  We argue that an imperfect criminal law procedure cannot be group-fair, if
'group fairness' involves ensuring the same chances of acquittal or convictions
to all innocent defendants independently of their morally arbitrary features.
We show mathematically that only a perfect procedure (involving no mistake), a
non-deterministic one, or a degenerate one (everyone or no one is convicted)
can guarantee group fairness, in the general case. Following a recent proposal,
we adopt a definition of group fairness, requiring that individuals who are
equal in merit ought to have the same statistical chances of obtaining
advantages and disadvantages, in a way that is statistically independent of any
of their feature that does not count as merit. We explain by mathematical
argument that the only imperfect procedures offering an a-priori guarantee of
fairness in relation to all non-merit trait are lotteries or degenerate ones
(i.e., everyone or no one is convicted). To provide a more intuitive point of
view, we exploit an adjustment of the well-known ROC space, in order to
represent all possible procedures in our model by a schematic diagram. The
argument seems to be equally valid for all human procedures, provided they are
imperfect. This clearly includes algorithmic decision-making, including
decisions based on statistical predictions, since in practice all statistical
models are error prone.
