To build intuition, let's consider a hypothetical society. We divide a group of people in two. The first group rolls a die with 100 sides labeled from 1 to 100 in even increments. The second group rolls a die with labeled 1 to 110 in even increments. The roll determines their annual ability level which is unknowable. If you were hiring, which group would you prefer? Is it fair to the second group?
Note that this isn't strictly equivalent for several reasons. The game above the error is part of the generating process instead of the measurement process. Also the distribution is district rather than continuous. The key commonality is that a weak predictor at an individual level can be a meaningful predictor at a group level.
Imagine if one group had 55% of its dice labeled up to 110 and 45% up to 100, and the other group had the opposite. So if all you use for hiring is the group membership, you're going to very arbitrarily miss out on a lot of 110s and hire a lot of 100s. And now also imagine that this process of identifying who is in what group is ridiculously expensive.
It's not contrary at all. It's the same point. The changing mix is merely a difference of degree. The key argument you raise is that discrimination has a cost. I don't have the numbers offhand to determine whether the cost of monitoring exceeds the benefit of higher quality education. Regardless, that's a different argument that should be backed by a different set of data.
It's contrary in that a naive interpretation leads to the opposite conclusion. That's all I meant. And I'm not sure how you define what is a different argument and what isn't. It's all about the value of these scores.
Note that this isn't strictly equivalent for several reasons. The game above the error is part of the generating process instead of the measurement process. Also the distribution is district rather than continuous. The key commonality is that a weak predictor at an individual level can be a meaningful predictor at a group level.