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Section: Application Domains
Law
There are now many ways in which mathematics are applied to law. They include the following approaches:
Given our expertise, we are concerned with approaches 1 and 2: our first applications are based either on a mix of economic and statistical methods, or on purely statistical ones. We will also develop original probabilistic models.
From a general point of view, the benefits of using actuarial models in law is twofold:
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mathematical models should allow for a more profound understanding of law structures and rules. Indeed, as explained in [47], law can be seen as an information technology in the sense that it provides information to the community about the content of legal norms and, in its common law form, elicits information about the world from the disputes before a court. In this two-way path, tension between law's potential for certainty and its capacity for discovery reflects in part the imperfect circulation of information. The joint use of adequate mathematical models and big data tools should greatly enhance this circulation, thus improving the efficiency of the system as a whole;
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in a more complex and more informed world, legal procedures are likely to become more frequent. However, the state resources devoted to law cannot increase without bounds. Making available tools that would facilitate amicable settlement is then of strong interest. In particular, models allowing one to estimate outputs of legal decisions, at least in certain areas and in a rough way, would certainly draw people to be more inclined to negotiate rather than go to court, thus reducing the burden put on the legal system. This tendency is already quite noticeable in particular in the USA, where so-called on-line dispute resolution systems gain popularity.
We contribute to both these goals, paying in addition extra caution to the performative aspects. Our first studies are detailed in the next sections.
Law-Mathematics correspondences
In order to root our subsequent studies on firm bases, we intend to start by evidencing some parallel notions in law and mathematics, and to study if they are profound enough to yield useful tools. While this will inevitably be sometimes rather qualitative, it will definitely shed some light on how to model legal reasoning in a mathematical way.
An example of such a qualitative link is the fact judges, as mathematicians, when faced with a question, often have immediately a intuition of their answer. In a second phase, lawyers try to find which legal texts or jurisprudence allow them to justify this answer, while mathematicians invoke a series of computations and known theorems to do the same. In both cases, if no path is found to the initial answer (that is, no legal texts or no valid sequence of computations), the practitioner tries to defend or prove the opposite one. We have no idea yet how to formalize this parallelism, but this will be a topic of study. More quantitative ones are the following:
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Judges often say that they weigh different factors when they need to make a decision. The obvious corresponding mathematical notion is the one of linear models, where variables are linearly combined to produce an output. We will choose some simple domains, such as for instance child support, to check whether the decided amount is indeed obtained by weighting the criteria that judges are supposed to take into account.
This requires to analyse a large amount of case law and assessing the fit of various linear or generalised linear models. State-of-the-art techniques in machine learning are used in this connection.
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Finally, an obvious and probably fruitful correspondence between both domains rests on the notion of causality. Determining which events are causes of others is clearly a crucial task in courts, since evidencing responsibilities is at the core of making informed judgements.
On the other hand, statisticians have, until rather recently, avoided to consider causal questions, concentrating on correlations. This is still true today, where most researchers and practitioners would claim that statistics can only evidence dependencies between random variables but cannot assess causal links, except when controlled experiments may be performed. It is hard to think of a situation in law where one could perform such experiments.
However, a growing community has started to develop what now seems to be a somewhat coherent theory, termed causality theory, that allows one to efficiently decide if a variable is indeed a cause of a variable under some conditions [49]. Apart from theoretical developments, this theory has been applied in various domains, and most notably in economy and biomedical studies. We are not aware of any applications in law.
We study this area in two ways:
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the most direct one is to choose a specific domain, analyse some decisions in it in light of the legal and jurisprudential criteria that are supposed to base them, and check whether they are indeed causes of the decision in the sense of causality theory. More generally, we try to construct the whole Bayesian network associated with a given field;
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a more ambitious goal is to question whether the way law sees and organizes causality is anything like what is performed in statistical causality theory. This task requires an abstract model of legal causality that must be constructed from scratch. This is a long term aim.
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Scales and performativity
We have just won a call “Droit, justice et numérique” of the “Mission de recherche Droit et Justice”, a “groupement d'intérêt public” created by the French ministry of justice and CNRS. Our proposal is a joint project with L. Godefroy (Faculté de droit et science politique, Nice University), who has expertise in the relations between the digital world and law, and F. Lebaron (Versailles St Quentin University). F. Lebaron is a sociologist and a specialist of performativity. We aim at studying the performative effects of scales from a general point of view by using our respective knowledges in law, sociology and statistics. More precisely, we will first choose some domains where scales have been introduced, like for instance child support or competition law. Statistical studies based on sociological insights will then be performed to measure how much these scales have performed as compared to the previous, scale-free, situation. This step will require to construct models in order to enhance the estimation step and thus the interpretation of the results. Based on the analysis of the current performative effects and our models, we will, if needed, propose modifications allowing one to reduce unwanted effects.
As a last step, we hope that a global pattern of how scales perform will emerge, maybe from a comparative analysis of the models in different areas. This could open the way to the construction a general theory.
Quantifying legal risk
Our most successful application to date is in the quantification of legal risk: once one is prepared to accept that a legal decision is a random variable, one realizes that legal risk, which is a special component of the global risk companies or even citizens face, may be treated as are other risks. In particular, financial risks have been the topic of extensive studies in recent years, partly in response to the several crises we have witnessed. One lesson from this area is that, although one cannot of course predict the future state of a market, one is able to estimate its probability distribution. This allows one for instance to compute Values at Risk and thus to control one's risk.
We have designed an approach that can quantify legal risk in the same way as financial risk: given a specific domain, e.g. spousal support or dismissal without fair cause, we carefully design a set of legal criteria and analyse a large amount of cases in light of these criteria. We then use refined machine learning techniques to produce a probability distribution that reflects the decisions that would be taken by the judges in our database. This probability distribution takes into account both inter- and intra-judges variability. The mathematical result is that, when the size of the database tends to infinity, the estimated probability distribution tends, under some assumptions, to the actual one.
We have applied this theory to two fields so far : spousal support and dismissal without fair cause. Our future plans include in particular areas in labour law.
In view of to the strong interest this tool has raised among professionals (lawyers, insurance companies, but also the french ministry of justice), we are thinking of creating a start-up company that would commercialize it. As a consequence, we are not able to detail the mathematics involved in this study.
Intellectual property
This project is conducted in the frame of an ISN-funded collaboration between Inria and CERDI (University Paris Sud). Its aim is to help judges make informed decisions concerning the amount of fines in cases of violation of intellectual property. Indeed, in this domain, the fundamental rule that the amount is fixed so as to make good the damage suffered is not adequate: a person who commits a fault with a view to gain can be condemned, in addition to compensatory damages, to pay punitive damages. This rule has been introduced in 2007 under the impulsion of European law. In practice, it seems that it has not been implemented with great success. Our contribution studies a Bayesian network model for understanding how judges compute such amounts. We construct two such networks, one based on law and jurisprudence from Canada and one from France. This project has started in the fall of 2015.