To Foul or Not To Foul?

To paraphrase Shakespeare:

To foul, or not to foul: that is the question:
Whether 'tis nobler in the mind to play defense,
or to foul thy opponent whilst not in the act of shooting.

Background

The rationale behind intentionally fouling your opponent when protecting a 3-point lead is simple.  By committing a non-shooting foul, you put your opponent on the foul line for at most 2 shots.  And no matter how good a free-throw shooter is, it's hard to score 3 points with only two shots.

ESPN college basketball analyst Jay Bilas¹ is adamant.  When you're leading by 3 points late in the game, don't intentionally foul your opponent if there are more than 2 seconds left². While there is a certain macho sentiment in saying that your defense can stop their offense, does it result in a higher probability of winning?  We consider two strategies for protecting a 3-point lead late in the game (let's say 5 seconds remaining):

1. Play straight-up defense.
2. Intentionally foul before your opponent can potentially attempt a 3-point shot.

We refer to these as the Bilas and non-Bilas strategies, respectively.

1. You used to be able to read his article here: http://proxy.espn.go.com/ncb/columns/story?columnist=bilas_jay&id=1709319.  But the link has since been removed.

2. His two second rule of thumb seems to be predicated on precluding the unlikely but devastating event of a made first foul shot, an offensive rebound after the intentionally missed second free-throw and a desperation three-point heave.  Under this scenario, you'd lose the game.  But then again, you'd potentially lose if you fouled a three-point shooter who made the shot.  Which is more likely?

Mathematical Modeling

We'll diagram the potential outcomes and determine the probabilities associated with each.  However, since the possible outcomes are numerous, we'll break them up and look at them separately.  Before proceeding, we'll have to make a few realistic assumptions about how the game can turn out:

1. Your team will not score again in regulation.
2. Your opponent will not initially attempt a 2-point field-goal when trailing by three.
3. Your opponent will not get more than one offensive rebound.
4. Your opponent is in the double-bonus (i.e., will shoot two free-throws on a non-shooting foul).
5. If your opponent trails by more than one point with one free-throw remaining, the free-throw will be missed intentionally.
6. If your opponent trails by less than three and gets an offensive rebound, he will attempt a tip-in and not a 3-pointer.
7. There's a 50% chance of winning in overtime.

Non-Bilas Strategy:

Based on the preceding assumptions, we can diagram possible outcomes to the game using a decision tree.  If you foul intentionally, there are four possible outcomes resulting in either a win in regulation or an overtime period.

This ignores Bilas's concern that your opponent will make a free-throw, rebound the intentional miss of the second free-throw, and hit a desperation 3-pointer.  While your opponent might employ this strategy, it lowers his probability of winning, so it's reasonable to assume that it's infrequent.  We'll denote the probability of winning using the non-Bilas (i.e., foul intentionally) strategy by p*.

Bilas Strategy:

Suppose, instead, you decide to play defense.  In this case, the number of potential outcomes is much larger.  (And remember, this is a simplified version of potential events.  Considering other behavior becomes incredibly complicated -- even when we're only looking at the last 5 seconds of regulation.)

Three of the terminal nodes result in wins in regulation, and two more result in overtime.  We're left with the two nodes in blue.  The one labeled FI (foul intentionally) is the same situation we'd be in if we employed the non-Bilas strategy, so the probability of winning here is simply p*.  The other blue node (2up+2FT) indicates you have a two-point lead with your opponent attempting two free-throws. It's obvious that this is a worse situation than fouling intentionally (where you'd be up three with your opponent shooting two).  So the probability of winning here is less than p*.

Comparing Strategies:

The probability of winning using the Bilas (straight-up defense) strategy is a weighted average of 1 (nodes resulting in wins), ½ (nodes resulting in overtime), p* and a number less than p*.  Since we're interested in comparing the probability of winning using the Bilas strategy to p*, we can bound the grayed portion of the tree by p* and restrict our attention to the red sub-tree.

After some algebra, we determine that the non-Bilas strategy is guaranteed to result in a higher probability of winning the game if:

1 + [2 Pr(Win in Regulation) / Pr(Overtime)] ≤ p*/(1-p*)    (EQN 1)

Pr(Win in Regulation) =  Pr(No Shot Attempt)
                       + Pr(Missed 3PT Shot) Pr(Def. Reb.)
                       + Pr(Missed 3PT Shot) Pr(Off. Reb.) Pr(Missed Desp. 3PT)

 Pr(Overtime Period) =   Pr(Made 3PT Shot)
                       + Pr(Missed 3PT Shot) Pr(Off. Reb.) Pr(Made Desp. 3PT)

                  p* =   Pr(Made 1st FT) {1 - 0.5[Pr(Off. Reb.) Pr(Tip-in | Off. Reb.)]}
                       + Pr(Missed 1st FT) {1 - 0.5[Pr(Off. Reb.) Pr(Desp. 3 | Off. Reb.)]}

In order to evaluate equation 1, we'll make some assumptions about various probabilities:

                  Pr(Made FT) = 0.75
    Pr(Off. Reb. | Missed FT) = 0.2
       Pr(Tip-in | Off. Reb.) = 0.9
Pr(Desperation 3 | Off. Reb.) = 0.2

These result in an estimate of p* = 0.9275 and p*/(1-p*) = 12.8.  Turning to the other ratio, we'll assume:

          Pr(No Shot Attempt) = 0.1
         Pr(3PT | Shot Taken) = 0.25
   Pr(Off. Reb. | Missed 3PT) = 0.2
Pr(Desperation 3 | Off. Reb.) = 0.05

So Pr(Win in Regulation) = 0.75, Pr(OT) = 0.25 and the left hand side of equation 1 is equal to 7.  Therefore, equation 1 is satisfied (7 ≤ 12.8) and the non-Bilas (foul immediately) strategy is preferable to the Bilas strategy (playing defense).

A few words about assumptions are in order.  We've made a bunch of them.  However, an approximate answer (in this case) is more useful than an exact answer in terms of unknown probabilities that you can't use to make a decision.  So the question is how much these assumptions influence the final answer.  Fortunately, in this case, not much.  First of all, consider that we bounded the probability of winning for the Bilas strategy.  So equation 1 is a best case scenario for the "play solid defense" approach.  If the inequality is satisfied, then we can feel certain that fouling is the better choice.

Furthermore, the assumed probabilities purposely favored the Bilas strategy.  For example, we ignored the potential for an opponent to make a 3-point basket and get fouled.  In addition, we assumed that the probability of your opponent successfully hitting a desperation 3-pointer after a missed free-throw was 20% (probably too high) while we assumed that same probability was 5% after a missed 3-pointer (probably too low).  When these assumptions are taken into account playing defense looks even less appealing.  Again, by implicitly favoring the Bilas strategy with our assumptions, we're confident that if the other option still looks better (and it does), then it's the choice to make.

Last, a word about the particular numbers.  Consider the probability of winning if you foul immediately, p*.  This value is relatively unaffected by the particular choices of probabilities we made.  For any reasonable values, p* ≥ 92% and p*/(1-p*) ≥ 11.  Similarly, altering the values of the second set of assumptions (used in computing the left hand side of equation 1) consistently results in a ratio of less than 8.  Therefore for any reasonable values of these probabilities, intentionally fouling your opponent increases your chances of winning.

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Last modified: 01/05/2008