Closed-Loop View of the Regulation of AI: Equal Impact across Repeated Interactions

TL;DR

The paper argues for AI regulation grounded in civil-rights notions of equal treatment (one-shot fairness) and equal impact (long-run fairness) within a closed-loop model of AI systems and users. It formalizes how a system’s outputs influence user behavior and, through retraining, feed back into future decisions, making ergodicity and a unique invariant measure central to ensuring stable, fair long-run outcomes. By defining precise conditions for equal treatment and equal impact and discussing guarantee properties via iterated function system theory, the authors connect regulatory goals with stochastic control concepts. A credit-scoring case study demonstrates that equal treatment can coexist with equal impact, achieving uniform long-run outcomes across individuals and racial groups under the proposed framework. The work offers a principled bridge between regulatory objectives, control theory, and practical fairness considerations, laying groundwork for future formal guarantees and constraint design.$

Abstract

There has been much recent interest in the regulation of AI. We argue for a view based on civil-rights legislation, built on the notions of equal treatment and equal impact. In a closed-loop view of the AI system and its users, the equal treatment concerns one pass through the loop. Equal impact, in our view, concerns the long-run average behaviour across repeated interactions. In order to establish the existence of the average and its properties, one needs to study the ergodic properties of the closed-loop and its unique stationary measure.

Figures (6)

• Figure 1: A closed-loop model of an AI system and its interactions with the users: the AI system provides some outputs, e.g., scorecards in credit scoring, matches in a matching market, or suggestions in a decision-support system. Users observe the outputs and take action in response. With some delay, their actions in response to the outputs are utilized in retraining the AI System.
• Figure 2: The 2020 annual income distribution of "BLACK ALONE", "WHITE ALONE" and "ASIAN ALONE" households in USA, with three races distinguished by colours. Data are sourced from Table A-2 of the Current Population Survey (CPS) of US Census Bureau.
• Figure 3: Solid curves depict the mean value of time series $\{\textrm{ADR}_s(k)\}_{k\in[N]}$, across five trials, with race information distinguished by colour. Error shades display mean $\pm$ one standard deviation.
• Figure 4: The time series $\{\textrm{ADR}_i(k)\}_{k\in[N]}$ for all users from five trials ($5\times 1000$ curves), with their race information distinguished by colour.
• Figure 5: The density of ADR$_i(k)$ at different time steps, with the race information ignored. Darker colours denote higher density.

Theorems & Definitions (7)

• Definition 1: Equal Treatment
• Definition 2: Equal Treatment Conditioned on Non-Protected Attributes
• Definition 3: Equal Impact
• Definition 4: Equal Impact Conditioned on Non-Protected Attributes
• Definition 5
• Definition 6
• Definition 7: Incremental ISS, Angeli2002