“Fairness” seems intuitive, but it has multiple formal definitions that conflict with each other. Consider a loan approval model: is it fair if it approves the same percentage of applicants from each group (demographic parity)? Or if it has the same accuracy for each group (equalized odds)? Or if a “70% approval score” means the same thing regardless of group (calibration)? These sound similar but are mathematically different — and in most real-world scenarios, you cannot satisfy all of them simultaneously. This is the central challenge of algorithmic fairness: choosing which definition of fairness to optimize for is a value judgment, not a technical decision.

Demographic parity (also called statistical parity or group fairness) requires that the model’s positive prediction rate is the same across all groups: P(Ŷ=1 | A=a) = P(Ŷ=1 | A=b) for all groups a, b. In plain English: the model should approve (or reject) the same percentage of applicants from each group. Pros: simple to understand and measure, directly addresses representation, aligns with affirmative action goals. Cons: ignores actual qualifications — if Group A has a 90% qualification rate and Group B has a 50% rate, demographic parity requires approving unqualified candidates from Group B or rejecting qualified candidates from Group A. It can reduce overall accuracy and may not be legally defensible in all contexts.

Equalized odds requires that the model has the same true positive rate (TPR) and false positive rate (FPR) across all groups: P(Ŷ=1 | Y=1, A=a) = P(Ŷ=1 | Y=1, A=b) and P(Ŷ=1 | Y=0, A=a) = P(Ŷ=1 | Y=0, A=b). In plain English: among qualified applicants, the approval rate should be the same across groups. Among unqualified applicants, the rejection rate should also be the same. Equal opportunity is a relaxed version that only requires equal TPR (same approval rate for qualified applicants). Pros: allows different base rates, focuses on accuracy rather than outcomes. Cons: requires ground truth labels, may still produce unequal outcomes if base rates differ.

Calibration (also called predictive parity) requires that a predicted probability means the same thing regardless of group: P(Y=1 | S=s, A=a) = P(Y=1 | S=s, A=b) for all scores s. In plain English: if the model gives someone a “70% risk score,” that person should actually have a 70% chance of the outcome, regardless of their group. Pros: intuitive (“scores are honest”), preserves the meaning of risk scores, important for decision-makers who use scores to set thresholds. Cons: can coexist with very different outcomes across groups (if base rates differ, calibrated scores will naturally produce different approval rates). Calibration is the metric most valued by actuaries, insurers, and risk assessors.

The impossibility theorem (Chouldechova, 2017; Kleinberg, Mullainathan & Raghavan, 2016) proves that when base rates differ between groups, demographic parity, equalized odds, and calibration cannot all be satisfied simultaneously (except in degenerate cases like a perfect predictor or equal base rates). This is not a limitation of current algorithms — it’s a mathematical impossibility. No future algorithm can overcome it. The implication: you must choose which definition of fairness to prioritize. This is a normative (value-based) decision, not a technical one. Different stakeholders may legitimately disagree on which metric matters most.

All the metrics above are group fairness metrics — they compare outcomes across demographic groups. Individual fairness takes a different approach: similar individuals should receive similar predictions, regardless of group membership. Formally: if individuals x and y are similar (according to a task-relevant distance metric), then the model’s predictions for x and y should also be similar. Counterfactual fairness is a related concept: would the prediction change if the individual belonged to a different group, all else being equal? Pros: doesn’t require defining groups, avoids the “which group?” problem. Cons: requires defining a “similarity metric,” which is itself a value judgment. What makes two people “similar” for a loan? For a job?

Choose your fairness metric based on the context: Use demographic parity when: the goal is equal representation, base rate differences are due to historical discrimination, or the application is resource allocation (advertising, opportunities). Use equalized odds when: accuracy matters equally for all groups, the application has high-stakes consequences (criminal justice, medical diagnosis), or you need to minimize both false positives and false negatives across groups. Use calibration when: the output is a risk score used for decision-making, stakeholders need to trust the meaning of scores, or the application is insurance, lending, or risk assessment. Use individual fairness when: group definitions are unclear or contested, or you want to avoid stereotyping within groups.

Several open-source tools make fairness measurement practical: Fairlearn (Microsoft) — Python library for assessing and improving fairness. Provides fairness metrics, visualization dashboards, and mitigation algorithms. AIF360 (IBM) — AI Fairness 360, a comprehensive toolkit with 70+ fairness metrics and 10+ mitigation algorithms. Aequitas (University of Chicago) — bias audit toolkit focused on decision-making systems. What-If Tool (Google) — visual interface for exploring model fairness without code. All of these compute the metrics we’ve discussed (demographic parity, equalized odds, calibration) and provide visualizations to help communicate findings to stakeholders.