Group Fairness Is Not Derivable From Justice: a Mathematical Proof

TL;DR

This paper mathematically demonstrates that imperfect criminal justice procedures cannot ensure group fairness unless they are perfect, non-deterministic, or degenerate, highlighting fundamental limitations in achieving fairness.

Contribution

It provides a formal proof that only perfect, probabilistic, or degenerate procedures can guarantee group fairness in imperfect systems.

Findings

Only perfect procedures guarantee group fairness.

Imperfect procedures can only be fair if they are lotteries or degenerate.

The argument applies broadly to all human and algorithmic decision procedures.

Abstract

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.