Up Three, Should You Foul? Eh, Maybe

There are few debates more contentious in basketball nerd circles than whether or not to foul when leading by three points at the end of games.

There's the conventional wisdom that you do not foul; you defend the three-point line and force the opposing team to make a tough shot to just force overtime. After all, fouling extends the game, and the introduces the infinitesimal -- albeit real -- chance that you lose in regulation.

Or you can prevent the other team from even attempting a three-pointer by fouling them before the shot. Then they have to a) make the first free throw, b) deliberately miss the second, c) gather the offensive rebound, and d) score on a putback to even the game.

Fortunately, the folks at the Harvard Sports Analysis Collective took a look at every such situation in college basketball last season to come up with an empirical answer. Results after the jump.

Turns out, it doesn't really matter. From the Harvard Sports Analysis Collective:

Of the 52 teams that committed a foul, six lost the game for a winning percentage of 88.46%. Of the 391 teams that did not foul, 33 lost the game for a winning percentage of 91.56%. Both a two sample t test of proportion and a Chi-squared test fail to reject the null hypothesis that there is a difference in winning percentage between the two strategies. In this sample, teams that did not foul won slightly more often. For the less statistically inclined, this means that there is no significant difference between the two strategies.

This is a puzzling result, to say the least. Intuitively, given that nearly everything that has to go right for the trailing team to tie the game after an intentional foul, it would seem like that would be the clear cut option for the leading team. And a quick thought experiment would seem to back that up.

Consider the following scenarios (or for the less math-inclined, just skip below and trust me):

Case 1: Team A does not foul and Team B attempts a three-pointer. WinProb=3PT%*0.5**

** You multiply by one-half in this case, because making a three-pointer would force overtime, and an extra period is statistically a 50-50 proposition despite what you might think about momentum, etc.

Case 2: Team A does foul. Team B makes the first free throw, intentionally misses the second, gets the offensive rebound, and makes a two-pointer to tie it up. WinProb=FT%*OReb%*2PT%*0.5

Case 3: Team A does foul. Team B makes the first three throw, misses the second on purpose, grabs the board and makes a three-pointer to win in regulation. WinProb=FT%*OReb%*3PT%

Case 4: Team A does foul. Team B misses the first free throw, deliberately misses the second, gets the offensive rebound and makes a three-pointer to send it to overtime. WinProb=OReb%*3PT%*0.5

Using some data and guesswork, we can fill our variables in with some decently close approximations. Admittedly cheating a little bit, according to the incomparable 82games.com, teams snag offensive rebounds 13.9% of the time off a missed free throw, and make 50.4% of their putbacks in those situations (granted, this is from NBA games, but it should be in the ballpark for college ones as well). Add in that college players made 34% of three-point attempts and 69% of free throws, and we're all the way there.

Or at least mostly. The fact that teams that did not foul won approximately 92% of the time implies that trailing teams only made potentially-tying three-pointers 16% of the time in end-of-game situations. That makes sense given that the winning team will guard the three-point line and essentially concede anything in close. For our purposes, we'll say it's a 20% chance the trailing team makes a shot from distance in this situation.

Now we can crunch some numbers. And it looks like fouling should be a slam dunk (10% vs 5.9%)...so what gives? User error. Again from the Harvard Sports Analysis Collective:

Interestingly, in 12 cases, the leading team fouled the shooter in the act of shooting a 3-point shot (nine of those times, the shooter made all three free throws). Some of these cases, like the Kansas State, are players who are instructed to foul simply fouling too late.

Of course, we don't exactly how many of those cases were instances where a team didn't mean to foul versus those where they did and mistakenly fouled during a shot, but the point remains: actually executing an intentional foul in this situation can be fraught with difficulty. Just ask Kansas State after their epic double-overtime win over Xavier last season. As SI's Luke Winn chronicled:

Clemente dutifully tried to foul Holloway shortly after he dribbled past halfcourt. But ... Clemente didn't foul hard enough to guarantee a whistle, and the refs remained silent. That caused Wildcats guard Chris Merriewether -- heretofore considered a heady defensive sub -- to reach in a second later, allowing Holloway to sell the act of shooting a three, and draw an improbable three-shot foul.

Fortunately for Frank Martin's squad, this miscue didn't end up costing them the game. But the difficulty of actually getting your players to foul effectively, as opposed to just playing defense and making the opposition make a play ends up making this debate essentially moot.

Which is a long-winded way of saying that it's complicated.
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