Machine Learning and Law and Economics: A Preliminary Overview

2.3.1 Supervised Learning

A supervised learning model makes predictions based on a training sample x1,  y1, x2,  y2,  … xN, yN of previously solved cases, where the joint values of all of the variables are known (Hastie, Tibshirani, and Friedman 2009, 485). A metaphor of ‘learning with a teacher’ can be used to explain the underlying mechanism (Ibid, 485). In this metaphor, the student ‘presents an answer y(i) for each x(i) in the training sample,’ and the teacher then ‘provides either the correct answer and/or an error associated with the student’s answer’ (Ibid, 485). Here, the error is characterized by a loss function L(y,  y^), which is to be minimized to approximate the answer (Ibid, 485). An example of the loss function includes L(y, y^)=(y – y^)2, which is used under the method of least squares. To formalize, supervised learning tries to discover a function h that approximates the true function f, given a training set of N example input-output pairs x1, y1,  x2,  y2, … xN,  yN where each y(i) was generated by an unknown function y = f(x) (Russell and Norvig 2010, 695). In terms of the underlying statistical logic, supervised learning is not much different from conventional methodologies employed for empirical law and economics.

2.3.1.2 Support Vector Machine (SVM) and the Kernel Trick

SVM is considered to be among the best off-the-shelf supervised learning algorithms (Ng 2018). The intuition behind SVM is simple: it separates two groups of data points by drawing the best borderline between them (Ibid). More accurately, SVM classifies data by finding the ‘best hyperplane (or boundary)’ that separates data points of different classes, where the best hyperplane means a hyperplane with the largest margin between the two classes (Ibid).

As an illustration, suppose that, in 20 precedent cases in a training dataset, courts decided whether a certain copper pipe product conforms to the buyer’s requirements and that the courts’ decisions served as a basis for the buyer’s right to reject non-conforming goods. Suppose further that the deviation of the diameter and thickness of each product from the buyer’s requirements, in percentage terms, is distributed as shown in Figure 6.

Figure 6:

Distribution of a Training Dataset.

A judge would wish to derive a consistent test regarding non-conforming products from these data compiled from precedents. An SVM model generally fits well with such legal line-drawing. SVM produces a separating hyperplane in the following steps. First, unlike a logit model, where 0 or 1 are assigned to each category, SVM starts with assigning −1 and 1 to each category (Ibid). Thus, in the above hypothetical example, the products that are judged to be conforming to the buyer’s requirements are assigned 1, and those judged to be non-conforming are assigned −1. We can then compile a training set {(x(i),  y(i));  i=1,…,N}, where y(i)=1 when x(i) is a conforming product, and y(i)=−1 when x(i) is a non-conforming product. Then, SVM’s job is, given such a training set, to get a decision boundary wTx+b=0 (where w is a parameter vector, and b is an intersection), which maximizes the margin (or distance) between the decision boundary and the nearest points (among x(i)) (Ibid). The decision boundary so calculated, which is in fact a line, is plotted in Figure 7. However, we find that this linear decision boundary does not separate two groups well, as it is severely underfitted. To make the decision boundary better fit with the distribution of data, SVM uses a ‘kernel trick.’ Mapping data to new features through the Gaussian kernel K(x,z)=e−∥x−z∥22σ2 before optimization, SVM produces the decision boundary as shown in Figure 8. This decision boundary can serve as a basis upon which the judge draws a line between ‘conforming’ and ‘non-conforming’ cases.

Figure 7:

Linear SVM Without Kernelization.

Figure 8:

SVM with Gaussian Kernel.

SVM had been, since its development in 1994, widely recognized as the best performer for multiple purposes among various ML techniques, until deep learning made a spectacular revival around 2004 (Ibid). A drawback for SVM, however, would be the costs associated with the burdensome computation (Ibid). On the other hand, SVM is well suited to handle multi-dimensional calculations. Considering that many legal doctrines require multi-factor tests and sometimes utilize not clearly defined concepts such as the ‘totality of circumstances,’ SVM could prove to be exceedingly useful.

2.3.2 Unsupervised Learning

If supervised learning is similar to ‘learning with a teacher,’ unsupervised learning is analogous to ‘learning without a teacher’ (Hastie, Tibshirani, and Friedman 2009, 486). We sometimes need to categorize (or, ‘cluster’) items into one or more groups based on the difference (or, more formally, ‘distance’) among these items without being told a standard for determining such difference (Ibid, 486). Whereas supervised learning works based on the premise that there is a clear measure of success or failure (or, more precisely, expected loss over the joint distribution P(X, Y)), there is no such a measure in unsupervised learning (Ibid, 486). As such, heuristic judgments are made in order to assess the quality of the result (Ibid, 486−7). To formalize, unsupervised learning aims to directly infer the properties of the probability density P(X) of observations (x(1), x(2), ..., x(N)) in X (Ibid, 486).

Since, in many cases, a legal judgment eventually leads to a yes or no determination, the usefulness of clustering techniques of unsupervised learning for law and economics might be limited. These techniques could, however, be usefully deployed for certain specialized purposes. For instance, the Principal Component Analysis is widely used for purposes of preprocessing datasets for dimensionality reduction before running a regression model.

2.3.3 Reinforcement Learning

The problem of supervised learning is that it could be difficult and sometimes unwieldy to provide explicit supervision for sequential decision-making and control problems (Ng 2018). Reinforcement learning is useful in overcoming such a problem. In order to do so, reinforcement learning uses observed rewards, instead of preexisting data, to learn an optimal or nearly optimal policy for the environment (Russell and Norvig 2010, 830). Ng (2018) illustrates this using a four-legged robot as an example: a programmer would like it to walk but it is all but impossible to use supervised learning to supervise its behavior and to make it walk (Ibid). In such circumstances, a reward function can be used. That is, the programmer can provide the four-legged robot with a walking algorithm in the form of a reward function. This algorithm would tell the learning agent which behavior is desirable or undesirable, and then the agent will choose its action over time for enhanced rewards through a trial and error process (Ibid).

Lots of contemporary reinforcement learning algorithms are modeled as a Markov Decision Process, a discrete-time state-transition system which finds an optimal policy that maximizes the expected value of the total discounted rewards (Ibid). Under this process, an agent is thrown into an environment, continues to perceive states from the environment, and takes actions based on the states, in turn affecting the environment (Ibid). The agent takes actions without any built-in or explicit strategy. The agent first explores the environment by making random decisions based on, for instance, a brute force algorithm. Yet it repeats trials and errors, and the reward function maps the agent’s actions and environment to payoffs. The agent continues to choose its actions over time for large rewards through a repeated game process. Reinforcement learning is particularly well suited with a game that is played within a closed environment. As such, it was only natural that reinforcement learning was effectively applied to the game of Go, beating world-class human Go masters.

From this explanation, law and economics scholars could realize that reinforcement learning is more akin to agent-based simulation than to conventional empirical analysis. In order to gain more useful law and economics insights, perhaps more attention should be paid to multi-agent reinforcement learning (MARL), which has the potential for significantly improving the prediction of multiple agents’ strategic behaviors, in particular in the game theory context.

2.3.4 Deep Learning

Deep learning refers to an ML methodology that learns from a hierarchical representation of data. It is a technique that stacks an interconnected group of nodes. To illustrate how it works, suppose there is an ML model which determines whether a particular use of a copyrighted material constitutes fair use under copyright law.

In Figure 9, each node represents a neuron and each arrow represents a connection from the output of a neuron to the input of another. In this hypothetical example, fair use ultimately depends on four derived features: substantiality, effect, purpose of use, and nature of work (see Ng 2018). This supposes that we already have the insight that these features may determine fair use under copyright law (Ibid). Yet a surprising aspect of deep learning is that we only need to know the input features x and the output y. Neural networks will, through a process called ‘end-to-end learning,’ figure out what would be in the middle by itself (Ibid). In Figure 9, five input features are connected to four hidden internal neurons. These five features are: the degree of similarity, change in the frequency of use, commercial? art?, and fictional?. These hidden neurons are connected to the output layer which outputs whether the use of copyrighted work constitutes fair use (1) or not (0). The goal of the deep learning model is to automatically determine the hidden features such that they can make a prediction about fair use and, in order to do so, we only need to have a sufficient number of training examples (x(i),  y(i)) (Ibid). Every junction between the layers has a parameter (or weight) which constitutes an element of the vectors of weights W, and the activation function g(z) (in most cases, g(z) = max(z,  0) (ReLU function) is used for hidden layers) converts the weighted sum z=WTx  to the values to be sent to the next layers (a=g(z)) (Ibid). We place training examples into the neural network one by one and compute the losses of the neural network based on the difference between the predicted output y(i)^ and actual y(i) (1 if courts recognized fair use in past cases, and 0 otherwise). After the final loss of the neural network is computed, the chain rule is recursively applied to compute gradients all the way back to the inputs and to update the weights in a manner that the loss is minimized and the neural network thus fits the data best (Ibid). This process is called backpropagation. Once the model is trained, it is tested on the test set to measure predictive accuracy.

Figure 9:

An Imaginary Deep Learning Model Predicting Fair Use.

Due to the difficulties in understanding the underlying features that deep learning models have created, they are often called a black box (Ibid). So, while deep learning has produced numerous promising and exhilarating results, the opaqueness and inexplicability of deep learning algorithms have raised concerns that the algorithms, if applied to affect legitimate human interests, may undermine human autonomy.

3.2.1 Algorithmic Transparency

A primary issue that these AI ethics guidelines try to address is that the opaqueness and inexplicability of AI algorithms (in particular, deep learning as a ‘black box’ algorithm) could, unless properly managed, undermine human autonomy and control. Some of these guidelines propose technological measures to make algorithm more explicable and grant a right to the users to request explanation as to how an algorithm works. Some also contain a proposal to audit the process of algorithmic decision-making, or to establish a mechanism to contest the outcome.

Several jurisdictions have gone further and legislated regulations over algorithmic transparency and explicability. Under Article 22 of EU’s General Data Protection Regulation (GDPR), the data subject is not subject to a decision based solely on automated processing, including profiling, without her consent, unless the decision is necessary for contracting or authorized by EU or member state laws. The data subject is also granted the right to obtain human intervention in the automated processing, to express her viewpoint, and to contest the decision. Under Korea’s Act on the Use and Protection of Credit Information (Credit Information Act) (amended in February and effective in August 2020), a data subject, who is subject to an automated credit scoring by personal credit bureaus or financial institutions, has the right to request the explanation of the outcome, standard, and underlying data of the automated scoring, and to contest the scoring by submitting advantageous information or requesting the correction, removal, or reevaluation of underlying data (Credit Information Act, Article 36-2).

A paradox in this type of approaches is that, in general, the more transparent an ML model is made, the less functional and less accurate the model may become. For example, if a complete formula for credit scoring is made public, loan applicants may try to submit the features which are found to have higher correlation with the outcome of credit scoring and which are conducive to enhance the outcome of credit scoring. Such adaptive and exploitative behaviors are likely to impair the functionality of the ML model as a classifier. This problem would be particularly serious when the ML model was introduced to expand the opportunity of the financially distressed (e.g., an ML model that analyzes social network service can be deployed to expand the opportunities of those having a thin credit file like the young generation). Moreover, in practical terms, conducting automated processing, at the full exclusion of human intervention, appears to be uncommon in practice and, as such, the actual scope of applicability of these regulations can be much more limited than initially expected.^[6] Therefore, we need more thorough law and economics studies to find an optimal point where the social benefit that can be derived from a well-functioning ML model is balanced against human autonomy and other fundamental social values.

3.2.2 Algorithmic Fairness

The data, on which ML models heavily rely on, are often biased and may not represent the whole population properly. An ML model trained on the biased data can cause direct discrimination (or disparate treatment) or indirect discrimination (or disparate impact) when applied to different groups of people. An AI agent trained on historical data can, for instance, overlook the recent growth of gender equality and reveal gender biases when deployed for automated recruiting or credit scoring. This resulted in the debates on how to ensure algorithmic fairness vis-à-vis the protected group, and many of the ethics guidelines mentioned above deal with algorithmic fairness.

The discussions on algorithmic fairness are truly transdisciplinary, and there is already extensive literature in computer science, law, economics, and public policy. From an economics viewpoint, the consideration of algorithmic fairness can be perceived as constrained utility maximization (Corbett-Davies et al. 2017). Numerous ways of defining this constraint have been proposed. Verma and Rubin (2018) categorize them into (i) definitions based on predicted outcome, (ii) definitions based on predicted and actual outcomes, and (iii) definitions based on predicted probabilities and actual outcomes. Among them, Corbett-Davies et al. (2017) identify the three most popular definitions: (i) statistical parity (an equal proportion in each group receives the same classification), (ii) conditional statistical parity (an equal proportion in each group receives same classification if a set of legitimate risk factors are controlled), and (iii) predictive equality (false positive rates are made even across different groups). The first and second definitions are based on predicted outcomes, while the third is based on predicted and actual outcomes. Paying attention to the definitions based on predicted probabilities and actual outcomes instead, Kleinberg, Mullainathan, and Raghavan (2017) identify three key elements of algorithmic fairness: (i) calibration within groups (people with the same predicted probability have the same probability to be classified in the positive class regardless of the group they belong to; for example, same acceptance rate across different sexes given the same merit); (ii) balance for the negative class (the members of the negative class from different groups have same average predicted probability; for example, male and female applicants rejected have the same merits); and (iii) balance for the positive class (the members the positive class from different groups have same average predicted probability; for example, male and female applicants accepted have the same merits), but at once prove that except in highly constrained special cases, no algorithm can simultaneously satisfy the three conditions. In fact, as there is a tradeoff between the ability to classify accurately and the fairness of the resulting data (Feldman et al. 2015), we need to pay attention to the marginal decrease in utility for an ML classifier in return for fairness. A more normative strand of the literature has paid attention to the due process aspect in the presence of the conscious ’masking’ of the discriminatory intent under the veil of opaque algorithm (Barocas and Selbst 2016, 692−3, 712−3).

As illustrated in the above example of the COMPAS system, this issue of algorithmic fairness is likely to develop in parallel with the increased use of algorithms in the judiciary or in the public administrative processes. That said, unlike the U.S. (see the Civil Rights Act of 1964) and the EU (see, for example, the Race and Framework Directives and Title III of the Charter for Fundamental Rights), most Asian countries do not appear to have enacted omnibus anti-discrimination legislation that inhibits discrimination in the private sector and, instead, some Asian countries have targeted regulations aimed at narrower areas such as equal employment. As such, this issue would be closely associated with the development of anti-discrimination laws that govern the private sector in general and the expansion of the constitutional principle of equality to civil relationships.

3.2.3 Algorithmic Accountability

There are ongoing debates on how to reform tort, product liability, and safety regulation regimes to effectively address the harms that could be caused by robots such as self-driving cars, medical robots, and drones or other AI agents by holding right persons accountable to the harm. Initial solutions have been sought from extending traditional liability regimes (such as respondeat superior liability theory, vicarious liability, or strict liability) to hold stakeholders liable or conversely shielding stakeholders from liability by granting an AI agent the status of electronic personhood. From the perspective of law and economics, however, the key task would be to identify which of various stakeholders (including developers, controllers, manufacturers, sellers, service providers, platforms, and users) can avoid relevant harms at the least costs and to allocate liabilities to the parties so identified. At the same time, to lower the costs of enforcing legal remedies by ensuring the traceability of accountable parties, appropriate technical governance mechanisms, industry standards, and audit systems should also be devised. Separate from this, in order to prevent undue chilling effects arising from potential liability burdens, discussions on algorithmic accountability should be coupled with the discussions on the structure of risk pooling (by way of insurance, for instance) and the scope of immunity.

3.2.4 Addressing Potential Economic Harm from Algorithmic Pricing Agents

Antitrust scholarship has debated on the potential anticompetitive effect from price discrimination by way of behavioral targeting and personalized pricing, In addition, economic harms from the ‘tipping’ or convergence between actions by multiple algorithmic agents, such as stock trading bots or dynamic pricing agents, has drawn attention. In particular, a concern that price-setting algorithms might facilitate collusion in oligopolistic markets (‘algorithmic tacit collusion conjecture’ (Ittoo and Petit 2017)) hard-hit antitrust scholarship, shortly after first proposed by Mehra (2014) and elaborated by Ezrachi and Stucke (2016). At the basis of their conjecture stand implicit suppositions: (i) a causation or correlation between a heightened price transparency or frequency/speed of interaction and a heightened risk of tacit collusion and (ii) a direct impact of the use of the same or similar algorithms or self-learning algorithms, leading to tacit collusion (without the mediating effect of market concentration) (Ibid). Their conjecture, however, has some theoretical weaknesses such as: (i) the transparency on the customer side (unlike that on the supplier side) can rather make it harder for the suppliers to collude; (ii) there is no theoretical or empirical ground for asserting that the use of the same or similar algorithms would facilitate tacit collusion; and (iii) in a heterogeneous product market, the agents can evolve in a way to effectuate price discrimination, even with no interests or intention to collude.

The literature has tried to run reinforcement learning models (in particular, multi-agent Q-learning models) to verify the algorithmic tacit collusion conjecture. The first actual implementation of the algorithm is found in Calvano et al. (2018), which concludes that their two-agent independent Q-learning model, built on the environment of logit demand and constant marginal costs, ‘systematically learn to collude’ after an average of 165,000 iterations. Klein (2019)’s experiments with a two-agent independent Q-learning model appears to show that Q-learning can learn to price above static levels in a sequential competition situation These experiments appear to support the algorithmic collusion conjecture at a first glance, but their findings are based on strong simplifying assumptions and thus are ‘largely suggestive’ (Deng 2018, 91). One of the overly strong assumptions of these experiments might be that there exist only two players, given that one of Ezrachi and Stucke’s key intuitions is that algorithmic agents can collude even in the absence of market concentration (Ezrachi and Stucke 2017). Overall, this conjecture remains to be a theoretical conjecture not based on solid empirical grounds. This conjecture and other related discussions, nonetheless, have made significant contributions in that reinforcement learning models have been designed and applied to analyze actual and potential social harms from these discussions. More broadly, the deployment of AI models in businesses and its impact on market dynamics have emerged as an important area of research.

3.2.5 Heightened Privacy Concerns

As ML-based image classifiers such as convolutional neural network achieve outstanding predictive accuracy, AI-based facial recognition through closed-circuit television, satellites, and drones has got the potential to be used for predictive policing – the ‘use of historic crime data to identify individuals or geographic areas with elevated risks for future crimes, in order to target them for increased policing’ (Asaro 2019) – or more direct surveillance over a specific group or person. This is just one example where the use of an AI model could have serious privacy implications. Since today’s AI is predicated upon the extensive use of data – often personal data – developing and deploying an AI model often have ramifications on privacy and, as such, how to find a balance is an important issue.