  We mathematically compare four competing definitions of group-level
nondiscrimination: demographic parity, equalized odds, predictive parity, and
calibration. Using the theoretical framework of Friedler et al., we study the
properties of each definition under various worldviews, which are assumptions
about how, if at all, the observed data is biased. We argue that different
worldviews call for different definitions of fairness, and we specify the
worldviews that, when combined with the desire to avoid a criterion for
discrimination that we call disparity amplification, motivate demographic
parity and equalized odds. We also argue that predictive parity and calibration
are insufficient for avoiding disparity amplification because predictive parity
allows an arbitrarily large inter-group disparity and calibration is not robust
to post-processing. Finally, we define a worldview that is more realistic than
the previously considered ones, and we introduce a new notion of fairness that
corresponds to this worldview.
