May 11, 2010

To Foul or Not to Foul?

Filed under: Uncategorized — wwinston @ 8:37 am

Suppose your team is up by three points. The other team has the ball and there are ten seconds left in the game. Should you foul or not? The April 23 New York Times (see

had a great analysis of this decision.

One of our top MBA”s, Sean Vinsel has done a great piece of research on this subject. Here is a summary of his results. The key insight is that the benefit of fouling depends on how good the other team is at shooting 3′s. Also it is important to note that  even with little time left in an NBA game several possessions can follow an intentional foul. We saw this on Saturday when the Lakers intentionally fouled the Jazz and the Jazz got the ball back with a chance to win the game.

Sean Vinsel’s Analysis of Fouling with a 3 Point Lead

In the Friday, April 23rd issue of The New York Times, Jonathan Abrams and Howard Beck wrote an article concerning teams up 3 points at the end of NBA games committing fouls to send the losing team to the free throw line instead of being able to attempt a game winning 3 pointer. In this article they included quotes from several coaches, most of whom oppose the idea, and statistics from Synergy Sports Technology tracking these situations over the past 2 years. According to their data, It came up 165 times. 19 times the leading team fouled, and 146 they didn’t. 18 of the 19 fouling teams won (94.7%, regulation and OT wins) and 135 of the 146 teams won after not fouling (92.5%). These samples are so small, however, that it is unclear which strategy is best, and under what scenarios.

Traditional analysis has been to simply think through this problem, leading to analysis like the following quote from the Times article:

“On the surface, it seems like an easy choice. Long-distance shooting has improved and teams cannot tie the score if they are not given the chance. Intentional fouls remove the surest route to a tie: the 3-pointer.”

However, there are so many variables in this situation (3 point %, free throw % for both teams, time to commit fouls, time to get shots, rebounding) that simply looking at actual situations won’t give you a large enough sample. But what if you could simulate many different scenarios thousands of times? We have done just that, and the results show some interesting delineations in strategy.


                Uising Visual Basic for Excel, we have created a model that allows the user to simulate the last seconds of an NBA game up to 100,000 times, with various parameters. To create a model to simulate the end of an NBA game, some randomness has to be simulated. The key factors included the amount of time to foul intentionally, the amount of time for a team to take a last-second shot, and the amount of time to get a rebound off of a free throw, 2 pointer, or 3 pointer. We collected data from the 2009-10 season from situations where a team was trailing by 0, 1, 2, or 3 points and fouled, or took a shot, in the last 15 seconds. Data was also collected on rebounds and the time it took to collect them. Because of precision was required to tenths of seconds,’s game logs were used. In addition, the user was allowed to specify shooting percentages for the leading or trailing teams on free throws, 2 pointers, and 3 pointers. Defensive rebounding percentages (76.7%) were gathered from the SacTown Royalty blog, and some small randomness was built in to account for situations where the leading team fouled during the act of a shot, including a made 3 pointer. Users can select whether to foul or not, and can specify a time at which they will stop fouling. To simulate the data used by Abrams and Beck, start times were randomized between 3 and 10 seconds. The user can also exercise an option to start at the same time for every simulation.


During the 2009-10 NBA season, through January 9, teams were hitting 36.7% on 2 pointers and 16.7% on 3 pointers in the last 10 seconds of games where they trailed or were tied. Notice, this includes multiple shots on one possession, and games where teams were down by less than 3 or tied, so it is a slightly different data set than Abrams and Beck. When applying this season’s shooting percentages to our model, we get a winning percentage of 92.24% over 50,000 simulations when not fouling, compared to Abrams and Beck’s 92.5% (regulation and OT wins) and 93.75% over 50,000 simulations when fouling compared to their 94.7%. Although their sample sizes are smaller, the results Abrams and Beck quoted seem reasonable when we run historical data through our model. It seems like these percentages are very close to the true shooting percentages for NBA teams in the last 10 seconds when trailing or tied. The model then indicates that there is some advantage to be gained by fouling in these situations, even though the 3 point shooting percentage is so low!

This 16.7% estimate for 3 point shooting does include last second heaves at the end of games, however…perhaps we should look at higher-quality 3 point shots in the last 10 seconds. With the same set of data through January 9 as mentioned above, NBA teams shot 37 3 pointers after timeouts in the last 10 seconds, and hit 9 of them, for 24.3%. Let’s take a look at how the model performs at 20, 25, and 30 percent estimates for 3 point shooting, with 50000 simulations each:

At 20%:

Fouling: 44322 wins, 5262 overtimes, 416 losses – 93.91%

Not fouling: 41358 wins, 8642 overtimes – 91.36%

At 25%:

Fouling: 43576 wins, 6058 overtimes, 366 losses – 93.21%

Not fouling:  39275 wins, 10725 overtimes – 89.28%

At 30%:

Fouling: 44896 wins,  4878 overtimes, 226 losses – 94.67%

Not fouling: 37644 wins, 12356 overtimes – 87.64%

The winning percentages when fouling fluctuate a little more because the remaining possessions can vary wildly depending on who hits their free throws, while not fouling basically is limited to the opponent shooting a 3, then possibly a rebound and some foul shots. However, we can see that increasing the other team’s three point shooting affects our winning percentage when not fouling much more than when fouling, because their 3 point shot is going to always be taken with a chance to tie the game. When we foul, we’ll sometimes gain points, sometimes lose them, but fewer 3 pointers will be taken with a chance to tie in that case. If you think the other team is a good 3 point shooting team, that makes the case for fouling them even stronger. As Abrams and Beck suspected, there seems to be an advantage to be gained by intentionally fouling teams with a 3 point lead!


  1. Seems to go against all logic but you can not deny the results. Fascinating discovery.

    Comment by Larry (IEOR Tools) — May 11, 2010 @ 9:09 am

  2. I wonder what the difference in 3 point shooting percentages is when the defense knows that they don’t have to guard against the two point shot (although I guess you sometimes end up with Jameer Nelson not guarding Derek Fisher).

    The Visual Basic for Excel thing made me smile. I did something similar for figuring out if Lebron should’ve passed to Donyell Marshall for 3 or went for the layup against the Pistons back in 2007.

    Comment by Chris K — May 12, 2010 @ 4:21 pm

  3. Doesn’t seem to go against all logic at all- free throws are high percentage shots, 3pointers less so. However, if up 3, fouling does not allow them to tie the game, and even if they make both, allows you to make 2 free throws to again go up by 3. It’s entirely logical.

    Comment by John Kenney — May 12, 2010 @ 10:42 pm

  4. wonder if the model takes into account turnovers. i would imagine so with all the thought into everything else, but didn’t mention it. reads like it is just factoring teams shooting %, time, and rebound rates.

    also, a better question is can or have they identified other statistically significant factors once the random game data was generated? for example, i would wager that the time left on the clock is significant: with 3 seconds left, you’re probably much better off fouling (by the time you shoot freethrows and miss the second intentionally, the game is over), whereas if there are 10 seconds left the chances of missing freethrows and turnovers might tip the scales the other way, at least in the 20% example.

    Comment by Herman — May 13, 2010 @ 1:35 am

  5. time left is key and we do consider turnovers

    Comment by wwinston — May 13, 2010 @ 7:12 am

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