The Pokémon Theorem and other Fairness Impossibility Results

2026-05-09

AI Summary

This work uncovers a unified geometric origin underlying the incompatibility of multiple fairness criteria under unequal base rates. By modeling fairness constraints as linear conditions on conditional mean embeddings in a reproducing kernel Hilbert space (RKHS) and leveraging an over-determined analysis of the law of total expectation, it elucidates the fundamental nature of fairness conflicts. The paper introduces the “Pokémon Theorem,” proving that any finite set of linear mean-based fairness criteria inevitably entails residual violations and revealing the unavoidable class collapse in fair representation learning. Building on RKHS theory, maximum mean discrepancy (MMD), Kolmogorov m-width, and spectral regularization, the authors derive a signal–error frontier under approximate fairness relaxations, with experiments validating the theoretical bounds on standard fairness benchmarks.

Abstract

Fairness impossibility results often look like distinct scalar incompatibility statements. We show that several share one RKHS geometry: fairness criteria are linear constraints on conditional mean embeddings, and unequal base rates make the law of total expectation overdetermine those constraints. This view yields four results. The Kleinberg--Mullainathan--Raghavan dichotomy needs only group-conditional unbiasedness, not full calibration. The \emph{Pokémon theorem} shows that a distinct group pair satisfying any finite collection of linear mean-fairness criteria leaves a residual violation witnessed by the MMD, decaying at the Kolmogorov $m$-width rate under spectral regularity. The same tools prove an impossibility for fair feature learning: parity and class-conditional separation in representation space force class collapse under unequal base rates. The approximate relaxations yield signal and error frontiers, allowing a trade-off between real-world estimators and fairness goals. Experiments on standard fairness benchmarks are consistent with our bounds.

Problem

• fairness impossibility
• conditional mean embeddings
• unequal base rates
• linear fairness criteria
• MMD

Innovation

• RKHS geometry
• conditional mean embeddings
• fairness impossibility
• maximum mean discrepancy (MMD)
• spectral regularity