  Among the various aspects of algorithmic fairness studied in recent years,
the tension between satisfying both \textit{sufficiency} and
\textit{separation} -- e.g. the ratios of positive or negative predictive
values, and false positive or false negative rates across groups -- has
received much attention. Following a debate sparked by COMPAS, a criminal
justice predictive system, the academic community has responded by laying out
important theoretical understanding, showing that one cannot achieve both with
an imperfect predictor when there is no equal distribution of labels across the
groups. In this paper, we shed more light on what might be still possible
beyond the impossibility -- the existence of a trade-off means we should aim to
find a good balance within it. After refining the existing theoretical result,
we propose an objective that aims to balance \textit{sufficiency} and
\textit{separation} measures, while maintaining similar accuracy levels. We
show the use of such an objective in two empirical case studies, one involving
a multi-objective framework, and the other fine-tuning of a model pre-trained
for accuracy. We show promising results, where better trade-offs are achieved
compared to existing alternatives.
