Critique and Reply---Perspectives on Rawlsian Justice

In this blog, I will reply to some of the comments I got on my previous blog: Justice 1---Why I disagree with Rawls?

The first critique and reply is addressed at the general audience, while the second is more technical and abstract (euphemism for boring for most people), in that it tries to connect the problem with the problem of statistical inference.

I sincerely thank my two great friends for their input, because you make me to think harder on the question. I recently saw the following statement on a blog:

I love to argue one on one, and common beliefs are not important for friendship — instead I value honesty and passion.

I think it is fitting to put it here. I still cling to my belief, and really appreciate that you are there to challenge me.

I will address two critiques:

Critique 1 

So would you say the Rawls argument would be favorable by someone who is extremely risk adverse? If the axioms of rationality doesn't hold then your argument wouldn't hold. the stock example that you use has such small loss that it makes the rationality argument seem more reasonable than it really is. what if it is like this- stock A: 0 all the time, B: 90% chance of 100, 10% chance of -800. or even: A: 0 all the time, B: 10% chance of 900, 90% chance of -9. the axioms of rationality surely doesn't describe how we actually make decisions. But I think it isn't even necessarily the way that we SHOULD make decisions, because in this case there is reason to be extremely risk averse, and that because of the human beings that we are, equity could be an intrinsic good such that the huge disparity in outcome would discount the expectation that you take. but the point you made about ex-ante and ex-post is very illuminating for me.

Reply:

Von-Neunmann Framework does accommodate risk aversion. Indeed, it was developed to accommodate risk aversion when people try to analyze the decision making of gambling, when they find the concept of expected value incapable to deal with risk aversion. The stock example you give can be analyzed in such a framework. Depending on our parameters, we will get different prescriptions, but I think this is a strength, in that is shows this framework is rich enough to allow different degrees of risk aversion. Of course, some degrees of risk aversion are reasonable, others are not, which could be revealed by the decisions people make. 

Over the years, this framework has been examined with different further restrictions---for example, utility should be bounded so that we will not run into some sort of St. Petersburg Paradox.  This is an example where our intuition of risk aversion help us refine this framework, in this case putting a lower bound on risk aversion.

As for the axioms of rationality. If you look at them, they are very very primitive---so primitive that it is shocking when we find empirical evidence that people could violate such principles when we make decisions---we just could not even pass such primitive requirements of rationality. But as I commented, those evidence point to the irrationality of human beings, not the logical coherence of this framework. It is a perfect model for how decision should be made, though a crappy model for decisions are actually made.

Critique 2:

I was trying to argue that the veil of ignorance extends to the distribution with which one takes the expected value to get expected utility. Analogizing to Bayes, when you don't have an actual prior, then you should do minimaxity and I think minimaxity will lead to Rawls's conclusion

When you say "aggregate welfare ex post" do you mean maximize over \theta given the observed distribution of types in the world? Say that I am of type i*, but this is unknown to me, so I assume that i* follows the observed distribution of types in the world (e.g. just tally how many rich/poor people there are in the world). I choose \theta to maximize my expected utility. Is this what you mean by maximizing aggregate welfare ex post? But I think Rawls is considering the following situation which is slightly different: I am of type i* (again, I don't get to observe this), but this time there is a veil of ignorance so that I do not get to observe the distribution of types (pretend someone deleted data on how many rich/poor are in the world). I can't do expected utility maxmization in this case. Depending on whether data on rich/poor is available, this could lead to drastically different outcomes. Suppose that I am told most people are rich in the world, then given this observed distribution, I'd choose \theta to benefit rich people, since most likely that will benefit me. But if I need to make a decision without knowing the relative number of rich and poor, I'd make sure that everyone is somewhat well off under my decision rule.

Reply

Despite its resemblance to minimax (in fact Wikipedia attributed an application of minimax to Rawlsian idea), I will argue the resemblance is only superficial, and the logic does not carry through.

So I will formalize the problem of inference and justice, in such a way to make the structure parallel.

Inference:
We have a type $i|\alpha\sim F_\alpha$. That is $i$ is the unknown parameter, and its distribution comes from a family of distribution (priors) with parameter $\alpha$. But we do not know which prior it is, that is we do not know $\alpha$.

We are looking for a rule $\theta$ according to which we can do inference. Every time we act according to the rule, we incur some loss, and this is captured by the perfomance function $U(\cdot)$ (negative of loss). Since the performance of the rule also depends on $i$ itself, $U$ is a function of not only the rule $\theta$, but also the type $i$.

As remarked, $i$ is a random variable, how $\theta$ performs in expectation depends on $\alpha$, that is the number we are concerned about is
\[
E[U(\theta,i)|\alpha]
\]
We would like to maximize that number, but we do not know $\alpha$. So we choose the rule such that it will gives us the best result in its worst case scenario.  In essence we are applying the minimax principle to $\alpha$ the parameter that determines the distribution, not to $i$, the type that we are interested in.

I have taken some time to set up the problem of statistical inference slowly, because I do not need to do this for social choice again. We are facing the very same problem, maximizing the following
\[
E[U(\theta,i)|\alpha]
\]
We can apply the minimax principle to $\alpha$, but not to $i$. What Rawls argue is that, we need to apply minimax principle to the type $i$, to maximize the welfare of the worst of individual, not to $\alpha$ to maximize the general welfare in the worst kind of society. This, is in my views, unjustified. While I can acknowledge the logic minimax estimator in statistical inference, that logic does not justify Rawls argument, which I believe is flawed.

Where does my utilitarian principle lie? It still fits this framework and it has deeper connection to statistical inference than I thought. Since this is not the focus, I will be brief, but it suffices to say that a utilitarian rule is always admissible (in the sense of Bayesian statistical inference).

Can we still reach the optimal utilitarian rule under this extension of veil of ignorance? The answer is yes.

First, I state the obvious. If people has the right prior (they know $\alpha$, they should choose the rule according to utilitarian view, that is
\[
\theta^*=argmax_\theta\: E(U(i,\theta)|F_\alpha)
\]
Now we consider that they do not have the prior, they would like to be able to choose the same rule, but they cannot. but they observe that they do not have to choose "the rule" now. They can postpone choosing "the rule" after the veil is lifted. All they need to do is to choose a rule for choosing "the rule" after the veil is lifted. After the veil is lifted, they will observe $\alpha$. They stipulate that the rule chosen after the veil is lifted will be the same rule if they were to choose the rule now with the actual prior. In essence they stipulated that the rule that will be chose after the veil is lifted will be in accordance with the utilitarian principle. Put another way, they choose a family of (utilitarian) rules $\theta_\alpha$, and say they will implement the rule $\theta_{\alpha_0}$ when they observe $\alpha=\alpha_0$ after the lifting of the veil.

Finally, from my point of view, I am not in favor of extending the veil of ignorance towards the distribution of types as a thought experiment. For me Rawls was more concerned that our knowledge of our own relative position in the type distribution will bias us (rich do not want too much welfare state). Thus the veil of ignorance is to avoid bias between the relative well off and relative worse off within a society, not to avoid bias resulting from knowledge of differences across societies. I now actually feel I should argue this point more forcefully because it does have implications. If we think the veil of ignorance should extend over the distribution of types as well, then it will imply that all societies, should only have one just rule. But if we only allow the veil of ignorance over types, then we would allow rules to differ across societies. For example, in a society where everyone is already so rich that they live comfortably, there should be less redistribution; while in a society when only some live very comfortably and some live miserably, we might need to be more aggressive in redistribution.

Justice 1---Why I disagree with Rawls

As I mentioned in the previous blog, this is a series of blogs that I will discuss about issues related to justice. I dedicate this series of blog to my US history teacher, Mock Trial coach, and mentor when I was an exchange student. He challenged me to read widely, think deeply about important issues.  It was during a discussion with him, when I first invoked a Rawlsian argument, forced to defend it, and think hard about it. Today, however, I intend to tear it down.

Rawls' framework

Rawls had a powerful idea---the veil of ignorance. The idea is that if we debate policy now, our opinion will be influenced by our current positions and status---a rich person might favor small redistribution, an African American might favor Affirmative action, and an healthy person might fight against universal health care. Rawls' idea is that, let us return to the "original position", that is we do not know who we will be, whether we are black or white, rich or poor, healthy or frail. We decide on what kind of policy we will want:

"no one knows his place in society, his class position or social status, nor does anyone know his fortune in the distribution of natural assets and abilities, his intelligence, strength, and the like. I shall even assume that the parties do not know their conceptions of the good or their special psychological propensities. The principles of justice are chosen behind a veil of ignorance."

Under this framework, Rawls reached the conclusion that a just society will maximize the utility of its worst-off citizen (subject to preserving basic liberty, and equality of opportunity):

Social and economic inequalities are to be arranged so that they are to be of the greatest benefit of the least-advantaged members of society.

Rawls' Fallacy:

That I argue is wrong. Yes, we are behind the veil of ignorance, and we are unsure who we will be, but why do you assume we will design the rules so as to make our worst outcome as good as possible. Consider an analogy, when you pick stocks, you do not choose a stock that will give you the best return in the worst case scenario, you will consider the average return and the variation. For an extreme example, one asset will give you 0 return no matter what. Another asset will give you 100% return 90% of the time, but -0.1% return 10% of the time. It will be crazy to think one will just choose the first risk-free asset over the second asset.

Utilitarian Argument:

What should be the conclusion? I argue, it is utilitarian. Choosing the rule under the veil of ignorance is like making ex-ante optimization. I will show that this ex-ante maximization problem is equivalent to an ex-post aggregate utility maximization problem, thus justifying a utilitarian framework. If you see this, you can skip the next two paragraphs.

 Let me formalize this a bit, let $i$ be a random variable of the type of a person. Under the veil of ignorance, we do not observe our type. We choose the rule for the society $\theta$. After we choose the rule, the veil of ignorance is lifted, and we learn our types, and we get a utility depending on both our types and the rule we choose $U_i(\theta)$. The question is ex-ante, how will we choose the rule $\theta$? I think at this point, it is fairly obvious to economics student, what the answer is.  We need to decide on a decision rule and if we impose some axioms of rationality (discussed in appendix), like one can compare two rules under the veil of ignorance, we have only one way of choosing---the Von-Neumann Expected Utility Framework. In other words, we will choose the optimal $\theta$ so as to maximize
\[
\max_{\theta} E ~U(i,\theta)
\]
where the expectation is taken over $i$.

There are dual interpretations to the objective function shown above. It could be interpreted as ex-ante optimization with uncertainty over type, or it could be interpreted as maximizing average ex-post utility of people in the society. This dual interpretation means that an ex-ante expected utility maximization problem is equivalent to an ex-post aggregate utility maximization problem.  So if you accept the veil of ignorance and the rationality assumption, you have to accept that a just society maximized the aggregate welfare of all its people, weighting everyone equally, at least in principle.

Critiques:

I do have some criticisms/cautions.

The link from theory to practice is far from obvious. I briefly discuss three.

In reality, the rule we impose will influence the distribution of the type $i$ for later generations. For example, if rich and poor people have different birth rate, the distribution of types will change. Another channel is genetics, for intelligence and disease. I do not want to get into this, as a truthful discussion makes people uncomfortable. It suffices to say, the theory suggests an overly static framework.

It is impossible to assign utility value to different people in practice, thus, the ex-post maximization problem is ill-defined in practice. We have to assign the utility ex-post, but we are already out of the veil of ignorance, so our assignment could no longer be innocent.

Finally, let us not forget the political economy. People who make the rules ex-post are not angels, why could we trust them to design the rule optimally even if the model is perfect and they can get the utility function?

Of course, there is another possibility, after reaching the natural implication of the veil of ignorance (with the aid of assumptions), we might come to doubt this very assumption/framework to begin with?

Appendix:

Axioms of rationality that will lead to Von-Neunmann framework (from wikipedia)
I need to point out that it is true this framework has been attacked as a model for human decisions, (most prominently by prospect theory). But those attacks point to the irrationality of human beings, not the logical coherence of this framework. It is a perfect model for how decision should be made, though a crappy model for decisions are actually made.

Completeness assumes that an individual has well defined preferences and can always decide between any two alternatives.
Axiom (Completeness): For every A and B either or .


This means that the individual either prefers A to B, or is indifferent between A and B, or prefers B to A.


Transitivity assumes that, as an individual decides according to the completeness axiom, the individual also decides consistently.
Axiom (Transitivity): For every A, B and C with and we must have .


Independence also pertains to well-defined preferences and assumes that two gambles mixed with a third one maintain the same preference order as when the two are presented independently of the third one. The independence axiom is the most controversial one.
Axiom (Independence): Let A, B, and C be three lotteries with , and let ; then .


Continuity assumes that when there are three lotteries (A, B and C) and the individual prefers A to B and B to C, then there should be a possible combination of A and C in which the individual is then indifferent between this mix and the lottery B.
Axiom (Continuity): Let A, B and C be lotteries with ; then there exists a probability p such that B is equally good as
• .

What do you deserve?

I have been listening to Harvard Open Course on Justice. It has been an awesome experience, though from time to time I get deeply troubled by some of the responses students give. What troubles me most is in the class where Rawlsian idea of justice was debated.

One guy named Mike argued that the society should be based on meritocracy. To illustrate his point, he gave an example of college admission, where he argued forcefully that he worked hard, and put in effort, so he deserve to be admitted based on his merit alone.

To begin with, I must confess I embrace meritocracy, a topic for another day, but I am uncomfortable with the fact that he is so confident that his performance and his admission is due solely to his own merit (you need to watch the video or at least hear the conversation to see that conviction), and oblivious to the fact that he is fortunate to have many who helped him during the process.

His intrinsic assumption is that his excellence today, which secured him a place in Harvard, is all his doing, his merit and his effort. Let us put aside whether merit or talent is arbitrary in themselves, it is a mistake to think our performance or excellence is a deterministic function of our decisions, talents or other merits. Whenever I reflect upon my past, I am often amazed at how fragile this path has been, and how much uncertainty it involves---I was extremely lucky to meet numerous people, who were so kind to give me a helping hand when I need them, provide guidance when I am lost, and share their wisdom in discussions.  They never had to. My teachers in high school and professors in college never had to discuss ideas with me to foster my interest and kindle my curiosity, they never had to spend time writing an enthusiastic recommendation letter which were essential for my admission to Williams, and then to Harvard and Stanford. Without their enthusiasm, things would be very different.

I doubt if anyone has a different story. To say the very least, everyone in Harvard got in with at least a strong recommendation letter. Your recommender does not have to write such a good one. It takes time. Even if you are extremely talented, he still does not have to. It is a favor done for you. Don't take it for granted.

I think when people are given some favor, even for an arbitrary reason, they get used to it, and think somehow they deserve it. It is because of this, we feel entitled to everything we earn. But we don't. We owe a lot to those who helped us.  We are lucky to get those help.

This blog is about uncertainty, which shapes our world more profoundly than we typically recognize. It is via recognition of its role, can we understand our proper role, can we shred undue arrogance, can we fully appreciate our fortune that makes us who we are today.

My hope is from now on, I can write a series of blogs on several issues and dedicate them to the people who mostly influenced my thinking, and inspired my interest. Ten years later, I can look back at those blogs, I will not lose perspective of who have helped me along the way, will not falsely think that I deserve everything, and will be reminded of the responsibility I have because I owe so many people.

A Letter from UC Berkeley Professor

In response to some strike activity at US Berkeley, a Math Professor sent the following letter to his students. It is a powerful letter, and it is worth reading through. I feel his message is so important, that I am obliged to summarize his point for those who do not have time to read.

He offers to cover the sections for his students since two teaching assistant will be on strike, and he refuses to cancel his class. He then moves on to give his rationals----The world we live in is complicated, and we need to solve lots of difficult and complicated questions now and in the future. Students in UC Berkeley are immensely talented, and they should have the responsibility to tackle those questions in the future. Because of that, their education is extremely important for the future task:

And do not fall into the trap of thinking that you focusing on your education is a selfish thing. It’s not a selfish thing. It’s the most noble thing you could do.
Society is investing in you so that you can help solve the many challenges we are going to face in the coming decades, from profound technological challenges to helping people with the age old search for human happiness and meaning.

Learn well, it is not just a luxury, it is a responsibility.


Dear All,

As some of you may have heard, there is some strike activity taking place on campus tomorrow.

I want to let you know that I will not be striking, which means that I will be, so-to-speak, crossing a picket line. Moreover, I know that two of your GSIs have decided to strike, but because I happen to be free in the afternoon when they teach, and because I enjoy teaching smaller classes from time to time and I haven’t had a chance to in a while, I’ll be covering those sections. If you were planning to see me at office hours tomorrow afternoon, then feel free to come to one of the sections I’ll be covering. I will be in Stephens 230c from 2:10 to 4pm, Cory Hall 285 from 4:10pm to 5pm, and Evans Hall 6 from 5:10pm-6pm.

The reason for me taking this decision is extremely simple: We have 7 class days left until the end of the course. Despite the fact that we've made good time and are likely to finish the syllabus with a few lectures in hand for review, class hours are valuable and your education is too important to just cancel a class if we don’t have to. Whatever the alleged injustices are that are being protested about tomorrow, it is clear that you are not responsible for those things, whatever they are, and I do not think you should be denied an education because of someone else’s fight that you are not responsible for. I say this with no disrespect whatsoever to the two GSIs who have decided to strike. Societies where people stand up for what they believe in are generally better than societies where people do not, sometimes dramatically so. Further, I cannot discount the possibility that I may be in the wrong on this and they may be right. I have certainly been on the wrong side of political judgements before and I’m sure I will be again. However from a practical point of view I’ve made my decision and you should all turn up to class and discussion tomorrow as normal.



Beyond practical matters, I think it’s also worth reflecting a little on the broader relationship between politics and your education, and I think I have some important things to share on this topic that may be helpful to you.

I do this with some trepidation. Normally I try to avoid talking about politics with my students and also my professional colleagues because people have a wide variety of views, sometimes held with great conviction and feeling. If I was to get into a political disagreement with one of you or one of my colleagues, it might get in the way of or distract us from the central mission we have of working together to give you a great education.



However sometimes political events reach into our lives without our invitation or control, and we have no choice but to engage with each other about politics. Many times in history it has done so with far more violence and disruption than a strike, and it is wise to be psychologically prepared for this fact.

If I’ve learned one thing about politics since I was your age, it is this: Politics, like most things in life worth thinking about, including mathematics, is very big, very complicated, and very interconnected. I’ve lived and worked in four countries on four continents, all with societies set up differently both politically and socially. I’ve discovered that there is no unique or obviously best way of setting up society. For every decision and judgement you reach, there are people who benefit and people who lose out. It’s the same with the way I teach my classes. I know that for every decision I make about how to teach you there are some of you who benefit and there are others who would do better if I did things differently. There is no way of getting around that. Every judgement you make in life is a question of balancing different interests and ideals. Reasonable good people can disagree on political questions like whether to strike or not, and they can disagree about far more contentious topics also.

All this may sound like speaking in platitudes. However it is a point worth making to all of you because you are so young. One of the nice things about being young is that your thinking can be very clear and your mind not so cluttered up with memories and experiences. This clarity can give you a lot of conviction, but it can also lead you astray because you might not yet appreciate just how complicated the world is. As you get older you tend to accumulate life experiences to learn from, and this is the source of wisdom, but the trouble is that the lessons we glean from life do not all point in the same direction. Sometimes it is hard to tease the correct learning from the experiences life throws at us.

So what are we to do with the fact that when we are young we lack a lot of the perspective we need to make definitive judgements about what is right, but that as we get older our judgements tend to be informed by our experiences, and these experiences guide us in contradictory ways, both between different people and within the same person?

I don’t know.

However one thing I do know is that you are not going to be able to avoid making these kinds of judgements, just as I cannot avoid making a judgment about whether to strike or not. Like it or not, I have to make a political choice, and I have to talk to you about it. For me, the choice not to strike is quite easy, but for you the kinds of judgements and choices you are going to face in your lives are going to be far from easy; they are going to be of a complexity and importance that will rival that faced by any previous generation. To an extent that you may not yet appreciate, the world is changing incredibly quickly. In just a decade, since I was your age, the internet and telecommunications has truly transformed the way we live, not just in rich countries but around the world. When I was an undergraduate, if I wanted to check my email I went to a little room in the basement to use a computer, and if I wanted to learn something I went to a library. The kinds of breakthroughs we are seeing in biotechnology remind me of the way people were talking about electricity in 1900. Of course I don’t know - nobody knows - but my guess is that biotechnology in the 21st century could be similarly transformative to the way the full power of electricity only hit prime-time in the 20th century. The recent controversy about the NSA has shown that the role of information technology on society can be, or at least might become, double edged. There is climate change, another controversial and difficult topic, the exact impact of which we do not yet know. These are just a few of the challenges we can see, and we should remember that history has a habit of throwing curve balls at each generation that nobody saw coming. And among all this tumult, our search for common human peace and happiness on some level becomes more difficult, though no less important. A previous generation dodged the bullet of nuclear armageddon when things looked bleak, but for your generation the bullets are coming thicker and faster than ever before. The potential all of you in your generation are going to have for both good and harm is tremendous.

I suspect many of you have heard sentiments along these lines before. However I also suspect that many of you will think something in response along the lines of `I know all that, but these things are for someone else to figure out, not me.’

That is a mistake.

One of the things you can lose track of when you attend a top tier university like Berkeley is just how exceptional and amazing you really are. I’m blown away every time I talk to you. The way you ask penetrating questions, the way you improved so much between midterm 1 and 2, the way you challenge me to be a better teacher, it just knocks my socks off. You really are amazing. I’ve taught students all over the world, and I’ve never seen a group of students so talented. I’m not just talking about some of you. I’m talking about all of you. It’s a privilege to be your professor. Sadly, however, I know many of you don’t feel that way. The difficulty you all face is that as you look around at all your fellow students, it’s easy to have your eye drawn by people doing better than you. Or rather, I should say people who look like they’re doing better than you. In reality the true extent of how much people are learning can be difficult to measure. Sometimes failures and adversity are better preparations for long term success than effortless progress.

Why am I telling you all this?

I’m telling you this because you all need to know that there is not some great pool of amazing people in some other place who are going to shape the way our species navigates the coming decades. The simple fact is that, like it or not, technology is going to change the way we live in the future, and you’re going to have to solve some very hard problems, as well as figure out how best to use new technology for good, while at the same time facing human dangers that have haunted humanity throughout history.

Part of the work of your generation is going to be technological, using scientific ideas to serve the interests of society, and part of the work is going to be fundamentally human, tied inexorably with qualities of the human condition - human emotion - that dominate the whole of history. These things are not separate, but are inexorably linked, and you are in a better place to understand that connection than me.

I can’t tell you what your particular role should be in the new realities of the 21st century. It’s up to you to decide if you want to make the focus of your life technological, focused on new innovations to drive society forward, or essentially human, focused on the age-old struggles of trying to get along, work together, and find happiness, or some combination of the two.

However I can tell you this:

Whatever you decide to do with your life, it’s going to be really, really complicated.

Science and technology is complicated. History and politics is complicated. People are complicated. Figuring out how to be happy, and do simple things like take care of our kids and maintain friendships and relationships, is complicated.

In order for you to navigate the increasing complexity of the 21st century you need a world-class education, and thankfully you have an opportunity to get one. I don’t just mean the education you get in class, but I mean the education you get in everything you do, every book you read, every conversation you have, every thought you think.

You need to optimize your life for learning.

You need to live and breath your education.

You need to be *obsessed* with your education.

Do not fall into the trap of thinking that because you are surrounded by so many dazzlingly smart fellow students that means you’re no good. Nothing could be further from the truth.

And do not fall into the trap of thinking that you focusing on your education is a selfish thing. It’s not a selfish thing. It’s the most noble thing you could do.

Society is investing in you so that you can help solve the many challenges we are going to face in the coming decades, from profound technological challenges to helping people with the age old search for human happiness and meaning.

That is why I am not canceling class tomorrow. Your education is really really important, not just to you, but in a far broader and wider reaching way than I think any of you have yet to fully appreciate.

Admission Bias?

Update: I realized there is a coding error in my previous version, and I have corrected it. The result now is not as dramatic, but I think it is still striking. I mistakenly plotted the conditional (on wealth) probability of admission, when we are interested in what is the implied percentage of student population based on wealth. I apologize for my mistake.

Motivation:

A friend of mine emailed me a note she wrote on Harvard's admission status. The main finding is that the admitted students tend to come disproportionately from wealthy families, suggesting that, despite the bind-need policy, there is still a selection bias towards the rich. For example, the author found:

About 150 students come from family with less than \$20,000 income, that is about 2.2\% in Harvard’s 6,700 undergraduate population. In the whole country, there are about 20% of families with household income below \$20,000 (Census, 2012)..... 40\% of Harvard graduates presumably come from families with annual income higher than [200,000]. But in the US, only 4.5\% families make more than \$200,000. 40\% of Harvard’s students come from the top 4.5\% of America’s family income spectrum.

While I do agree certain selection criterion like so called leadership, and versatility are unfair to the poor, I think we need to dig deeper into the selection process and data to get a better sense. This phenomonon, could be the result of fair admission process with no bias towards the poor. What do I mean by this?

Theoretical Framework:

Let me illustrate with a model.

Let $Y_i$ be the relevant performance of an individual for admission process. We have
\[
Y_i=\beta_x X_i+\beta_z Z_i+e_i
\]
where $X_i$ is talent, $Z_i$ is work ethics, and $e_i$ is pure luck.
Now we consider the family wealth of that student $T_i$:
\[
T_i=\alpha_x X'_i+\epsilon'_i
\]
where $X'_i$ is the talent of the parents, and $\epsilon$ is all the other factors. We would expect there to be a inter-generational correlation of talents. We could decompose $X_i$ into the inherited part ($E(X_i|X'_i)$) and the innovation part $\xi'_i$. For simplicity, however, we will decompose in the following way:
\[
X_i=E(X_i|T_i)+\xi_i=\delta T_i+\xi_i
\]
This means that talent is not independent of family wealth simply because
1) talent has positive inter-generational correlation;
2) higher talent means higher wealth level in expectation.
Now we can write the performance in the following way
\[
Y_i=\beta T_i+e'_i
\]
Note that $\beta=\beta_x\cdot \delta$, and $e'_i=\beta_z Z_i+e_i+\xi_i$.

A calibration and result

Let us do some calibration. We will scale $e'_i$ so that $\beta=1$. We let $e'_i\sim N(0,3)$ and $T_i\sim N(0,1)$.

Technical fuss: I know wealth does not come from Normal distribution, but consider a function acting on wealth to make the output normally distributed. For example, log-normal distribution is considered to be a good approximation for the distribution of wealth, so instead of saying $T_i$ is wealth, we could say $T_i$ models the log of wealth, which then becomes normally distributed. If you work through the argument, you will find none of these matters.

So the rest is a Bayesian exercise. Conditional on observing getting admitted to Harvard $Y_i>cutoff$, what is the conditional distribution of $T_i$? I calculated the cutoff and did the corresponding calculation, (no worry, I will show you the code and how I calculate the cutoff in the end for transparency). Here is what I find.

The conditional distribution is highly skewed to the right---that is conditional on admission through this fair process, we would expect most of the students come from rich families. Here is the distribution:

The above plot is the percentage of students admitted as a function of income. On the y-axis, it is the percentage of admitted student with respect to total college student population---to find the population percentage among admits, you need to multiply the number by 1000. Note that unconditional $T_i$ comes from $N(0,1)$. Notice the lack of admission in the lower income group until 1 standard deviation below mean.

Another graph. What is the probability of admission in each income group?

As you can see, as the family income goes up, the chance of getting admitted is monotonically increasing. According to our model, this correlation is not causal (family wealth does not cause better performance), and is a result of the centrality of talent.

Ok, let me give you a number in the end. What will the predicted percentage of students coming from the top 4.5% of income distribution? 49.4238%

Discussion

What is the point? The point is not that the education system is fair. The point is we need to be more careful in our diagnosis. Let me give some far-fetched general comments, not validated by the work I did in this blog. The problem might very well lie in the inequality in elementary education and secondary education, and if that is the case, targeting higher education admission process is unlikely to be effective. Efforts like affirmative action aiming to change the result, without addressing the root of the problem, is not only doomed to fail but also likely to create new problems.

I am of the minority view that in empirical work, we need to think harder about the structure of the problem. As those tiny details matter for policy. We need to know the industry we are talking about, understand its structure, and use data to give as precise a picture of the industry. We could gain a lot from reduced-form estimation, but in some applications, without forcing us to be clear the data generating process and the structure, and link them with data, the policy implication could be very limited.

Robustness Check and Technical addendum:

How I choose the parameters for the random variables? Well, I did some exercise finding a reasonable range for the parameters. As it turned out, they tell similar stories. so I choose the parameters that is simple for expository purposes.  I hate blackbox, and I will disclose the workings, which lead to a different set of parameters (slightly more complicated), and show you that the result is basically the same.

So for the first equation

\[
Y_i=\beta_x X_i+\beta_z Z_i+e_i
\],
I let $\beta$'s to be 1 and all variables come from iid $N(0,4)$. That is talent accounts only 1/3 of the performance. (I am using variance 4 to reduce fractions later in calculations)

then for family wealth,
\[
T_i=\alpha_x X'_i+\alpha_\epsilon \epsilon'_i
\]
I let $\alpha$'s to be $\frac{1}{\sqrt{2}}$, and both random variables come from $N(0,4)$. That is talent accounts for only half of the variation in wealth.
Finally, for inherited talent, I assume the correlation is only 1/2.
\[
X'_i=\frac{1}{\sqrt{2}} X_i+\frac{1}{\sqrt{2}}\xi_i
\]
The weird scaling is to make sure that the variance of talent is steady over generation.
If you work through the algebra, you will find the following regression function:
\[
X_i=\frac{1}{2} T_i+\frac{\sqrt{3}}{2}\epsilon'_i
\]
substituting in, we have
\[
Y_i=\frac{1}{2}T_i+\frac{\sqrt{3}}{2}\epsilon'_i+Z_i+e_i
\]
We can pack the last three terms as one error with variance 11, and $\frac{1}{2}T_i$ as scaled wealth $\sim N(0,1)$. In essence we have
\[
Y_i=\tilde{T}_i+\sqrt{11}\tilde{\epsilon}_i
\]
where both random variables are iid standard normal.
 What will the predicted percentage of students coming from the top 4.5% of income distribution be? 22.56%.
Feel free to try different numbers.

Appendix:

 So to make this transparent, I will show you what I did.

How to calculate cutoff? The number of students enrolled in Harvard is 6700, which means 6700/4 students each year. In US about 25 m students enroll in colleges each year. Of course there are other great universities, like Princeton, Williams (yes!), Yale, Stanford. So let me be generous, and say Harvard get the top 0.1%. So with this we can find the 99.9 quantile in terms of Y, which is about 6.18. The rest is coding it up (I used Mathematica):
In[49]:= cutoff = Quantile[NormalDistribution[0, 2], 1 - 10^(-3)];

In[50]:= N[cutoff]

Out[50]= 6.18046

In[51]:= F[x_] := 
 NIntegrate[
  PDF[NormalDistribution[0, 1], x] PDF[NormalDistribution[0, Sqrt[3]],
     y - x], {y, cutoff, +\[Infinity]}]

In[52]:= Plot[F[x], {x, 0, 5}]

In[43]:= wealth = Quantile[NormalDistribution[], 0.955]

Out[43]= 1.6954

In[54]:= ratio = NIntegrate[F[x], {x, wealth, +\[Infinity]}]/0.001

Cleaning the Wound

I had a bike crash yesterday. Something dirty got under my skin. Fearing pain, I only did a superficial wash and applied some antibiotics.

Unfortunately, this morning, I discovered that the site is swollen with pus coming out---a clear sign of ongoing infection. I had to cut open the wound and thoroughly clean it with a scrub. It was quite painful.

I knew I could not just bury those things away. I need to clean it eventually, and it gonna be more painful if I do it later. But I was too much of a wimp to cut it open and clean it yesterday. I am not even sure if I cleaned it thoroughly enough today.