đź”¬ Research summary by **NicolĂ˛ Cangiotti and Michele Loi**, who are postdoctoral researchers in the Department of Mathematics at Politecnico di Milano.

[Original paper by NicolĂ˛ Cangiotti and Michele Loi]

**Overview**: Many law procedures are nowadays based on algorithms, and all the algorithms are based on mathematics. However, it seems not trivial to introduce in this formal context the ethical concepts of justice and fairness. This paper exploits a mathematical structure â€śto show that theories of justice do not provide a sufficient normative grounding for reasonable accounts of group fairnessâ€ť.Â Â

**Introduction**

The debate around the concepts of justice and fairness is becoming more and more relevant especially in relation to the spread of algorithmic decision-making in many human fields, including AI in law. In our paper we propose an abstract mathematical language introducing a formal distinctinction between fairness and justice. This formal framework highlights analogies between predictive algorithms and the way any legal procedure works. Then we argue that if we consider an imperfect criminal law procedure (mistakes are possible) we cannot be fair with respect to all possible groups of individuals.

In particular, we present a coherent mathematical argument for deterministic procedures leading to binary decisions (an individual is either convicted or acquitted in a trial). It turns out that â€śunless the procedure is perfect, one can always identify at least two morally arbitrary groups, relative to which the procedure is not fair.â€ť

## Is group fairness derivable from justice?

### A World Between Justice and Fairness

The concepts of *justice *and *fairness *may be considered interchangeable if we do not assume that they actually represent different ideas. Starting from the point that these two notions are different one could try to define fairness with the same normative elements that are used to define justice. However, it turns out that group-fairness can only be achieved in an absolute sense by procedures involving a *non-deterministic* element. This interesting result could be regarded as an insight about the impossibility of group fairness or it could be regarded as a reason to provide the definition of group-fairness basing it on approaches alternative to the theories of justice.

Our work focuses on imperfectly just procedures, those that do not guarantee a perfectly just distribution. The property of group fairness as a property of procedures, in the definition from which we start, is an intuitive one. Indeed, it seems intuitive that a procedure is group fair only if it does not favor in a morally arbitrary way (intentionally or unintentionally) any individual, who belongs to a group over an individual who belongs to a different group.

### Computational Thinking in Procedural Law

Letâ€™s try to think about the problem of justice and fairness as a computer should do, namely in mathematical terms. At first, we define a procedure as a rule leading to the allocation of benefits, burdens or harms of various kinds. More precisely, by a procedure we mean a sequence of actions such that, when a given criterion is satisfied, a given outcome for the individual is produced. One could observe that some procedures are extremely complicated: they involve a large number of criteria that jointly determine a decision. However, it is not so restrictive to suppose for the sake of simplicity that the criterion can correspond to a combination of criteria, being satisfied, to such an extent that their satisfaction is sufficient and necessary for the decision concerning the individual.

To provide a more intuitive point of view, we exploit an adjustment of the well-known ROC (Receiver Operating Characteristic) space, in order to represent all possible procedures in our model by a schematic diagram involving the positive and the negative rate of convicted people. This pictorial trick allows us to evaluate case by case the possible procedure computing the relative outcome, when a criterion comes into play.

### What is the price of group fairness?

The mathematical formalization described by the ROC diagrams led us to the unintuitive conclusion that if we want to require the fulfillment of group fairness with respect to all imaginable morally arbitrary groups (which could also include the singular individual as a group), this can only be achieved by avoiding determinism. Clearly, in a non-deterministic procedure, a given initial state of the procedure may correspond to some outcomes in one case, and to other outcomes in another, without any change in its inputs, in normal operating conditions. This is problematic since this random element should represent exactly the opposite of what justice should try to achieve, where there is a clear moral ground to justify an outcome.

Thus, could one take this as a reason to reject group-fairness altogether? Certainly one could assess procedures that are imperfect, and simply ought to minimize injustice, disregarding group-distributive effects altogether. But, a less radical option is, for instance, to amend our account of group-fairness. This requires a different theory of what makes certain groups â€śmorally arbitraryâ€ť in the sense that the concept of the morally arbitrary is not reducible to â€śsomething other than what, morally speaking, justifies inequalityâ€ť. This is the direction in which we want to push our argument. Indeed, we believe that our result is best interpreted as an argument supporting the quest for such a philosophical view.

**Between the lines**

The relation between justice and group-fairness (understood as quality of imperfect procedure) requires in-depth examination as our work tries to highlight. In fact, starting from reasonable definitions we show that such a relation is normally problematic. In fact, an intrinsic incompatibility seems to arise already at our first step of approximation for a given procedure (namely a binary deterministic procedure). Our paper aims to stimulate the discussion around group-fairness in imperfect procedures, regarded as a general type of procedures, conceptually broader than statistical models.