Regulating Fairness in Adaptive AI Systems via Closed-Loop Ergodic Control
Ramen Ghosh · HAL (Le Centre pour la Communication Scientifique Directe) · 2025
The long-term regulation of machine learning systems deployed in high-stakes decision-making demands fairness guarantees beyond static, single-shot metrics. In this work, we introduce a closed-loop framework for population-level fairness in AI systems, where algorithmic decisions influence user outcomes, and these outcomes recursively update the system through retraining. We formalize a dynamical fairness criterion-equal impact-requiring convergence of long-run group-averaged outcomes, irrespective of initial states or protected attributes. Modelling the AI-user interaction as a stochastic process with endogenous feedback, we characterize it when such systems admit a unique invariant distribution, ensuring ergodic fairness. In this context, Ergodic fairness refers to a state where the system's behaviour over time, as observed through its invariant distribution, is fair and unbiased. We derive new sufficient conditions for the stationary measure's existence, uniqueness, and attractivity in both linear and nonlinear settings with non-continuous action spaces. Furthermore, we design a class of feedback-based fairness regulators that steer the system toward equitable steady states and prove that memory-dependent policies may disrupt ergodicity, leading to unfair equilibria. We validate our theoretical results in university admissions, where admission policies are periodically updated using observed student retention data. Our simulations demonstrate how feedbackinduced bias can lead to long-term group disparities even under seemingly neutral policies and how the proposed regulators mitigate these effects. This work lays a mathematical foundation for fairness-aware regulation of adaptive AI systems operating over populations and time.