The Hanover Collaboration

Looking at the Michigan case again through the lense of Ricci I think I’ve decided how to do affirmative action.

Notice that we simultaneously want:

1. diversity
2. fairness
3. the best candidates
4. Oh, and no “quota” systems, those are too rigid.

There are a number of obvious inconsistencies here, but I’ve got an idea of how to tackle them. We relax 3 by saying that we would like N of the top M candidates. To assess the impact of this we define a loss function L(M) that quantifies the impact of this relaxation.

We quantify 1 by making up a diversity function, D, which tells us how much we value diversity, and is a function of the mix of races in the selected group.

To ensure 2, let’s require that all candidates of a particular race in the top M are selected with equal probability. However for different races, the acceptance rates may differ. This will be our knob: we’ve got to select M and these rates. This defines a relatively straightforward optimization problem.

But there’s a problem: At the optimum there is probably some race for which every candidate in the top M was selected. This looks like it runs afoul of 4. I would argue that, strictly speaking, this is not a “quota” because it depends on L, and hence varies depending on the relative quality of the candidates.

For example, if MIT had to substantially lower its standards to get more whites that would be reflected by making M much closer to N.

Suppose that in the top N is 10% white and also that the 200th ranking white is the 2000th candidate. Assume that the relationship between L and D is such that the resulting class of 1000 people is 20% white. Taking a random white in the top 2000, we see that under this system their odds of acceptance go from 50% to 100%. For a random non-white in the top 2000 we see their odds going from 50/50 to 37.5%. Under this analysis, the reduction in odds for the biased-against race is always going to be less than the increase for the biased-towards race.

At MIT we’re naturally pretty diverse (and the functional difference between the first and Mth guy might be quite a ways). But for the New Haven Fire Department, I think that diversity is easily attainable. Let’s just assume that L=0 as long as all the captains passed the exam, (an assumption that ought to hold up in court, I’d imagine). If we assume that equally diverse is ideal, we could take the passing hispanics and blacks with probability 2/3¹ and whites with probability 3/16. Compare this to what they did in which passing blacks had zero chance of selection, hispanics 2/3 (unchanged!) and whites 1/2. I’d say that’s a pretty modest bias away from fairness (1/2 to 3/16) for an entirely justifiable goal. Seriously, that should have been pretty obvious.

¹ Actually, you could get the same result with a slightly higher average score by setting the passing grade at the grade of the second highest black person. This is what the optimization I described would probably do.