  Recent discussion in the public sphere about algorithmic classification has
involved tension between competing notions of what it means for a probabilistic
classification to be fair to different groups. We formalize three fairness
conditions that lie at the heart of these debates, and we prove that except in
highly constrained special cases, there is no method that can satisfy these
three conditions simultaneously. Moreover, even satisfying all three conditions
approximately requires that the data lie in an approximate version of one of
the constrained special cases identified by our theorem. These results suggest
some of the ways in which key notions of fairness are incompatible with each
other, and hence provide a framework for thinking about the trade-offs between
them.
