The Pok\'emon Theorem and other Fairness Impossibility Results

TL;DR

This paper introduces a geometric framework for fairness impossibility results, revealing fundamental limitations and trade-offs in achieving fairness criteria under unequal base rates.

Contribution

It unifies fairness impossibility results using RKHS geometry, introduces the Pokémon theorem, and analyzes trade-offs in fair feature learning and estimation.

Findings

Fairness criteria are linear constraints on conditional mean embeddings.

Unequal base rates overdetermine fairness constraints, leading to impossibility results.

Experiments support the theoretical bounds on fairness trade-offs.

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{Pok\'emon 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.