  Statistical parity metrics have been widely studied and endorsed in the AI
community as a means of achieving fairness, but they suffer from at least two
weaknesses. They disregard the actual welfare consequences of decisions and may
therefore fail to achieve the kind of fairness that is desired for
disadvantaged groups. In addition, they are often incompatible with each other,
and there is no convincing justification for selecting one rather than another.
This paper explores whether a broader conception of social justice, based on
optimizing a social welfare function (SWF), can be useful for assessing various
definitions of parity. We focus on the well-known alpha fairness SWF, which has
been defended by axiomatic and bargaining arguments over a period of 70 years.
We analyze the optimal solution and show that it can justify demographic parity
or equalized odds under certain conditions, but frequently requires a departure
from these types of parity. In addition, we find that predictive rate parity is
of limited usefulness. These results suggest that optimization theory can shed
light on the intensely discussed question of how to achieve group fairness in
AI.
