# Fairness Impossibility Results Between 2016 and 2017, several groups working independently proved that the standard mathematical definitions of algorithmic fairness are **mutually incompatible**. This is a theorem, not an empirical finding, and it holds regardless of how good the data is or how carefully the model is built. ## The three criteria **Calibration within groups.** Among everyone assigned a predicted probability *p*, the actual outcome rate is *p*, in every group. A score of 0.7 means the same thing for everyone. **Balance for the positive class (equal false-negative rates).** Among people who do experience the outcome, the average score is equal across groups. **Balance for the negative class (equal false-positive rates).** Among people who do *not* experience the outcome, the average score is equal across groups. Each is individually a reasonable rendering of "the system treats groups fairly." Each has been argued for in good faith. ## The theorem Kleinberg, Mullainathan and Raghavan (2016) and Chouldechova (2017) established, by different routes: > **When base rates differ between groups, no predictor can satisfy all three criteria simultaneously — except in the degenerate cases of a perfect predictor or equal base rates.** Chouldechova's version makes the arithmetic transparent: for a binary classifier, positive predictive value, false-positive rate, false-negative rate, and prevalence are algebraically linked. Fix prevalence at different levels in two groups and equalize any two of the error measures, and the third is forced apart. There is no room to maneuver — it is an accounting identity, not a modeling limitation. Since real base rates virtually always differ between groups, and no socially consequential predictor is perfect, **the impossibility binds in essentially every real deployment.** ## What this actually established The result is frequently misdescribed in both directions, and the misdescriptions are politically useful to different parties. **It does not say fairness is impossible.** It says three *specific formalizations* cannot hold at once. A system can fully satisfy any one, or trade off partially among them. **It does not excuse discriminatory systems.** "You can't have all fairness properties, so criticism is naive" is the standard bad-faith reading, and it does not follow. A system can fail *all* three criteria, and many do. The theorem constrains the ideal; it says nothing exculpatory about the actual. **It does not make the choice technical.** This is the substantive point. Because the criteria cannot be jointly satisfied, **someone must decide which one this system will honor — and that decision is irreducibly a value judgment about who should bear the cost of error.** ## Who bears the error That last framing is the one worth carrying, because it converts an abstract theorem into a distributive question with an obvious political character. - Prioritize **calibration** and you are saying scores must mean the same thing everywhere, accepting that groups with higher base rates will see more false positives. - Prioritize **equal false-positive rates** and you are saying no group should bear disproportionate wrongful adverse treatment, accepting that identical scores will carry different meanings across groups. In criminal justice, a false positive is a person wrongly detained. In lending, a person wrongly denied. **The impossibility theorem is, in effect, a proof that someone must be wronged and a demand that we say who** — a question that used to be settled implicitly by discretion and is now settled implicitly by an engineering default. That is the deeper political finding: the choice is being made either way. Before the theorem it was made invisibly. The theorem's real contribution is making it impossible to pretend otherwise. ## The governance implication The result argues against a specific and popular move: outsourcing distributive judgment to technical staff. If the fairness criterion determines who bears wrongful harm, then selecting it is a policy decision requiring the legitimacy of a policy process — public, contestable, on the record. In practice it is nearly always made by a vendor, in a model specification, disclosed to no one. Procurement documents specify accuracy. They do not specify a fairness criterion, which means the criterion is chosen by default and by whoever wrote the loss function. That is the [[Mathematics and Power Hub]] pattern exactly: **a genuine mathematical result, correctly understood, showing that a political decision has been relocated into a technical artifact where no one is accountable for it.** ## Related [[Criminal Risk Assessment Instruments]] · [[Statistical Discrimination]] · [[Predictive Policing]] · [[Credit Scoring]] · [[Actuarial Science and Risk Classification]] · [[Algorithmic Bias]] · [[Machine Learning]] · [[Mathematics and Power Hub]]