The Impossibility Theorem of Fairness

“The impossibility theorem” is shorthand for several results about incompatible statistical fairness criteria. Kleinberg, Mullainathan and Raghavan’s 2016 paper formalizes three score-level conditions: calibration within groups; balance for the positive class, meaning the average score among people with a positive outcome is equal across groups; and balance for the negative class, meaning the average score among people with a negative outcome is equal across groups. Except in constrained cases—including equal base rates or perfect prediction—one cannot satisfy all three simultaneously. These positive- and negative-class balance conditions compare average scores conditional on outcomes; they are not directly false-positive or false-negative rate parity after thresholding. Calibration means that among people assigned a given score, the observed outcome frequency matches that score within each group. A classifier applies a threshold to a score and assigns a discrete decision. Chouldechova’s related 2017 result addresses a thresholded classifier: with differing outcome prevalence and an imperfect predictor, predictive parity and equal false-positive/false-negative rates cannot both generally hold. That classifier-level error-rate balance is related to, but distinct from, the score-level positive- and negative-class balance conditions in Kleinberg et al. These results depend on the target, base rates, score quality and chosen criteria; they do not establish a universal impossibility of fairness. The theorem does not tell a decision maker which criterion to prioritize, establish that observed labels are valid ground truth, or guarantee that optimizing any one metric produces just outcomes. Context matters: false positives and false negatives may have different costs, scores may be used as rankings rather than probabilities, and group labels or outcomes can themselves reflect structural disadvantage. Woodworth and colleagues study conditions under which one can learn fair representations while navigating trade-offs, further illustrating that conclusions depend on the formal setup. Responsible use requires stating the assumptions and policy choice, reporting resulting harms, and considering procedural and causal approaches beyond the statistical parity metrics.