  We study the problem of learning classifiers with a fairness constraint, with
three main contributions towards the goal of quantifying the problem's inherent
tradeoffs. First, we relate two existing fairness measures to cost-sensitive
risks. Second, we show that for cost-sensitive classification and fairness
measures, the optimal classifier is an instance-dependent thresholding of the
class-probability function. Third, we show how the tradeoff between accuracy
and fairness is determined by the alignment between the class-probabilities for
the target and sensitive features. Underpinning our analysis is a general
framework that casts the problem of learning with a fairness requirement as one
of minimising the difference of two statistical risks.
