Win probability FT% threshold Pre-game OT 1-and-1 2-shot .900 .756 58.2 39.9 .800 .671 62.2 44.7 .700 .607 65.4 48.8 .600 .552 68.4 52.9 .500 .500 71.4 57.3 .400 .448 74.4 62.0 .300 .393 77.9 67.6 .200 .329 82.4 74.8 .100 .244 88.2 84.2

For example, if we estimated that a team had a 40% chance of beating its opponent before the game started, that team would be justified in fouling a free throw shooter worse than 74.4% if the opponent was in the single bonus and 62.0% if they were in the double bonus.

If you’re interested in the methodology that produced the figures in the chart, you’ll have to read parts 1 and 2. My apologies in advance. Obviously, we shouldn’t treat these numbers as gospel. There are assumptions involved in these calculations and those can affect the final numbers. But I think these are solid ballpark figures from which to make a decision, and the decision to is more beneficial—and less disputable—for underdogs.

In part 1, I promised to show that Jim Valvano’s decision to foul Alvin Franklin was not only justified but appropriate, and through the chart we can see that it was. Brent Musburger helpfully notes after the game that the Wolfpack were an eight-point underdog, which would put them at about a 20% chance of winning (or a 32.9% chance of winning in overtime). Franklin was a 63% free throw shooter which is well below the listed threshold of 82.4%.

Here’s how the fouling worksheet looks using this figure and Franklin’s season-long shooting percentage.

So State’s chance of winning was increased from 25 percent to 33 percent by using this strategy. It’s hard to imagine another single strategic decision that could affect a team’s chances so dramatically.

However, there is little interest in pursuing such a strategy in 2014. Two contemporary examples that illustrate this point occurred on the same night last season. According to FanMatch, both Auburn and Boston College were given a four percent chance of beating their respective opponents on February 19, Florida and Syracuse. Both underdogs faced the last-possession situation described here.

Based on my win probability model, teams with a four percent chance of winning have about a 16 percent chance of winning in an overtime scenario. Let’s say my model was missing some information—betting markets were slightly more bullish on each underdog’s chance of winning—and each team actually had a 25 percent chance in overtime.

In this case, playing straight-up defense on the final possession would give each squad just a 20 percent chance of winning (because the possibility of overtime is so large and the chance of winning in overtime is so bleak). Therefore, the FT% shooting threshold for fouling goes up significantly. In a single bonus situation, fouling anyone that was worse than a 87.7% shooter would improve the team’s win chances. Both the Eagles and Tigers were in the double-bonus situation, but that case was viable, too, requiring fouling a shooter worse than 83.4% to improve the win probability.

In BC’s situation they could have fouled Tyler Ennis, a 75% shooter, and improved their chances of winning from 20.6% to 23.3% using my calculations. BC played straight up D, survived to see overtime, and eventually handed Syracuse its first loss of the year in one of the season’s most unlikely outcomes.

Auburn, on the other hand, accidentally fouled its way into a great situation when Asauhn Dixon-Tatum hugged Patric Young after an Auburn missed free throw. Given that Young was a 57% free throw shooter for his career, there may not have been a foul all season that improved a team’s chances more than this one. According to the math, sending Young to the line and reducing the possibility of overtime bumped Auburn’s chances of winning to 28.9 percent from the 20.6 percent they would have faced by playing defense. It’s the kind of improvement that doesn’t occur in even the most cherry-picked of fouling-up-three scenarios.

Young made both free throws, Auburn turned it over on the subsequent throw-in, and just like that the Tigers were on their way to a regulation loss. Of course, despite the improvement in win probability the brilliant accident didn’t come anywhere close to guaranteeing victory, just as after Valvano’s decision, N.C. State was still a significant underdog, as well. (Until Franklin missed the first free throw, that is.)

But we’re a long way from 1983, and Dixon-Tatum’s happy accident was strongly criticized by the SEC Network’s on-air crew and at least one national writer. And this reaction is always going to be a factor to a coach that considers fouling. Despite the fact that this strategy can provide a large benefit to a team trying to pull off an upset, that team will still lose most of the time when fouling. And the losing coach will have employed an unconventional strategy which will get criticized in a way that playing straight up defense will not.

So we’re only likely to see this strategy implemented by a coach who isn’t concerned about his reputation. However, it’s a strategy that can be useful, especially for underdogs who figure to struggle in five more minutes of action.

]]>Factoring in the possibility of an offensive rebound to this analysis makes it somewhat more complicated, but it’s necessary to determine the merits of fouling. My examples will all use the single bonus situation. Not surprisingly, the math is much more favorable for fouling when the offense is in the single bonus since the possibility of making zero free throws increases.

Allowing for an offensive rebound, I’ll list the possible situations that could result from a foul in a tie game, and the associated win chances based on history. (Assuming the possession after a foul begins with between 20 and 25 seconds left.) Unlike the data I presented in Part 1, I have removed all possessions that began with a steal since we want to know the chances of winning from a situation where the defense is set.

Free throws by Situation for fouled team fouling team W L OT W% ---------------------------------------------------------- miss-DR Tied on offense 202 34 327 64.9 miss-OR Tied on defense 34 202 327 35.1 make-miss-DR Down 1 on offense 113 217 48 36.2 make-miss-OR Down 1 on defense 29 300 65 15.6 make-make Down 2 on offense 38 281 110 21.7

Armed with these numbers, we can figure out the breakeven point for implementing the fouling strategy. Unfortunately, this isn’t defined by a single number, but two: the chances of giving up an offensive rebound and the ability of the free throw shooter. Actually, the relative strength of the two teams is also an important consideration, but I’ll stick to the case of equality for now in order to keep the discussion as simple as possible.

Let’s assume that the offense has a 15 percent chance of rebounding a missed free throw. This is reasonable since that’s the national average over the past five seasons. I’m comfortable with using that as an average, but there are reasons why this number may be generous. First, last season’s rate was closer to 13 percent. Second, given that nobody fouls with the game tied, teams may be more foul-averse in these situations and consequently may crash the glass with less urgency than usual.

Using the above probabilities, sending a 71.4 percent free throw shooter to the line will produce a win probability for the fouling team of 35.1 percent, identical to the historical chances of not fouling in this situation. (For a two-shot situation, the cut off is 57.3 percent, which makes you wonder why the strategy is apparently used more frequently overseas, where the single bonus situation does not exist.)

For people who like doing those handy worksheets during tax time, here’s one for determining the viability of this strategy:

Obviously, the worse the free throw shooter, the more the odds tilt in the favor of the fouling team. And since primary ball-handlers tend to be very good from the line, it’s not always possible to foul a sub-70 percent shooter early in the possession. However, as long as this strategy is a novelty, which will probably be for many, many years, it might not be *that* difficult.

Consider the case during the West Coast Conference tournament when San Francisco was tied with BYU. Coming out of a timeout with 33 seconds left, Matt Carlino (71 percent FT shooter) dribbled out most of the clock while the Cougars set up a play for a last-second shot. Carlino was a realistic fouling option once you consider that BYU was in the single bonus and its offense against USF’s defense was a battle of strength vs. weakness that Rex Walters’ should have wanted to avoid.

But USF might have done even better. What if USF had tried to trap Carlino out of the timeout? I doubt it’s something that would have been expected. And with any luck, the ball would have ended up in the hands of the Cougars’ other primary ball-handler, Kyle Collinsworth, a 57 percent shooter for his career. To the worksheet!

Sending Collinsworth to the line would have given USF a 41.2% chance of winning assuming the average win probabilities shown above, compared to a 35.1% chance by not fouling. But that assumes that the Dons would have had a 50% chance of winning in overtime, which probably wasn’t true.

My method had given them a 38% chance of winning the game at tip-off, which would translate to a 43.7% chance of winning in overtime. Here’s one more look at the worksheet using that assumption. The win probabilities decrease for each case and so the chance of winning playing traditional defense drops to 31.4%. Thus, there’s more of an advantage to fouling.

The issue of team strength is an important consideration. In playing straight up defense, most games will go to overtime. Historically, 58% of such games have gone to an extra period. Therefore a team’s chances of winning while defending are very sensitive to its strength relative to the opponent, while the chances of winning while fouling are relatively insensitive to team strength because overtime becomes more rare in these cases.

To give you an idea of this effect, here are the free throw percentage limits given a particular win probability for the fouling team. Fouling a shooter *worse* than these percentages improves the fouling team’s chance of winning using my assumptions.

Win probability FT% threshold Pre-game OT 1-and-1 2-shot .900 .756 58.2 39.9 .800 .671 62.2 44.7 .700 .607 65.4 48.8 .600 .552 68.4 52.9 .500 .500 71.4 57.3 .400 .448 74.4 62.0 .300 .393 77.9 67.6 .200 .329 82.4 74.8 .100 .244 88.2 84.2

For a team considered a heavy favorite, there aren’t many cases where the strategy can be implemented. But for underdogs, fouling when tied is definitely something to consider. In the final installment, I’ll talk about two of the most famous cases that occurred last season, summarize the proper use of this strategy, and discuss other pesky nuances that should be considered when thinking about this approach.

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Neither member of the CBS broadcast team, Gary Bender or Billy Packer, criticized the idea. Actually, the normally disagreeable Packer was fully supportive of the strategy. And he should have been. Franklin was a 63 percent shooter from the line, and—spoiler alert— I’ll show the math that supports that Valvano gave his team a better chance of winning by giving the foul, assuming Franklin truly had a 63 percent chance to make his free throws.

Today, observers would not be so kind to the Valvano strategy. While popular opinion has seemingly turned the corner on championing the dubious philosophy of fouling while up three with a few seconds remaining, for whatever reason there are few observers who would endorse fouling when tied. Perhaps the introduction of the shot clock changed the thinking. Anyway, I suspect Packer himself might sing a different tune if he were still calling games.

This is not an uncommon situation, by the way. Over the past five seasons, there have been 615 cases where a game was tied with a possession beginning with between 25 and 35 seconds remaining. It happened twice in the 2014 Elite Eight with both Arizona and Michigan on the defensive end of this scenario (and ultimately losing).

In this situation it feels like overtime is a victory for the defense, and that’s may be how coaches, media, and fans view it. For that reason, there’s little discussion about an alternate strategy. If the goal is to get to overtime, then it would be foolish for the defense to foul. But if the goal is to win, then there are cases where the math justifies fouling.

Especially for an underdog that would be expected to lose in overtime, there are times when the possibility of winning in regulation more than compensates for the chance of giving points to the opponent. Over the past five seasons, the defending team won 34.5 percent of the time when faced with the scenario described above. Given that the regulation win is largely a pipe dream for the defensive team, it shouldn’t be surprising that they are significant underdogs in this situation.

Unlike the fouling-up-three situation, where there are plenty of cases of teams utilizing one of two possible strategies (either fouling or playing traditional defense), the fouling-when-tied idea is to my knowledge not used by anyone intentionally. So investigating this situation is not as simple as looking at history and comparing how two strategies have fared.

What we can do is use history to figure out the win probabilities that would result after the fouling strategy was implemented. In the following calculations of expected winning percentage, I am assuming that a team has a 50 percent chance of winning a game that goes to overtime. I’ll eventually discuss the ramifications of changing that number.

First, here’s the data on the defensive team winning with a tie score and the shot clock off:

Score W L OT W% ------------------------ Tied 37 228 350 34.5

Assuming the teams are of comparable ability, any alternative strategy one uses has to be successful more than 34.5 percent of the time for it to be justified.

If the defense chooses to intentionally foul, they figure to get the ball back either tied, down one, or down two. It’s clearly best to give this foul early in the possession, so the assumption will be used here that the fouling team gets the ball back with between 20 and 25 seconds left. Here are the historical win chances for the fouling team in those cases.

Score W L OT W% ------------------------ Tied 228 37 350 65.5* Down 1 132 234 58 38.0 Down 2 40 310 123 21.5

Through this information, one can start to see the appeal of fouling. If the opponent doesn’t make a free throw, the fouling team has flipped the script and made itself the last-possession bully. Even if the opponent makes one free throw, the fouling team has improved its chances slightly. It’s only the case of making two where the fouling team’s chances of winning are harmed.

At this point, the alert reader realizes this analysis is missing a key factor that hurts the fouling math a bit - the offensive rebound. For both the tied and down-1 scenarios, we must investigate whether the offense or defense rebounds the missed free throw. I’ll get to that in the next installment, as well as some specific scenarios where this strategy could have been used.

**I’ve simply reversed the chances of winning when tied with 25-35 seconds left. Over the past five seasons, the defensive team actually has a slightly better winning percentage (35.8) when tied and the possession begins with between 20 and 25 seconds than if the possession began with 25 to 35 seconds (34.5). This is probably due to the amount of noise in the data, and therefore I’ve applied to latter figure to both cases.*