Matthew Yglesias » Chart of the Day

Jun 16th, 2009 at 8:28 am

Chart of the Day

Here’s a fascinating histogram charting home team margin of victory in NBA games from 1997 to 2009. You see the predictable skew toward the home team, and also a big spike around zero:

What explains this? A star PhD student at Northwestern, Toomas Hinnosaar, and I have been thinking about this. Our focus in on the dynamics and strategy at the end of the game.

Of course another possibility would be that officiating is biased somehow toward the tie outcome.

Comments
32

Filed under: Basketball, NBA, Sports

32 Responses to “Chart of the Day”

Brent Says:
June 16th, 2009 at 8:39 am
I never gamble, but is there a way to place a bet on whether or not a game goes to OT? I would not have guess that is the most likely outcome at the end of regulation.
mercurino Says:
June 16th, 2009 at 8:49 am
this data can likely be explained by the rule that allows teams to inbound the ball at half-court after a time out at the end of games. the NBA specifically designed the rules like this to get as many last-second game-tying and -winning shots as possible. mission accomplished, Mr. Stern.
rapier Says:
June 16th, 2009 at 8:53 am
What is needed is an analysis of foul shots by home and visiting teams, and or perhaps just foul totals.

As a Bulls fan who lived in Michigan during the Bad Boy era, thus having a very long string of winners to cheer for, it is obvious that stars and winners get a better shake from officials. Home or away.

Which doesn’t segue into my favorite officiating story but I will anyway. In the first round of the 86 NCAA playoffs Ewing’s Georgetown played Loyola Chicago. A mismatch if there ever was one. Gtown looked and played like the pros in terms of size and physicality. Loyola was a bunch of mites who were good shooters. At halftime Loyola was only a few points back. In the second half as I recall but don’t hold me exactly to the numbers, Loyola was called for 22 more personal fouls than Gtown. While Loyla was playing a scattered aggressive swarming defense Gtown was laying the wood to the little Ramblers in brutal fashion. All in all it was the most absurd case of officiating bias I’d ever seen. Georgetown went on to win the tournament in a style that looked just like the Pistons of the era.
SP Says:
June 16th, 2009 at 8:54 am
Wait, there were a lot of games that ended with a “margin of victory” of 0? Are you talking about end of regulation, or are they still playing all those games?
tom veil Says:
June 16th, 2009 at 8:54 am
Don’t blame the ref. It’s rules and strategy that are biased towards ties. Specifically, the idea that you can FOUL your way back into contention is unique to basketball.
doofman Says:
June 16th, 2009 at 8:57 am
@tom veil: You, sir, have apparently never watched the Denver Broncos offensive line.
JPKK Says:
June 16th, 2009 at 8:58 am
Its actually real simple, games that are not heading for a tie (single possession difference) go to fouling, which while a higher chance of winning than letting the other team hold the ball while up is still a low probability to win and often results in a bigger margin of victory. I remember watching a lot of NCAA tournament games that were upsets that I picked that almost happened. In almost all cases they were 2 possession games with a minute left that ended up with 10+ point gaps at the end once fouling started.
Brian Weatherson Says:
June 16th, 2009 at 9:09 am
JPKK’s explanation sounds very plausible from casually watching the game, and the graph tends to bear it out. A refereeing bias towards ties wouldn’t produce the strikingly low number of 1 point wins and losses compared to, say, 4 point wins and losses. The fact that the fouling strategy decreases one’s expected winning margin, while increasing one’s win(/tie) probability probably explains the odd structure of the data fairly well.
Jim W Says:
June 16th, 2009 at 9:10 am
mercurino,

Your explanation doesn’t account for why there are so many more ties at the end of regulation than one point wins by one team or the other.

I think the advent of the 3-point shot helps explain why teams are more likely to either lose by 2 or 3 points, or tie the game, and less likely to lose by 1 point. It would be interesting to see the stats before the 3-point shot.

The unique (and annoying) NBA feature of coming from behind via fouling helps explain why 6-12 point victories are more likely than close games, as it usually allows the winning team to expand their margin late.
Casper Says:
June 16th, 2009 at 9:11 am
Strategy leads to more ties. Teams down by two try a game-tying two-pointer on the last possession if possible; teams down by three try a three-pointer. 30% of the time the game comes down to this last shot situation and gets in position for a decent shot, they make it. It’s all last-possession strategy. If anything, I think there is evidence that officiating tends to favor the trailing team late, but then officials tend not to call fouls on the final play. So I don’t think officials are the reason for non-random scoring distribution on the last possession.
Jim W Says:
June 16th, 2009 at 9:13 am
To be more explicit, I predict that similar statistics from before the 3-point shot would result in 1 point victories having a similar likelihood as 2 and 3 point victories, and would result in a smaller spike for tie games.
Domino Says:
June 16th, 2009 at 9:17 am
Brent,

Don’t be too quick to put your money on a tie. The data shows just 6% of games go to ties. This is higher than any other point margin, but you still have almost 95% of games decided before OT, according to this data.
SLC Says:
June 16th, 2009 at 9:23 am
Re rapier

To expand on Mr. rapiers’ comment on foul shots, what would be interesting would be a comparison of free throw percentages on home court vs away court. A finding that teams have a higher free throw percentage on their home court would explain, in part, the home court advantage.
Adam Villani Says:
June 16th, 2009 at 9:35 am
To be pedantic I should point out that the spike isn’t “around zero,” it’s precisely at zero.
Brent Says:
June 16th, 2009 at 9:42 am
Domino,

Sure, but if I could only place a bet on the outcome at the end of regulation, then I’m still almost twice as likely to win than with the next most probable outcome at the end of regulation.

Of course 94% is still a lousy number to be up against. But so is 97%, 98%, 99%, etc. I won’t be gambling in the first place. But if I did, and if it were possible to do so, I’d place a bet that the game will go to OT.
Dan Kervick Says:
June 16th, 2009 at 9:46 am
Once either team has achieved a tie or is within two points in the last 24 seconds of a game, the strategy for the team with possession is to hold the ball and play for the last shot. Coaches are very risk averse. The chief aim for the team with possession in a tie game in that last 24 seconds is to avoid fouling or turning the ball over and giving the other team a chance to win in regulation. The secondary aim is to win outright. The result is usually a pressured jumpshot with relatively low risk of miscue, but also a relatively low probability of making the shot. The most likely possible outcomes in these scenarios are that somebody wins by two, with the risk aversion skewing results away from a normal distribution toward a tied conclusion, and away from a two-point margins.

Short answer: tie scores in close games are sticky. Once ties are achieved, risk aversion tends to preserve them until overtime.
jimmy Says:
June 16th, 2009 at 9:53 am
You guys are kidding, right? Check out the home court advantage before versus during David Stern’s reign as commish. The NBA under Stern has become a more expensive & glamorous version of professional wrestling.
Dan Kervick Says:
June 16th, 2009 at 9:56 am
Of course another possibility would be that officiating is biased somehow toward the tie outcome.

Very possible. But this doesn’t seem completely plausible to me. Near the end of a game, what a ref probably wants most is to take a shower and go home.

On the other hand, the closer a game is, the more worried a ref becomes that the result might turn on his own, no doubt controversial, call. Perhaps that means the the closer the game, the stronger the motive for whistle-eating, and the more difficult it becomes for offensive teams to score. So maybe there is something to this. But it would only seem to explain why there is a departure from a normal distribution in favor of a higher-than expected peak of close games. It wouldn’t seem to explain why there are so few one-point games.
Jim W Says:
June 16th, 2009 at 10:31 am
“It wouldn’t seem to explain why there are so few one-point games.”

Right. Again, this can be explained by the 3-point shot. Consider a team with the ball at the end of the game. Assume, for the sake of argument, that it is equally likely a team is tied, down by 1, by 2, or by 3. Also assume there is a 50% chance of making a 2 pointer and 33.3% chance of making a 3 pointer. Finally, assume the outcome is decided without fouling (yes, I know, its a big assumption). Here are the outcomes with associated likelihoods:
down by 3: lose by 3 (.667); tie (.333)
down by 2: lose by 2 (.5); tie (.5)
down by 1: lose by 1 (.5); win by 1 (.5)
down by 0: tied (.5); win by 2 (.5)
Adding it all up, we get:
lose by 3 (.667)
lose by 2 (.5)
lose by 1 (.5)
tie (1.333)
win by 1 (.5)
win by 2 (.5)

Note that there are additional likelihoods I have not included in which a team is down by 4, 5, 6 with the last shot, which will add additional probability mass into the lose by 3 or 2 results, to the detriment of the lose by 1 result.

Before the advent of the 3-point shot, on the other hand, the lose by 3 team would have had a .5 chance of losing by 1, and a .5 chance of losing by 3.
If they’re down by 3, they will either lose by 3 or tie. If They are down by 2, they will either lose by 2 or tie. If they are tied, they will either win by 2 or tie. The only scenario in which they will win or lose by 1 is if they start in that situation.
Jim W Says:
June 16th, 2009 at 10:35 am
Oops. Ignore the last 3 sentences of my comment above. I forgot to edit them out.
mpowell Says:
June 16th, 2009 at 10:38 am
This has nothing to do with the refs. Look at the video at the link. The peak only starts showing up around 22 seconds. I would bet dollars to donuts that is a specific function of the shot clock. When teams take shots that they know will result in a final possession by their opponent, they pursue very specific strategies.

1) You rarely attempt a 3 to take a 1 point lead since this forces your opponent to take a high percentage shot to win the game. You would rather go for the 2 since it is higher percentage and your opponent would prefer overtime to looking for a high percentage shot that would give you the ball back.

2) You rarely attempt a 2 to get to within 1 since you will then be forced to foul, which is not favorable.

3) If you are down by 1 and the other team has possession, you will generally foul immediately.

I think the last factor is the strongest and subsequent post at the linked website verifies this.
zyxw Says:
June 16th, 2009 at 10:42 am
With little time remaining if a team is down by 3 they’ll always opt to take the 3-point shot and tie. If they’re down by 2 they’ll often take the 2-point shot as a tie is better than a missed shot. If they’re down by 1 the opposing team is very likely to foul in order to prevent the shot, so there is a good chance only one free throw will be made resulting in a tie. In other words, a tie is a better outcome than a loss when you’re down in the last few seconds, and there are a lot of good ways to get the tie. However, if you’re in the lead, a tie is not very desirable and depending on how the other team is shooting it is often better to let them shoot rather than foul them as not every team has LeBron or Kobie taking that final shot. Hence, a lot of 1, 2, or 3-point leads stand at the end of games because the final shot just misses.
zyxw Says:
June 16th, 2009 at 10:52 am
Shorter me: in close games the team with the lead is the most risk averse, therefore they play conservatively to at least get a tie from that point, while the team behind has a lot of incentive to fight really hard to tie rather than lose. So the incentives are all on the tie.
Njorl Says:
June 16th, 2009 at 10:57 am
Professional sports coaches usualy labor under the delusion that if a game goes to overtime, they will probably win.
cj_n_pa Says:
June 16th, 2009 at 11:50 am
How do you get a negative delta? The point difference should be a zero or a positive integer.
AKL Says:
June 16th, 2009 at 11:55 am
JPKK basically figured it out back at comment #7. In one-possession games with less than 24 seconds to go, the team that is behind usually fouls. They do this because it gives them a slightly better chance to win than not fouling, but this strategy makes it more likely that they will actually lose by a bigger margin.

The other side of this coin is that when the game is already tied, with less than 24 seconds, the team with the ball will usually hold for the last shot. They do this because they think it will give them a better chance to win, but this strategy makes it more likely that the game will end in a tie.

These two extremely common strategies, when combined, exert statistical pressure toward ties and away from 1 and 2 point margins.
Jim W Says:
June 16th, 2009 at 1:02 pm
AKL,

Yes, that’s fine, except that 2 point margins actually have a high likelihood in the histogram. It is only the 1-point margins that are unexpectedly low. As I stated above, I think a major reason for this is the existence of the 3-point shot.
Max424 Says:
June 16th, 2009 at 1:11 pm
@10 Casper “officials tend not to call fouls on the final play.”

I think it is understood around the league, officials will almost never call a foul on the final play in a tie game. Nobody, refs, players, coaches, fans or David Stern, wants a game to end at the free throw line. Even the superstars don’t expect to get a call on a last shot in a tie game.

I think this qualifies as officials being “biased toward the tie outcome.”

@18 Dan Kervick: “Once ties are achieved, risk aversion tends to preserve them until overtime.”

I think Dan hits on the other point. You see an awful lot of bad shots on the last possession of a tie game, simply because teams are primarily concerned with getting off the shot as the clock is about to strike zero.
Brian Says:
June 16th, 2009 at 1:19 pm
it’s an example of statistics giving us the bird
pseudonymous in nc Says:
June 16th, 2009 at 2:40 pm
In a way, a basketball game ought to be seen in reverse — start with the presumption of a buzzer-beater (to win or tie) and work from that. That’s what coaches consider at the end of games, since you don’t need to coach garbage time on a blowout, and that’s how you get the spike at zero.
Njorl Says:
June 16th, 2009 at 3:17 pm
I wonder if you did a fit to a gaussian, discarding the data from -10 to 10, if the magnitude of the zero spike would fill in the space of the deficits around it. If not, then that means that other scores, +12, -11 etc are contributing to the unusual likelihood of a 0 outcome. That would be weird.

An what’s with the big discontinuity between +13 and +14?
santamonicamr Says:
June 16th, 2009 at 6:29 pm
It seems to me that a rational team down 2 points in the last possession would go for a three point shot. Say the odds are (as suggested above) 50% for a 2 and 33% for a three. If you go to overtime, your odds of winning are likely to be around 50% (I mean, you *are* going for a tie here at the end of regulation, after all!) That means your chances of winning when shooting a 2 at the end are 25% (.5 x .5). Your chances of winning by shooting a 3 are 33%.

The team would have to figure its odds of winning in overtime are 67% to go for the 2 (.5 x .67 = .335) That seems a stretch.

So that’s the rational thing to do. Of course, it’s also rational to go for it on fourth and 3 on your opponents’ 40 yard line. Doesn’t seem to happen very much.

I conclude from this that most coaches are risk averse and the risk they worry about is being criticized for excessive risk taking.

Matt Yglesias

Chart of the Day

32 Responses to “Chart of the Day”

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