketball games that are close are fascinating not only to the pure fans of the game but also to the analytics people who are interested in the rationality of the decisions involved in the game. There are various scenarios that are of interest but in this post I will focus on two very specific ones that can be (somehow) tracked through simple math. Apart from that they also appeared on both men’s national championship semi-finals between Gonzaga-South Carolina and UNC-Oregon. This will hopefully also make you think in an analytical way next time your time faces a similar situation.

### Offense: Trailing by 2 points on the last possession

This is a scenario where your team trails by two points, there are 10 seconds left and your team has the ball. What is the rational decision? The ultimate goal is to win the game so there is some deciding to be made. In this scenario there are two options; (a) attempt a 2 point FG and try to send the game to OT, or (b) attempt a 3 point FG and try to win the game in regulation. Each of these options is associated with a corresponding win probability. Let’s examine the first one: with the average 2 point FG% being about 51% this year in the NBA, option (a) wins the game for your team with an approximate 0.51*0.5 = 0.255 probability (we have assumed equal chances between the two teams to win in OT — while this might not be exactly true for all the matchups it is a fairly good approximation). With the second option, you will win the game as long as you score the 3 point shot which happens approximately 36% of the times this year on average in the NBA. This clearly means (without considering the associated variances) that option (b) (i.e., taking the three-point shot) gives the trailing team higher chances of winning the game.

However, in a basketball game there are two teams that are playing and in this setting the team that is ahead in score is playing defense. Based on the above analysis one must expect that the defense tries to avoid having the trailing team shot the 3. One could even go one step further and even suggest that the defense should foul the offense in order to avoid the three-point shot (and even have their own chance down the stretch if there are a few seconds left). By no means I am suggesting this but it is worth thinking this option too (?). In fact, if the defense purposefully foul the trailing team, will win the game with a probability around 55% (considering league average for FT% and OFR% — see below), which is smaller compared to the 64% probability if they let the trailing team attempt the three-point shot.

In conclusion, in this scenario the defense usually (possibly correctly) waits for the offense to make the move and from a probability point of view – on average – it seems that trying to win the game in regulation with a three-point shot gives the trailing team the best chances of winning (we will use this in the following, slightly more complicated scenario).

### Defense: Being ahead by 3 points on the last possession

An even more interesting situation can appear when a team is leading by 3 points but the opposition has the last possession of the game. This was the scenario in this year’s first NCAA semi-final between Gonzaga (up 3 points) and South Carolina (last possession with 12 seconds left). Similar situation appeared in the other semi-final between Oregon and UNC. It is clear that for the trailing team there is no other option but to attempt a three-point shot to tie the game. Of course, this is debatable too since a team can attempt a quick 2 point shot and then play the “free throw game”. In fact this is what Oregon did in the second semi-final of the NCAA tournament and while part of the plan worked (UNC had 0/4 FTs in the last 6 seconds), Oregon failed to get a defensive rebound twice and ultimate lost the game by a point. So in order to keep things simple and fairly tractable let’s assume that the offense is set to go for the tying three. After all this is what happens in the majority of the cases when the time is winding down (e.g., less than 10-12 seconds) and/or no time-outs are left that would advance the ball after the oppositions FTs (this is another problem that Oregon would have to face with their strategy because not only they did not have any timeout left but also in college basketball the ball is not advancing down the court after a timeout). Anyway, let’s crunch the numbers.

The defense has two options: (a) let the offense take the potentially tying three-point shot, or (b) foul the offense as the clock is winding down (e.g., 6-7 seconds left) in order to not let them take the three but lead them to the free throw line. Let’s examine every case and calculate the win probabilities.

For option (a), the win probability is easily calculated since the trailing team wins 18% of the times (0.36*0.5) – assuming that the shot will be taken close to the end of the game so the other team has not time for a shot.

Option (b) is a little more complicated but with some realistic assumptions we can still keep it tractable. Our assumptions are as follows:

- If a team trails by two and has the last shot, they will attempt a three-point shot (as per our analysis above)
- If the trailing team is led to the free throw line there is still enough time for fouling
*once*the opponent in order to stop the clock - In case the trailing team gets the offensive rebound from their own free throw shot and still trailing by three, they can move the ball quickly outside the three-point line and take a three for tying the game. Same is true if the trailing team in bounds the ball after the leading team scored FTs.
- If at any point during this last 6-7 seconds a team takes the lead by 4 points or more, then we consider the chances of the team losing negligible

Someone might oppose these assumptions but we are making them in order to find the best case scenario for the trailing team (or the worst case scenario for the leading team) when choosing option (b).

So when the leading team decides to foul the trailing team with a few seconds left (e.g., 6-7), then there is a large number of possible “paths” that the game can take. Only a handful of them gives the win to the trailing team and we are going to calculate the probability for these ones:

- Trailing team scores the first FT, misses the second one, gets the offensive rebound and scores a three to win the game
- Trailing team misses both FTs, gets the offensive rebound, scores a three and wins the game in OT
- Trailing team scores both FTs, fouls the opponent who misses both free throws, the trailing team gets the defensive rebound and they score a 2 point FG to win the game
- Trailing team scores both FTs, fouls the opponent who scores the first free throw, but misses the second one, the trailing team gets the defensive rebound and they score a 3 point FG to tie the game and go to OT
- Same scenario as above but the leading team scores the second FT
- Trailing team misses the first FT, scores the second FT, fouls the leading team, which misses both FT, the trailing team gets the defensive rebound and scores a 3 point shot to send the game to OT.

As you can see even with the more optimistic of the assumptions for the trailing team, the “paths” to their win are fairly complicated. Of course, we can calculate the total probability by using the league averages for 3 and 2 point FG%, offensive rebound rate as well as the FT%:

- FT% = 0.771
- 2FG% = 0.512
- 3FG% = 0.358
- OFR% = 0.233

Using these numbers the chances of the trailing team winning are just a little over 6% (0.063). Looking it from the perspective of the leading team, letting the opponent try to tie the game wins them the game with a probability 82%, while fouling them wins them the game with a probability 93.7%. Of course, changes in the value of the above variable will change this probability. We run a grid search over the values of FG% and OFR% (ranging from 10%-60%) and we found that the best chances of winning for the trailing team in this range is only around 12% (when all variables are equal to 60%).

If we consider the OFR% to be the league average (i.e., 23.3%) then the probability for the trailing team winning changes as per the following figure:

In general it seems (again without considering the variability around the average of these percentages) that the leading team will have even better chances to win by fouling the trailing team instead of letting it attempt to tie the game.

## Conclusions

- When down by two points, with the ball possession, shooting the three gives the offense better chances of winning the game.
- When a team is trailing by three, the
*optimal*strategy for the defense is to foul the offense.

**Limitations: **Of course this analysis does not come without limitations. First, we used the league averages, but teams differ in FG%, FT% and OFR%. Therefore, for a specific matchup the actual win probabilities will be different. However, we saw that there is not a lot of variation in the probability when we change these parameters and hence, the same decision might ultimately apply. Furthermore, we have assumed that there are a few seconds left on the clock and that the trailing team can thus, stop the clock only once. More complicated scenarios certainly appears in practise and decision-making might be different in these cases.

Despite the limitations I believe that this is a good path towards making data-driven decisions at the end of the game and certainly an interesting problem to work on given the different complex situations that you can have. Data can also help by identifying games where this setting appeared and what teams did. I have not looked yet into that but my feeling is that the data for these situations (at least situations that are virtually identical) would be very sparse and hence, it will be hard to make any statistical conclusion.

Would like to see references, very similar to a chapter in mathletics!

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The first scenario of trailing by 2 indeed is in the Mathletics book (the author of which is a friend of mine and we exchange ideas). However the main part of the article (which is also given in the title) is an example that is not worked out in the book. So I am not sure what exactly the point that you made is.

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