**I. The Issue**

It’s often claimed (at least by politicians, journalists and their ilk) that affirmative action tends to “stigmatize” succcessful members of the favored group, in the sense that a Harvard professorship is less prestigious when it’s held by someone who might not have made it to Harvard without an affirmative action boost.

It’s approximately equally often claimed that this is effect is too small to worry about.

I’m not aware of anyone on either side of this argument having attempted to settle the question with arithmetic.

In their defense, the question **can’t** be settled by arithmetic, because it’s pretty hard to quantify the difference in “prestige” between a professorship that reveals you’re likely to be in the top one-one-hundredth of one percent of the population and a professorship that only reveals you’re likely to be in the top two-one-hundredths of one percent of the population. But we can at least give some answers contingent on different assumptions about this issue. (And contingent also, of course, on various modeling assumptions.)

**II. A Model**

To that end, here is a primitive first-pass model:

- There is a prestige-conferring quality called “talent”.
- All of the most talented people are funneled into one occupation, say “professor”. This means, in particular, that if you and I are identical except for the fact that you’re better at basketball, then I am (by definition) more talented than you, since you are more likely to become a basketball star and hence less likely to become a professor. Nevertheless, talent confers prestige.
- Talent is distributed normally (or lognormally; it doesn’t matter because we’re going to fix the number of professorships, so only relative rankings matter).
- The population divides into two groups. Let’s call them the Eloi and the Morlocks. Each has its own separate normal distribution of talent.
- Although the Eloi constitute only half the population, they hold more than half the professorships. This is because there are more Eloi in the upper tail of the talent distribution than there are Morlocks.
- A person’s prestige depends on that person’s expected percentile talent rank among the entire population, conditional on two observed characteristics: Is this person an Eloi? and Is this person a professor? We’ll call this statistic the person’s
**expected percentile**for short.

Now we have a modeling decision to make: Why are there more Eloi at the top of the population-wide distribution than there are Morlocks? The two choices are:

**The Sigma Model**: The Eloi and the Morlocks are equally talented on average, but the Eloi have a higher variance.

**The Mu Model**: The Eloi and the Morlocks have the same variance in talent, but the Eloi have a higher mean.

It’s going to turn out (at least mildly surprisingly, I think) that the choice between these two models makes almost no difference.

**III. Introducing Affirmative Action**

Now we introduce an affirmative action program, requiring half of all professors to be Morlocks. This lowers the talent threshhold for a Morlock to become a professor (and raises the talent threshhold for an Eloi). Therefore the expected percentile of a Morlock professor falls, and so does the prestige of being a Morlock professor.

**Question 1:** **By how much** does affirmative action lower the **expected percentile** of a Morlock professor?

**Question 2:** **By how much** does affirmative action lower the **prestige** of a Morlock professor?

Question 1 is amenable to calculation. In fact, it’s a straightforward exercise (assignable to strong undergraduates!) to compute the answer to Question 1 in terms of three parameters:

- The fraction of Eloi (or Morlocks) in the population. Call this p. We fix the value of p (for now) at 50%.
- The fraction of the population who are professors. Call this variable t.
- The fraction of professors who are Morlocks. Call this variable r. We have assumed (see
**5)**above) that r < p.

(See here for some hints toward completing these calculations.)

**IV. Prestige Models**

Question 2, however, can’t be answered until we supplement our model with some assumption about exactly how prestige depends on expected percentile.

I find it easier to think about this question in terms of a variable I’ll call A, which is one **minus** the expected percentile. That is, if A equals, say, .01, then the average Morlock professor is in the top 1% of the population. If A equals .0001, then the average Morlock professor is in the top 1/100 of 1% of the population.

How does prestige depend on A? Here are two simple models:

**Prestige Model 1:** Prestige is measured by 1/A. For example, being in the top 1/100 of a percent of the population is 100 times more prestigious than being in the top 1 percent.

**Prestige Model 2:** Prestige is measured by 1-A. For example, being in the top 1/100 of one percent of the population is about 1% more prestigious than being in the top 1%.

Model 1 fits my intuition better than Model 2, but if your intuition begs to differ, I can’t prove you wrong.

It turns out (not surprisingly) that the choice of Prestige Model makes a **huge** difference for Question 2.

**V. Some Results**

If we go with Prestige Model 1, here are some graphs that answer Question 2:

The labels on the horizontal axis correspond to various assumptions about the fraction of professors who are Morlocks in the pre-affirmative-action world. (We called this r.) The lower this number, the more dramatic will be the effects of affirmative action.

Each group of four bars represents the prestige loss to Morlock professors, on various assumptions (namely t=.00001, t=.0001, t=.001, t=.01) about the fraction of the population that are professors (and hence the amount of prestige that attaches to a professorship in the first place). The two graphs show results according to the Sigma Model and the Mu Model as described above; as you can see, the choice between these two models is pretty much inconsequential.

The main takeaway from these graphs is that for a Morlock professor, the prestige cost of affirmative action program is pretty enormous. When the Morlocks start out under-represented by half (so holding only 1/4 of the professorships instead of half), affirmative action cuts the prestige of a Morlock professor by nearly half. (Perhaps interestingly, this means that the total prestige of all Morlock professors stays about constant, since there are now twice as many of them.) Even if the Morlocks start out with 45% of the professorships (so that the affirmative action program is quite small), Morlock professors still lose about 10% of their prestige.

(This seemed surprising at first, but much less so in retrospect — if you’re roughly doubling the number of Morlock professors, you’re going to, on average, roughly double the value of A and hence roughly halve the prestige of being a Morlock professor. In the absence of some calculation, that’s not a complete argument, but it does indicate why the outcome makes sense.)

**VI. A Different Prestige Model**

On the other hand, if we adopt Prestige Model 2 (where being in the top x% of the population gives you a prestige proportional to (100-x)), the prestige cost of affirmative action is near zero. For an illustrative example, take r=.25 and t=.001 (so that Morlocks hold 1/4 of the professorships initially, and you’ve got to be in the top 1/1000 of the population to be a professor). Then in the sigma model, the average Morlock professor is in the 99.9452 percentile before affirmative action and in the 99.8829 percentile afterward. On Prestige Model 1, that’s a prestige loss of about 53%. On Prestige Model 2, it’s a loss of about .06%. (The mu model gives very similar numbers.) So really, everything depends on how you measure prestige.

In an earlier thread, Jonathan Kariv observed that we might be able to hone in empirically on a prestige model by investigating the distribution of, say, Instagram hits among top celebrities. This seems to me like a good idea.

**VII. Minority Groups**

We’ve assumed that the Eloi and the Morlocks are equinumerous. What if we assume instead that the Morlocks are a minority, composing, say, 20% of the population? Here are some results for the Mu Model:

Here the horizontal labels correspond to the fraction of professors who are Morlocks (always under 20%) and we contemplate an affirmative action program that requires 20% of professors to be Morlocks. As before, the four bars in each grouping correspond to fractions of professors in the population of .00001, .0001, .001 and .01. And we get hits to prestige (measured by Prestige Model 1) extremely similar to what we get in the earlier case, where Morlocks are half the population. Conclusion: On this model, the “prestige cost” of affirmative action depends almost not at all on either the fraction of professors in the population (at least as long as it’s under about .01) and almost not at all on the fraction of Morlocks in the population.

**IX. A Caveat**

I’ve computed Prestige(Expected Percentile). It might have been better to choose a prestige model and then compute Expected Prestige(Percentile), which is not the same thing. I think the current choice is defensible, and I prefer it at the moment partly (but not exclusively) because it defers the choice of a prestige model to the last possible moment.

**IX. Concluding Words**

I started this exercise with an essentially flat prior over what might turn up. (Translation: I had no clue what to expect.) I got a lot of surprises.

**Surprise #1.**It makes essentially no difference whether the under-represented Morlocks are 50% of the population or 20% of the population.

**Surprise #2.**It makes essentially no difference whether the under-representation is caused by cross-group differences in **average** talent or in **variance** of talent (i.e. the Sigma Model and the Mu Model give essentially the same results).

**Surprise #3.**At least above a certain threshhold, it makes essentially no difference how prestigious professorships are to begin with; varying t (the fraction of the population qualified to serve as professors) between .01 and .00001 has almost no effect on the (percentage) prestige losses.

**Surprise #4.**At least on Prestige Model 1, which I tend to think is reasonable (but could be talked out of), the prestige cost of affirmative action is much larger than I’d have guessed.

**Non-Surprise #5.**On Prestige Model 2, there is almost no prestige cost to affirmative action. (This, I think, should have been obvious even before calculating anything.)

**X. What Next?**

What have I left out?

(Please note: There are dozens of arguments one could make for affirmative action, pro and con. This post is relevant to exactly one of those arguments. Please stay on topic.)

I’ve thought about this type of question before and the model I decided I like best for what you’re calling “prestige” would be phrased in your terms as “log((1-A)/A)”. I think without the log, this approximates your first metric at the top and your second metric at the bottom. I like the log form though because it makes both total prestige in the population and the median prestige 0, which feels intuitively right.

Your first prestige metric seems to drop too quickly at the top; your second metric too slowly. The log form of the combined metric drops somewhat in between. Or if you want a strictly positive metric, try using the log of your first metric (log(1/A)).

Anyway, something to think about.

The model choice question is actually brilliant, and I think reflects part of the divergence over whether affirmative action has a high or low cost in prestige.

Model 1 (high cost) is the intuition of an insider. If academia is your world, being at the top matters a great deal.

Model 2 (low cost) is the intuition of an outsider: it doesn’t matter if you’re a Harvard professor or a UMass professor, both sound nice, pay well enough and keep you in polite society. You could get similar low-cost results if Model 1 prestige was multidimensional and the losses were limited to one dimension.

Whether one cares a lot about in-group status or broader social status may determine how one feels about the prestige costs (and gains) of affirmative action.

This is going to take a bit of digesting.

Initial comments. Given the mu model, it is reasonable that there should be loss of prestige. It is talent that confers prestige, but the public cannot discern talent directly. Being a professor is an outward signal that you have talent. The signal has become less good at signalling talent, so prestige is reduced. By extension, it seems reasonable that the sigma model should produce almost the same result because the difference in size of the areas at the top of the distribution are the same. perhaps easy to say when the results are in.

Although, your counter-intuitive point at the start is interesting. I am less talented than you if I am just as good at literally everything you do, but I can do more things. This surely depends on what this prestige-conferring quality is. If we arbitrarily say it includes basketball, then I am more talented than you. We could amend the model by saying basketball stars get professorships too. Your model, with the example, specifically excludes basketball ability in the prestige-conferring quality. This suggests that basketball ability confers no prestige, which seems unreasonable.

Bu we can say we are only talking about prestige that is conferred by professorships, not basketball or anything else.

Part of the problem here is that the term “talent” provokes assumptions. We naturally assume that this quality you have called “talent” resembles in some way what we usually think of a “talent.”

On first glance my objection is that there is that there is a third model. Morlocks and Eloi are equally talented but Morlocks are discriminated against. That is, their talent as judged by the people awarding the chair as lower than their actual talent. This has the same effect as the mu model on the number of professorships as their perceived talent has a lower mean. However the signal is now under-representing Morlok’s talent, rather than under-representing it. You have decided not to include this in your model and I am sure your last point about staying on topic is to cut off such discussion. However I think it is important and can be on-topic also.

I have a solution. We re-name the prestige-conferring quality “looking good to people who appoint professors” rather than “talent.” We can then recognise that possessing this quality does not necessarily mean you will actually make a better professor, but it still fits the model just as well without leading the reader to believe that prestige is a measure of what we usually call talent.

#2 “Model 1 (high cost) is the intuition of an insider. If academia is your world, being at the top matters a great deal.”

But those actually in the discipline will be able to judge you by your actual output. Except at the start of your career, where perhaps it matters most.

Harold #3/4: Several good insights here. Thanks.

Curious … how much does affirmative action raise the prestige of the Eloi who still remain as professors?

Phil: The percentage gains to men run slightly lower than the percentage losses to women. For example, with r=.2 (i.e. the initial fraction of profs who are women is .2), women suffer about a 54% prestige loss and men earn about a 38% prestige gain. For higher r, the gap between womens’ loss and mens’ gain is smaller. At r=.45, there’s about a 9.8% loss to women and about a 9.2% gain to men.

Makes sense. Since the overall prestige total is negative, a man could benefit more than a woman lost only if there were more women than men. At 50/50, it’s not possible.

Under this model, where there is no information other than “man/woman” and “professor/not professor”, you’d expect the highest-ability male grad students to favor the affirmative action program, and the lower-ability male grad students to be against it. (Assuming existing professors are grandfathered in.)

Intuitively, I expected less gain than 38%. Interesting. Thank you!

Oops, the total change in prestige is zero, not negative. The men must gain less percentagewise than the women lose percentagewise because they are starting from a higher base, on average.

I’d suggest a third prestige model based on rank order: will affirmative action result in all the Morlocks underperforming all the Eloi?

This corresponds most closely to how people vview their status. We don’t go around thinking about our percentile, we think about who’s up and who’s down.

I am not sure that a mathematical model is that useful in this context.

I mean, it’s a given that an affirmative action preference will cause the professors from the preferred group to be from a lower percentile of “talent” than the non-preferred group.

The “prestige” difference generated by this percentile gap in “talent” thus depends entirely on the “prestige” value that you arbitrarily assign to the gap. Is it a big difference, or a small one? The output of the model merely returns the same definition of “prestige” that you feed into it.

And in the real world “prestige” is obviously subjective and context dependent, anyway. It would seem that prestige is something that, by its nature, must be empirically measured rather than mathematically modeled.

#7 When did the Eloi and Morlocks become men and women? I thought we were talking about height!

I am not sure what the gains and losses mean. Before affirmative action all profs have equal prestige. What if we tweak the model a bit and say that Eloi and Morlocks are exactly the same and have exactly 50% representation. We then introduce affirmative action so that Morlocks get 60% representation. Morlock professors will lose prestige and the Eloi professors that remain will gain prestige, but how much of each?

I don’t know if this helps but I’ve heard, Anthropologically speaking, people care about local relative rankings. They don’t care about global, or universal rankings.

A female nurse is going to care whether a similarly skilled male colleague is getting equal wage as her, but she won’t care if some guy in finance is getting a higher wage than her.

Off-topic follow-up to Thursday Puzzle/Science Lesson:

Ever wonder what a quantum entanglement looks like?