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.)

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.

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.)

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.

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.)

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.

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.

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.

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.)

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.)

]]>You live in a world with 1000 other people, only one of whom can beat you at chess. You can beat the other 999. This gives you great prestige, because this is a world where chess skill is exalted above all else.

Now your chess skills atrophy, and all of a sudden you find that 100 people can beat you at chess; you can beat the other 900. You’ve lost some prestige.

I want to quantify the fraction of your prestige that’s gone missing. Of course the answer could be anything at all depending on how you choose to quantify “prestige”, but I’m looking for a definition that most people will agree captures their intuitions (or at least doesn’t grate too harshly against their intuitions).

**Attempt One:** If N people can beat you, then your prestige is measured by 1/N. Therefore your prestige has fallen from 1/1=1 to 1/100 = .01. You’ve lost 99% of your prestige.

**Attempt Two:** Your prestige is measured by the number of people you can beat. Therefore your prestige has fallen from 999 to 900. You’ve lost just under 10% of your prestige.

Which of these seems more “right” to you? And do you have an “Attempt Three” that seems even better?

In a few days, I’ll tell you why I asked.

From Frank Harris‘s first-person account of the Great Chicago Fire:

]]>By the early morning the fire had destroyed over a mile deep of the town and was raging with unimaginable fury. I went down on the lake shore just before daybreak. The scene was one of indescribable magnificence: there were probably a hundred and fifty thousand homeless men, women and children grouped along the lake shore. Behind us roared the fire: it spread like a red sheet right up to the zenith above our heads, and from there was borne over the sky in front of us by long streamers of fire like rockets; vessels four hundred yards out in the bay were burning fiercely, and we were, so to speak, roofed and walled with flame. The danger and uproar were indeed terrifying and the heat, even in this October night, almost unbearable.

I wandered along the lake shore, noting the kind way in which the men took care of the women and the children. Nearly every man was able to erect some sort of shelter for his wife and babies, and everyone was willing to help his neighbor. While working at one shelter for a little while, I said to the man I wished I could get a drink.

“You can get one”, he said, “right there”, and he pointed to a sort of makeshift shanty on the beach. I went over and found that a publican had managed to get four barrels down on the beach and had rigged up some sort of low tent above them; on one of the barrels he had nailed a shingle, and painted on it were the words, “What do you think of our hell? No drinks less than a dollar!” The wild humor of the thing amused me infinitely and the man certainly did a roaring trade.

I’m a little surprised that this, from one of my all-time favorite bands, hasn’t been getting more airplay lately:

Every day, a man comes to my door with a United States nickel in his hand. He asks me whether I’d prefer to examine the heads side (which is always painted either black or white) or the tails side (which is always painted either red or green). I choose each day according to my whims.

And the same thing happens to my sister. Different man, different coin, but each day he’s there with a painted nickel, offering to let her examine either the heads side or the tails side.

Sometimes we call each other to compare notes on the colors we’ve seen. Here’s what we’ve concluded:

- Our heads sides are never both white.
- Whenever one of our tail sides is green, the other one’s heads side is white.

We have thousands of observations to support these conclusions: On days when we both examine our heads sides, we never both see white. On days when we examine opposite sides and one sees a green tail, the other always sees a white head.

**The Brain Teaser**: Today we both chose Tails and both saw green. What colors were on our Heads sides?

**Solution**: By point 2) above, they were both white. But by point 1) above, that can’t happen. So….?

So what now?

**Possible Resolution I.** There are no such coins. You’re right. There are no such coins. But there are subatomic particles that behave exactly like these coins. It’s easy to set up an experiment where every day, my sister and I each receive an electron. We can examine the spin of our electrons in either the up/down direction or the left/right direction. And here’s what we find:

- Our electrons are never both spin down.
- Whenever one of our electrons is spin right, the other is spin down.
- There are days when both of our electrons are spin right.

By points 1) and 2) and the same simple logic we used for the coins, there can never be a day when both electrons are spin right. Nevertheless there are such days. In fact, on those days when we both choose to make left/right measurements, our electrons are both spin right about 8.3% of the time. That’s not a huge percentage, but it’s sure not zero either.

So Possible Resolution I doesn’t work, at least if we replace the coins with electrons.

**Possible Resolution II:** The coins are very very sneaky and they like to screw around with our minds, so they change their own colors depending on the choices we make, just to fool us. Sometimes they **are** both white on the heads side, but not on the days when we’ve both decided to **check** the heads side — so that Rule 1 is false but we get fooled into believing it.

Aside from its inherent implausibility (intelligent coins? intelligent coins with nothing better to do than to mess with our heads?), this resolution falters on the fact that **my** coin has no way of knowing what choice my **sister** is making.

**Possible Resolution III:** Neither side of either coin **has** a color until we decide to examine it, so that on a day when I examine my tails side, it **makes no sense** to ask about the color of the heads side in the first place.

Therefore I am **not allowed** to pose this brain teaser in the first place.

If that strikes you as implausible, I invite you to devise another Possible Resolution. Plenty of people have done so. None, though, has ever devised a Possible Resolution more plausible than Possible Resolution III.

Welcome to quantum entanglement.

If you liked this example, you’ll love Chapters 14 and 15 of The Big Questions.

**Technical Appendix**: For the cognoscenti, I wrote R=B+W, G=B-W, and assumed the coins are prepared in the initial state BB+BW+WB. Rule 1 follows. You can check that this state is equal to both 2BR+WR+WG and to 2RB+RW+GW. Rule 2 follows. You can check that this state is also equal to 3RR+RG+GR-GG, so that when both parties make R/G observations, GG comes up 1/12 (about 8.3%) of the time.

Should oxycontin be legal? Here’s what the back of my envelope says:

In the U.S., there are about 50 million prescriptions a year for oxycontin, most of them legitimate and for the purpose of alleviating severe pain. I’m going to take a stab in the dark and guess that the average prescription is for a two-week supply.

There are also (at least if you believe what’s on the Internet) about 20,000 deaths a year in the U.S. related to oxycontin abuse. If we value a life at $10,000,000 (which is a standard estimate based on observed willingness-to-pay for life-preserving safety measures), that’s a cost of 200 billion dollars a year, or $4000 per prescription.

If those were all the costs and benefits, the conclusion would be that oxycontin should be legal if (and only if) the average American is willing to pay $4000 to avoid two weeks of severe pain. I’m guessing that might be true in some cases (particularly when the pain is excruciating) but not on average. So by that (incomplete) reckoning, oxycontin should either be off the market entirely or regulated in some entirely new way that will dramatically reduce those overdose deaths.

But of course what this overlooks on the benefit side is all the “abusers” whose lives have been enriched by oxycontin. This includes the vast majority who use and live to tell the tale, and also some of the OD’ers, for whom a few years of oxycontin highs might well have been preferable to a longer lifetime with no highs at all. Relatedly, what this overlooks on the cost side is that the average “abuser” is likely to value his life at considerably less than the typical $10 million — as evidenced by the fact that he’s electing to take these risks in the first place. Also relatedly, it overlooks the likelihood that many of those who overdose on oxycontin would, in its absence, be killing themselves some other way.

If the back of your envelope is larger than mine and you make those corrections, I’m reasonably confident that your bottom line will come out pro-oxycontin. (Please share that bottom line!) I am however, mildly surprised (and — both as a blogger who prefers slam-dunk arguments and as a libertarian who prefers to come down on the side of freedom — mildly disappointed) that the first quick-and-dirty calculation comes out the other way.

]]>Jamie Whyte, whose has been at various times an academic philosopher (and winner of the *Analysis* prize for the best paper by a philosopher under 30), a consultant to the banking industry with Oliver Wyman, a foreign currency trader, the leader of New Zealand’s ACT political party, the research director at the Institute for Economic Affairs, the author of several books that every thinking person should read, a frequent contributor to the European edition of the Wall Street Journal and other publications of that ilk, the incoming editor of Standpoint Magazine, an occasional guest poster on this very blog — and the deliverer of one of the most thought-provoking and entertaining lectures I’ve ever heard when he visited Rochester a few years back — will be here again next week, with two events open to the general public. They are:

Is There Too Much Social Mobility?, Tuesday April 9 at 5:30 in Goergen 101 at the University of Rochester

and

How to Make the Case for Liberty, Wednesday April 10 at 3:30 in Schlegel 102 at the University of Rochester.

If you’re in the vicinity, I hope you’ll stop in.

]]>My return trip from Lubbock to Rochester took almost 36 hours, due to maintenance issues on three separate aircraft. This leads me to wonder whether American Airlines is erring too far in the direction of safety and too little in the direction of getting people where they want to go — perhaps even recklessly so.

Here’s what the back of my envelope shows:

First, a standard ballpark figure for the value of a life is about ten million dollars. What this means is that empirically, people are willing to pay about $1 to avoid a one-in-ten-million chance of death, about $2 to avoid a one-in-five-million chance of death, about $10 to avoid a one-in-one-million chance of death, and so on for various other small probabilities. (Theory tells us that willingness-to-pay to avoid a probability of death should be some constant times that probability, as long as the probabilities are small. Data tell us that the constant is somewhere around ten million dollars.)

Next, I’ve noticed that when airplanes are overbooked and people are offered $600 to give up their seats, there are very few takers. That suggests that for most fliers, getting to their destinations on time is worth more than $600.

Next, 10 million divided by 600 is a little under 17,000. The value-of-life calculation suggests, then, that people are willing to pay no more than about $600 to avoid a one-in-17,000 chance of death.

(The above is arguably iffy, since you might argue that 1 in 17,000 does not count as a small probability. With a little effort, I’m sure one could dig up some data indicating the size of any necessary correction.)

Put that together, and you’ll find that a flight should be canceled only when it has a greater than one-in-17,000 chance of crashing. If the risk is any lower than that, then people would rather take that risk than lose $600, and would rather lose $600 than wait for the next flight — so we can infer that they would rather take the risk than be delayed.

The actual risk of crashing in a U.S. domestic flight seems to be somewhere around one in seven million. Seven million divided by 17,000 is a little over 400. That means that a flight should be canceled only when the plane appears to be 1/400 as safe (i.e. 400 times more likely to crash) as the average plane in the sky.

(I’ve assumed here that all flights are fatal to all passengers, whereas in fact it seems that well over half the passengers survive an average plane crash. But I’m going to bias this calculation in American’s favor by assuming that surviving a plane crash is roughly as bad as dying in one.)

I am happy to believe that some planes in some circumstances are more than 400 times as dangerous as the average plane. I am skeptical that three planes, all of which I was booked on over a single two-day period, all met that criterion.

If I’m right about that, then American Airlines is way too safety-conscious. And I missed teaching my classes for no good reason.

(Click picture for more info.)

If you get accepted to college because you faked being a sports star, pretty much everyone is outraged. I get that.

If you get accepted at college because you **are** a sports star, almost nobody seems to mind. That’s what I don’t get.

Either way, you’ve climbed the ladder by prevailing in a largely meaningless zero-sum (and hence socially useless) game, thereby signalling a dollop of narcissism together with a few mostly irrelevant talents or advantages. What’s the difference?