Equalized Odds

A predictor Ŷ satisfies equalized odds with respect to protected attribute A and true outcome Y if Ŷ is conditionally independent of A given Y. Formally: P(Ŷ=1|A=a, Y=y) = P(Ŷ=1|A=b, Y=y) for all a,b and y∈{0,1}. This decomposes into two simultaneous constraints:

• True Positive Rate Equality: The fraction of actual positives correctly identified must be identical across groups.
• False Positive Rate Equality: The fraction of actual negatives incorrectly flagged must be identical across groups.

Equalized Odds is a separation-based metric, not an independence-based one. Unlike Demographic Parity, which ignores ground truth labels, Equalized Odds permits a model to use the protected attribute if it is genuinely correlated with the outcome. This makes it compatible with optimal classifiers when base rates differ across groups—a critical distinction from the "levelling down" critique often leveled at Demographic Parity.

Achieving Equalized Odds often requires trade-offs with overall accuracy. When base rates differ between groups, the Bayes-optimal unconstrained classifier will naturally exhibit different error profiles. Enforcing Equalized Odds may require:

• Deliberately degrading performance on the majority group to match the minority group's error rates.
• Calibrating thresholds independently per group, a practice known as group-specific thresholding. This tension is formalized in the impossibility theorem of fairness, which states that Equalized Odds, Calibration, and Demographic Parity cannot all hold simultaneously except in degenerate cases.

A common implementation strategy is the Hardt et al. (2016) post-processing approach, which derives group-specific randomized thresholds from the model's ROC curves. The algorithm:

• Finds the intersection of feasible (TPR, FPR) pairs across all protected groups.
• Constructs a derived classifier that flips a subset of decisions to equalize error rates while minimizing accuracy loss.
• Produces a Pareto-optimal trade-off curve between fairness and performance, allowing practitioners to select an operating point.

Equalized Odds has several known weaknesses:

• Ignores calibration: A model can satisfy Equalized Odds while producing wildly different probability estimates across groups, undermining trust in the score itself.
• Self-fulfilling prophecies: If historical bias is encoded in the ground truth labels Y, equalizing error rates against biased labels perpetuates the original discrimination.
• Sample size sensitivity: Reliable estimation of false positive rates requires sufficient data per group; small subgroups yield high-variance fairness assessments.
• Does not address individual fairness: Two identical individuals from different groups can receive different outcomes as long as aggregate error rates match.

Equalized Odds aligns with the disparate impact testing frameworks required under the EU AI Act and NYC Local Law 144. Auditors operationalize it by:

• Computing disparity ratios for TPR and FPR across protected categories.
• Flagging systems where the ratio falls below a threshold (commonly 0.8, the "four-fifths rule" adapted from EEOC guidelines).
• Requiring documented justification when Equalized Odds is deliberately not enforced due to legitimate business necessity or base rate differences. The metric is typically reported alongside Equal Opportunity (which only constrains TPR) and Predictive Parity in a comprehensive fairness audit.

A fairness criterion requiring that a model's positive prediction rate is equal across all protected demographic groups, regardless of the ground truth. Unlike Equalized Odds, Demographic Parity ignores actual outcomes and focuses solely on equalizing the proportion of individuals receiving a favorable prediction. This can lead to lazy fairness where a model simply approves a fixed quota from each group, potentially introducing error disparities. It is formally expressed as P(Ŷ=1|A=a) = P(Ŷ=1|A=b) for all groups a, b.

A fairness metric that compares the rate of favorable outcomes for a protected group to that of a reference group, identifying potential indirect discrimination. The standard threshold is the 80% rule: if the ratio falls below 0.8, adverse impact is presumed. Unlike Equalized Odds, this metric does not condition on the true label, making it a measure of outcome distribution rather than error rate parity. It is widely used in U.S. employment discrimination law and lending compliance.

A causal fairness definition stating a prediction is fair if it remains the same in a counterfactual world where an individual's protected attribute was changed. This requires a structural causal model to compute what would have happened under a different demographic identity. Unlike Equalized Odds, which is purely observational, Counterfactual Fairness addresses the root causal pathways of discrimination. It is computationally intensive and depends heavily on the correctness of the assumed causal graph.

A non-protected feature that inadvertently encodes a protected attribute, leading to masked discrimination in a model. Common examples include:

• Zip code as a proxy for race
• Credit product type as a proxy for gender
• Browser language settings as a proxy for nationality Even if a model satisfies Equalized Odds on explicit protected attributes, proxy variables can reintroduce bias through redundant encodings. Detecting proxies requires mutual information analysis and domain expertise.

A fairness criterion requiring that predicted probabilities have the same semantic meaning across groups. Formally, for any predicted score s, P(Y=1|S=s, A=a) = P(Y=1|S=s, A=b). A well-calibrated model ensures that a score of 0.8 represents an 80% chance of a positive outcome regardless of group membership. Kleinberg et al. (2016) proved that Calibration and Equalized Odds are mutually exclusive unless the base rates are equal across groups or the predictor is perfect.

A relaxation of Equalized Odds that only requires equality of False Positive Rates (FPR) across groups, ignoring the True Positive Rate constraint. This is appropriate when the primary concern is avoiding unjustified interventions—for example, in pretrial detention algorithms where false positives mean unnecessary incarceration. Predictive Equality is a one-sided error rate parity that trades off against Equal Opportunity, which focuses solely on equalizing True Positive Rates.