  We study a two-stage model, in which students are 1) admitted to college on
the basis of an entrance exam which is a noisy signal about their
qualifications (type), and then 2) those students who were admitted to college
can be hired by an employer as a function of their college grades, which are an
independently drawn noisy signal of their type. Students are drawn from one of
two populations, which might have different type distributions. We assume that
the employer at the end of the pipeline is rational, in the sense that it
computes a posterior distribution on student type conditional on all
information that it has available (college admissions, grades, and group
membership), and makes a decision based on posterior expectation. We then study
what kinds of fairness goals can be achieved by the college by setting its
admissions rule and grading policy. For example, the college might have the
goal of guaranteeing equal opportunity across populations: that the probability
of passing through the pipeline and being hired by the employer should be
independent of group membership, conditioned on type. Alternately, the college
might have the goal of incentivizing the employer to have a group blind hiring
rule. We show that both goals can be achieved when the college does not report
grades. On the other hand, we show that under reasonable conditions, these
goals are impossible to achieve even in isolation when the college uses an
(even minimally) informative grading policy.

Computer Science and Game Theory;Machine Learning;Theoretical Economics