Parity in predictive performance is neither necessary nor sufficient for fairness

TL;DR: Parity in Predictive Performance (PPP) holds that a machine learning model is fair if and only if its predictive performance (by some measure) is (approximately) equal across groups of interest. We argue that this assumes that groups are equally difficult, which is unlikely to hold in practice. Absent this assumption, a model could be fair but not satisfy PPP, or be unfair yet satisfy PPP. Thus, PPP is neither necessary nor sufficient for fairness. We propose a new definition of fairness, Relative Realised PPP (R2P3), to account for these situations.