A Machine Learning Theory Perspective on Strategic Litigation

Problem Definition
The paper investigates strategic litigation (also known as impact litigation) through the lens of machine learning theory. In a common-law legal system, lower courts are bound by the precedents set by higher courts. The authors model this dynamic as a learning problem: lower courts learn a decision rule (classifier) from the historical rulings of a higher court. A strategic litigator seeks to influence future legal outcomes not just by winning a single case, but by manipulating the set of precedents available to the lower court.

The core problem is defined as follows: Given a high court with a fixed labeling function $f^*$, a lower court using a deterministic learning algorithm $\mathcal{A}$, and a strategic litigator with a goal function $g$ (representing the desired decision rule for future cases), the litigator must select a set of case fact patterns $S_l$ to bring before the high court. The high court labels these cases according to $f^*$, and the lower court updates its learned function $f \leftarrow \mathcal{A}(S_h \cup S_l)$, where $S_h$ is the existing historical precedent. The litigator's objective is to minimize the expected classification error between the learned function $f$ and the goal function $g$ over a distribution $D$ of future cases.

Methodology and Model
The authors propose a highly stylized, abstract model of a two-level court system (high court and lower courts) and analyze it under two specific instantiations:

1. 1-Dimensional Nearest Neighbor Setting: Case facts are points in $\mathbb{R}$. The lower court uses a Nearest Neighbor (NN) classifier. The hypothesis class consists of Boolean functions with finite alternations.
2. $d$-Dimensional Linear Separator Setting: Case facts are points in $\mathbb{R}^d$. The lower court uses a Support Vector Machine (SVM) to find a maximum-margin linear separator. The hypothesis class consists of linear separators.

In both settings, the litigator knows $f^*$, $\mathcal{A}$, and the goal $g$. The study explores whether the litigator can induce a specific $g$ or an approximation of it, and what the optimal set of cases to bring is. The paper also extends the model to include a mechanism for overturning precedent, where the high court removes historical data points that are inconsistent with its current reasoning $f^*$ when a new case forces a contradiction in the hypothesis class.

Key Results and Contributions

• Counterintuitive Structural Phenomena (1D NN):

  • Losing Cases as Strategy: It can be beneficial for a litigator to bring a case they are certain to lose (where $f^*$ rules against them) if that ruling shifts the NN boundary closer to $g$.
  • Myopia is Suboptimal: Greedy strategies (selecting the single best case at each step) or even 1-step lookahead (selecting the best pair) can lead to irreversible errors or fail to achieve the goal, even when the goal is theoretically achievable.
  • Short-term Worsening: A litigator may need to bring cases that temporarily increase the discrepancy between $f$ and $g$ to enable a sequence of moves that eventually achieves $g$.
  • Algorithmic Solution: The authors provide a dynamic programming algorithm that optimally selects a subset of cases from a finite pool $P$ in $O(|P|^2)$ time for the 1D NN setting.

• Achievability in $d$-Dimensions (SVM):

  • Characterization of Achievable Goals: The paper characterizes exactly which goal functions $g$ can be induced by an SVM. A function $g$ is achievable if and only if:
    1. The angle $\theta$ between the normal vectors of $g$ and $f^*$ satisfies $0 < \theta < 90^\circ$, there is no historical data in the disagreement region of $f^*$ and $g$, and no points lie directly on $g$'s boundary; OR
    2. $\theta = 0$ (parallel and same direction), and no historical data lies within a specific distance $\delta$ of $g$.
  • Efficiency: If a goal $g$ is achievable, it can be induced using at most 2 cases (support vectors).
  • Approximation: If $g$ is not achievable, the litigator can find an achievable proxy $g'$ that minimizes error using a brute-force enumeration over a sample from $D$, leveraging the Sauer-Shelah Lemma to bound the number of effective classifiers.
  • Finite Pool Selection: When restricted to a finite pool $P$, any achievable function can be induced by a subset of size at most $d+1$. The authors provide an algorithm to enumerate these subsets.

• Overturning Precedent:

  • The authors model a scenario where the high court removes inconsistent historical data when a new case creates a contradiction.
  • In the 1D NN setting, this is trivial: the litigator simply re-litigates the specific point to flip its label.
  • In the $d$-dimensional SVM setting, the authors prove that if $g$ is achievable under the constraints of Theorem 4, it remains achievable even with conflicting historical data, provided the litigator brings at most $2d + 1$ points. These points are constructed to force the removal of all disagreeing historical points while simultaneously teaching $g$.

Significance and Claims
The authors position this work as an initial foray into studying strategic litigation using machine learning theory. They explicitly state that their model is "highly stylized" and does not aim to capture the full complexity of the US court system.

The paper claims significance in three main areas:

1. Theoretical Novelty: It is the first exploration of strategic impact litigation within a machine learning setting, connecting legal precedent dynamics to learning theory.
2. Connections to Existing ML Theory: The work bridges concepts from teaching dimension (where a teacher selects examples to teach a learner) and clean-label data poisoning (where an attacker injects correctly labeled data to manipulate a model). However, it distinguishes itself by having the "teacher" (litigator) constrained by the "student's" (court's) existing data and the "teacher's" inability to label data directly (labels come from $f^*$).
3. Insight into Dynamics: By demonstrating counterintuitive phenomena (such as the utility of losing cases or the necessity of short-term degradation), the paper suggests that strategic case selection is a complex optimization problem that cannot be solved by simple heuristics.

The authors conclude by suggesting future directions, such as modeling uncertainty in the high court's function, incorporating noisy decisions, or analyzing multi-agent settings with competing litigators, but they do not claim to have solved these problems.