  Consider an actor making selection decisions using a series of classifiers,
which we term a sequential screening process. The early stages filter out some
applicants, and in the final stage an expensive but accurate test is applied to
the individuals that make it to the final stage. Since the final stage is
expensive, if there are multiple groups with different fractions of positives
at the penultimate stage (even if a slight gap), then the firm may naturally
only choose to the apply the final (interview) stage solely to the highest
precision group which would be clearly unfair to the other groups. Even if the
firm is required to interview all of those who pass the final round, the tests
themselves could have the property that qualified individuals from some groups
pass more easily than qualified individuals from others. Thus, we consider
requiring Equality of Opportunity (qualified individuals from each each group
have the same chance of reaching the final stage and being interviewed). We
then examine the goal of maximizing quantities of interest to the decision
maker subject to this constraint, via modification of the probabilities of
promotion through the screening process at each stage based on performance at
the previous stage. We exhibit algorithms for satisfying Equal Opportunity over
the selection process and maximizing precision (the fraction of interview that
yield qualified candidates) as well as linear combinations of precision and
recall (recall determines the number of applicants needed per hire) at the end
of the final stage. We also present examples showing that the solution space is
non-convex, which motivate our exact and (FPTAS) approximation algorithms for
maximizing the linear combination of precision and recall. Finally, we discuss
the `price of' adding additional restrictions, such as not allowing the
decision maker to use group membership in its decision process.
