Racial bias and NBA referees -- Part 2
This is part 2 of my comments on the NBA racial discrimination study. Part 1 is here. The NYT article on the study is here.
The study found a difference in the rates of how white and black referees call fouls against white and black players – notably, white referees call fewer fouls than expected against white players.
From this, the authors of the study conclude that, because most of the referees in the league are white, it would be advantageous to hire more white players and fewer black players. I don’t think that's correct, for several reasons, and I will explain why.
I'll start by describing the paper's approach. There were many different tests and regressions, but the first one, and the simplest one, is not that different from the more complicated ones. So I'll use that one.
This is the first chart in Table 3, which is below. The numbers are fouls called per 48 minutes, by race of the (majority of the three) referees and the perpetrator:
black players white players majority white refs 4.33 4.95 majority black refs 4.33 5.02
You'll notice that where black players are concerned, the race of the referees didn't matter: fouls were called at almost exactly the same rate by both races. ("Almost" because the numbers are rounded.)
But white players were caught much less often by white referees than by black referees, 4.95 fouls to 5.02 fouls, a difference of .07 fouls every 48 minutes.
And so, you can make this argument: in the left column, which represents black players, the top number is exactly the same as the bottom number. If white referees were as consistent for white players as the were for black players, the second column would look the same, and the top number there would also be equal to the bottom number. The chart would then look like this:
black players white players majority white refs 4.33 4.95 5.02 majority black refs 4.33 5.02
This new chart represents how you expect it "should" be in the absence of bias. The top-right cell (changed, in red) represents white referees making calls on white players. So translating the change of chart into English, you could say:
1. "White referees didn't call as many fouls on white players as they should have."
The "1." in front of the statement is because, as you've probably figured out, there's more than one way to adjust the chart. Instead of changing the top-right cell, we could change the bottom-right cell, so that the black ref number becomes equal to the white ref number:
black players white players majority white refs 4.33 4.95 majority black refs 4.33 5.02 4.95
This is just as consistent as the first way. The interpretation here is:
2. "Black referees called more fouls on white players than they should have."
These two statements, number one and number two, are equally consistent with the data. There is no way to determine which one is true. Indeed, they could both be true: maybe white refs were too lenient on white players, AND black refs were too hard on white players. Or it's possible that all refs are more lenient with white players. Maybe the "real" number should have been even higher than 5.02. Maybe blacks let lots of fouls go, but whites let even more fouls go! Or maybe both refs were tougher with white players than black players – but white refs were not as much more tough as black refs were more tough.
We can reduce that mess to one principle, which we can say at least three different ways:
(a) white referees were more lenient towards white players than black referees were;
(b) black referees were tougher on white players than white referees were;
(c) regardless of whether black referees were too tough or too lenient on white players, the white referees were either less tough or more lenient.
The statements are three different ways of saying the same thing. The first way kind of implies that it's the white referees who are biased. The second way implies that it's the black referees who are biased. And the third way implies that maybe all the referees might have been biased.
Any of those is possible. There's evidence of bias, but absolutely no evidence of who is biased.
But, wait, we're not finished: there are still other interpretations of the chart. This time, start with the right column. White refs gave white players 0.07 fewer fouls than black refs. If white refs are more 0.07 more lenient towards white players, they should also be 0.07 more lenient towards black players. That is, the left column should look the same as the right column! And so, instead of giving them 4.33 fouls per game, they should have given them only 4.26:
black players white players majority white refs 4.33 4.26 4.95 majority black refs 4.33 5.02
This gives rise to a third interpretation:
3. "White referees called more fouls on black players than they should have."
And there's one more interpretation, which happens when you change the bottom-left cell:
black players white players majority white refs 4.33 4.95 majority black refs 4.33 4.40 5.02
4. "Black referees called fewer fouls on black players than they should have."
(Digression: to some, interpretations 3 and 4 are perhaps less intuitive than 1 and 2. In the first two charts, black referees and white referees treated black players exactly the same. This satisfies our wants and expectations for NBA officiating, that the race of the referee should make no difference. It seems like those numbers somehow must be right, and making them unequal is the wrong thing to do.
But I think that prior expectation is unrealistic. Why shouldn't there be differences in refereeing style between whites and blacks? There are differences in basketball style. There are differences in culture, in politics, and many other areas. That's one of the reasons employers like "diversity" – because people of different races can bring different attitudes and methods to the business. So why shouldn't referees be a bit different by race?
Sure, the NBA tries to enforce consistency, so you wouldn't expect any large differences. But isn't it acknowledged that some referees call a tighter game, and some call a more lenient game? Perhaps the division isn't exactly 50-50 between white and black.
And, indeed, other data show a difference: in one of their subsequent breakdowns, it was found that black referees were slightly more strict than whites even in fouls called against blacks.
Anyway, I include this digression because the authors of the study don’t really consider the possibility of non-bias-related differences between the two groups of referees. But if you do consider such differences, you can come up with explanations for the data other than racial bias (which I'm saving for part 3). The authors didn't do that, and neither did most of the opinions on the subject that I've read.)
So what can we conclude, not knowing which of the four possibilities is correct?
First, that there's some bias going on. But does it favor white players or black players? And is it by white referees or black referees?
Under 1 and 2, it's bias for and against white players; under 3 and 4, it's bias against and for black players. Under 1 and 3, it's bias by white referees. Under 2 and 4, it's bias by black referees.
So we don't know who benefits, and whose 'fault' it is.
What we do know is that, under any of the four possible interpretations,
-- black players benefit relative to white players when you add more black referees.
-- white players benefit relative to black players when you add more white referees.
And that's why, in the study, the authors repeatedly refer to "own-race bias" instead of "white bias" or "black bias." Every player benefits when more referees are of his own race.
(And one quick point: there is a fifth interpretation, which is that some combination of all four of the others are right! Maybe black refs both favor blacks and stiff whites. Maybe white refs both favor whites and stiff blacks. And when you put them all in a chart, you get what we got.)
What is the magnitude of the "own-race" bias? You'll notice that in every one of the four cases, we had to change one of the cells by exactly the same amount: 0.07. (It always has to work out that way, that no matter which of the four cells you change, it will have to change by the exact same amount.)
If you can change the race of a single one of your (48-minute) players, so that it now matches the majority of the referees that game, you should reduce your fouls by 0.07 (or about one foul per 15 such games). If, somehow, you could change all five players, it would be one foul every three games – or about one point per game. That's a lot. But it's hard to have two sets of five players who are exactly equal in every respect but race, which renders this strategy fairly unworkable.
Can you gain an advantage by switching race for an entire season, instead of by the game? I will soon argue that you cannot. But it sure seems like you can. Here's the logic:
Since two-thirds of NBA referees are white, most games (in theory, about 74%) would have a majority of white referees. So if you replace a black player with a white player all season, he'll have the advantage in 74% of games, and a disadvantage in 26% of games. 74% minus 26% is about half, so the advantage of that is about 0.035 fouls per game – one foul in 30 games. If you assume that the victim benefit equals the perpetrator benefit, double that to one foul in 15 games. That's about six fouls over a season. What's that – maybe a half a win or so?
The New York Times article recaps the issue like this:
Their results suggested that for each additional black starter a team had, relative to its opponent, a team’s chance of winning would decline from a theoretical 50 percent to 49 percent and so on, a concept mirrored by the game evidence: the team with the greater share of playing time by black players during those 13 years won 48.6 percent of games …
But I don't think that's right. The overall tendency of players to avoid and draw fouls will be accounted for in their statistics and evaluations. And so if teams consider fouls for and against when making personnel decisions, black teams should be exactly as likely to win as white teams. The GM may not know why player W takes fewer fouls than player B, but he doesn’t care – he'll hire player W even without knowing.
Teams are rational. If you had two players, one white and one black, equal in talent (not including the effects of racial bias), the white player would have better stats than the black player – because he plays against white refs more often! Therefore, he'd be contributing more to the team, and he would already have the job.
That is: even though the racial bias is invisible, the results are not. And teams would already have made the optimal player moves to correct for it, just as they correct for every other player characteristic.
That's in theory: in practice, that might not be entirely correct. It's possible that some of the effect has gone unnoticed – most likely, a player's ability to induce fouls on the other team. But if that's the case, then the problem is that teams aren't measuring it. The standard economic assumption is that teams are rational, and would notice it. As an empirical question, my gut says that teams would notice a fair bit of it, maybe most of it.
Look at it this way. Suppose that 25 years ago, a racist demon started giving every white baby an extra dollop of basketball talent. Nobody knew that until today, when a study came out proving it. Does that mean that teams should immediately rush out and get white players? Of course not. The white players' extra talent was obvious, even if the cause wasn't, and all the demon-gifted great young white players were already snapped up and are now playing.
Similarly, the effects of race-biased refereeing have already been factored into which players have jobs today. Which race got the extra jobs, blacks or whites? We don’t know, because we don’t know which way the bias goes. If the hidden truth is that referees are biased in white players' favor, then some white players have jobs they wouldn’t have otherwise. If the bias is in black players' favor, then the black players have the extra jobs. Just as you would expect.
But there's no way to tell which way it goes. The study provides insufficient evidence to tell whether the bias favors white players or black players.
In Part 3, I'll argue that the data can be explained by other theories than racism. One such possibility – and I think it's a good one – is discussed by Guy here, and by Andrew and Guy in the comments to this post.