  We investigate the problem of designing optimal classifiers in the strategic
classification setting, where the classification is part of a game in which
players can modify their features to attain a favorable classification outcome
(while incurring some cost). Previously, the problem has been considered from a
learning-theoretic perspective and from the algorithmic fairness perspective.
Our main contributions include 1. Showing that if the objective is to maximize
the efficiency of the classification process (defined as the accuracy of the
outcome minus the sunk cost of the qualified players manipulating their
features to gain a better outcome), then using randomized classifiers (that is,
ones where the probability of a given feature vector to be accepted by the
classifier is strictly between 0 and 1) is necessary. 2. Showing that in many
natural cases, the imposed optimal solution (in terms of efficiency) has the
structure where players never change their feature vectors (the randomized
classifier is structured in a way, such that the gain in the probability of
being classified as a 1 does not justify the expense of changing one's
features). 3. Observing that the randomized classification is not a stable
best-response from the classifier's viewpoint, and that the classifier doesn't
benefit from randomized classifiers without creating instability in the system.
4. Showing that in some cases, a noisier signal leads to better equilibria
outcomes -- improving both accuracy and fairness when more than one
subpopulation with different feature adjustment costs are involved. This is
interesting from a policy perspective, since it is hard to force institutions
to stick to a particular randomized classification strategy (especially in a
context of a market with multiple classifiers), but it is possible to alter the
information environment to make the feature signals inherently noisier.
