Monday, November 28, 2011

Meteors and Gender Differences

A meteor falls out of the sky and destroys your house.  "Why me? Life isn't fair!" you might be heard to cry.  There would probably be a lot of nodding from all directions.

Well yeah, it's pretty unfair from the perspective of after the meteor hit your house.  Everyone on the face of the planet has a non-meteor-kablooeyed house except you!  But on the other hand, before it happened, everyone had an equal chance of destruction.  From the perspective of beforehand, everyone is in the exact same situation, so it is in some sense fair.

The basic idea of fairness is something like "equality between people."  But equality from what perspective?


To accommodate randomness, let's say a situation is fair if two people are drawing from the same distribution. The question is: which distribution are we requiring to be the same?  The distribution of outcomes conditional on what point in time, or more generally what facts about the world?  What do we take as given, and what is still up in the air?  Fairness depends entirely on what you condition on.

Disagreement here is widespread.  That's okay.  But once in a while, it's nice to step back and notice what we're conditioning on, and perhaps question it.



To that end, here's a thought exercise: Why do we worry so much about unequal outcomes between groups of people?  Let's take a step back.

Consider a world where each person simply has a life outcome and a label.  You can think of the life outcome as income or any other measure of "success," and the label could be something like race or sex.  Our moral starting point is that the label shouldn't matter; skin color or gender should be irrelevant to how much we care about a person.

Now, there is a joint distribution over outcome and label.  Associated with it, we have densities for outcome and label alone, as well as conditional densities for each given the other.

Now consider these two scenarios:
  • Scenario 1: The label is drawn first, and then the outcome is drawn from the conditional distribution (conditional on the label that was already drawn).
  • Scenario 2: Life outcome is drawn first, and then the label is drawn conditional on the outcome.  
Mathematically, of course, the end product is the same in either case, namely (outcome, label) pairs drawn from the joint distribution.  (Or, if we are label-blind, we just see a bunch of people drawn from the same outcome distribution in each case).  But to most people, I think scenario 1 seems potentially much less fair than scenario 2.  Why?

Because fairness depends on what you condition on. In the first scenario it seems sensible to condition on the label, and say: The label shouldn't matter, so the distribution of outcomes conditional on the label should be the same for different labels. But in the second scenario, everyone draws from the same distribution of outcomes, and afterward there is an irrelevant draw from a label distribution. (In this world, people don't care about the label per se, so once their outcome is drawn, nothing else matters).

But here's the kicker.  Should the scenario we're in really make any difference to fairness?  Do we really want our notions of fairness to depend on things like the order events actually unfold?  If the outcome is what matters, and the outcome is the same in each scenario, what does it matter how we got there?

Personally, I don't want to care about anything I don't care about!  I care about the pool of realized outcomes, not labels, and to be label-blind means to have no preference over how labels are split among those realizations.  I don't want to get sucked into finding differences between scenarios 1 and 2; I want my notion of fairness to be robust to the order of irrelevant events such as the assignment of irrelevant labels.



Is it unfair to women that there aren't so many good athletic career opportunities?  As a point of fairness, how do they compare to the 99.99% of men who just aren't good enough at sports?  Is there a difference between a woman who can't be an NBA all-star because she's drawing from a distribution that doesn't have support over the upper tail of the rankings, versus a man who can't be an NBA all-star because he happens to just not be awesome at basketball?


Tell me, what do you think?  Is there a coherent way to argue that labels should and shouldn't matter at the same time?


[Discrimination, by the way, is orthogonal to this post.  To the extent that unequal outcomes across groups is evidence of discrimination, you might be upset indeed, the above notwithstanding!  But even putting discrimination aside, many people will still observe the unfairness of inequality across groups.  Is there something else going on here?]

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