# The impossibility of "fairness": a generalized impossibility result for decisions

arXiv: https://arxiv.org/abs/1707.01195 · HTML: https://mlsanity.com/paper/1707.01195/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jul 2017.

## Built on by (most cited first, 12 of 12)

- [On the Independence of Association Bias and Empirical Fairness in Language Models](https://mlsanity.com/paper/2304.10153/) (2023)
- [Pushing the limits of fairness impossibility: Who's the fairest of them all?](https://mlsanity.com/paper/2208.12606/) (2022)
- [Fairness in Reinforcement Learning: A Survey](https://mlsanity.com/paper/2405.06909/) (2024)
- [ACROCPoLis: A Descriptive Framework for Making Sense of Fairness](https://mlsanity.com/paper/2304.11217/) (2023)
- [Towards Standardizing AI Bias Exploration](https://mlsanity.com/paper/2405.19022/) (2024)
- [A Human-in-the-Loop Fairness-Aware Model Selection Framework for Complex Fairness Objective Landscapes](https://mlsanity.com/paper/2410.13286/) (2024)
- [SoK: What Makes Private Learning Unfair?](https://mlsanity.com/paper/2501.14414/) (2025)
- [Achieving Counterfactual Fairness with Imperfect Structural Causal Model](https://mlsanity.com/paper/2303.14665/) (2023)
- [Leveraging Contextual Counterfactuals Toward Belief Calibration](https://mlsanity.com/paper/2307.06513/) (2023)
- [Differential Privacy, Linguistic Fairness, and Training Data Influence: Impossibility and Possibility Theorems for Multilingual Language Models](https://mlsanity.com/paper/2308.08774/) (2023)
- [Biased Heritage: How Datasets Shape Models in Facial Expression Recognition](https://mlsanity.com/paper/2503.03446/) (2025)
- ["Patriarchy Hurts Men Too." Does Your Model Agree? A Discussion on Fairness Assumptions](https://mlsanity.com/paper/2408.00330/) (2024)

## Abstract (arXiv)

Various measures can be used to estimate bias or unfairness in a predictor. Previous work has already established that some of these measures are incompatible with each other. Here we show that, when groups differ in prevalence of the predicted event, several intuitive, reasonable measures of fairness (probability of positive prediction given occurrence or non-occurrence; probability of occurrence given prediction or non-prediction; and ratio of predictions over occurrences for each group) are all mutually exclusive: if one of them is equal among groups, the other two must differ. The only exceptions are for perfect, or trivial (always-positive or always-negative) predictors. As a consequence, any non-perfect, non-trivial predictor must necessarily be "unfair" under two out of three reasonable sets of criteria. This result readily generalizes to a wide range of well-known statistical quantities (sensitivity, specificity, false positive rate, precision, etc.), all of which can be divided into three mutually exclusive groups. Importantly, The results applies to all predictors, whether algorithmic or human. We conclude with possible ways to handle this effect when assessing and designing prediction methods.