The Pokémon Theorem and Other Fairness Impossibility Results in Machine Learning

A new academic paper titled "The Pokémon Theorem and other Fairness Impossibility Results" has been published on arXiv by researchers Daniel Matsui Smola and Alex Smola. The study addresses fundamental challenges in algorithmic fairness, demonstrating that various fairness impossibility results share a common geometric structure within Reproducing Kernel Hilbert Spaces (RKHS). The authors argue that fairness criteria act as linear constraints on conditional mean embeddings, which become overdetermined when base rates differ across groups. Key contributions include the "Pokémon theorem," which reveals that satisfying any finite set of linear mean-fairness criteria still leaves residual violations detectable by Maximum Mean Discrepancy (MMD). Additionally, the paper proves that fair feature learning faces impossibility under unequal base rates, leading to class collapse. The research provides approximate relaxations to balance real-world estimator performance with fairness goals, supported by experiments on standard benchmarks. This work offers significant theoretical insights for the machine learning community regarding the inherent trade-offs in designing fair AI systems.