Equalized Odds

Equalized Odds requires simultaneous equality of True Positive Rates (TPR) and False Positive Rates (FPR) across all protected groups. This is stricter than Demographic Parity, which only cares about prediction rates. A classifier satisfies Equalized Odds if, for any two groups A and B:

• P(Ŷ=1 | Y=1, A) = P(Ŷ=1 | Y=1, B)
• P(Ŷ=1 | Y=0, A) = P(Ŷ=1 | Y=0, B) This ensures the model's mistakes are not systematically biased against one group.

In the fairness taxonomy, Equalized Odds is formally known as the Separation criterion. It mandates that the model's score distribution be independent of the protected attribute, conditional on the true outcome. This means the classifier's output can only correlate with group membership through the target variable itself. Any residual correlation after conditioning on the true label constitutes a violation.

The COMPAS recidivism prediction tool became the canonical example for Equalized Odds. A ProPublica investigation found that while the tool had similar AUC across racial groups, it failed Equalized Odds:

• False Positive Rate for Black defendants: 44.9%
• False Positive Rate for White defendants: 23.5%
• False Negative Rate for White defendants: 47.7%
• False Negative Rate for Black defendants: 28.0% This disparity meant Black defendants were nearly twice as likely to be incorrectly flagged as high-risk.

A fundamental theoretical result shows that Equalized Odds and Calibration (predictive value parity) are mutually incompatible outside of trivial cases where base rates are equal across groups. If a classifier is well-calibrated—meaning a score of 0.7 represents a 70% chance of recidivism for all groups—it cannot simultaneously satisfy Equalized Odds when base rates differ. Practitioners must choose which fairness definition aligns with their ethical and legal context.

Equalized Odds can be enforced through post-processing techniques that adjust decision thresholds independently for each group. The seminal approach by Hardt et al. (2016) derives optimal ROC-based threshold classifiers that minimize error while satisfying the Equalized Odds constraint. This method does not require retraining the model, making it suitable for auditing and correcting existing black-box classifiers without access to the training pipeline.

Equal Opportunity is a relaxed variant of Equalized Odds that only constrains the True Positive Rate across groups, ignoring the False Positive Rate. Formally:

• P(Ŷ=1 | Y=1, A) = P(Ŷ=1 | Y=1, B) This is appropriate in scenarios like hiring or college admissions where the primary concern is ensuring equally qualified candidates from all groups have the same chance of being selected, without necessarily equalizing rejection errors.

A relaxed variant of Equalized Odds that focuses only on the True Positive Rate (TPR) across groups. It requires that qualified individuals have an equal chance of being correctly identified, regardless of group membership. This metric is specifically concerned with equality in benefiting from a positive prediction, ignoring disparities in False Positive Rates. It is mathematically expressed as: P(Ŷ=1|Y=1, A=a) = P(Ŷ=1|Y=1, A=b).

The complementary constraint to Equal Opportunity, focusing only on the False Positive Rate (FPR). It requires that the probability of an unqualified individual being incorrectly classified as positive is the same across all groups. This metric ensures that no group is disproportionately burdened by erroneous positive predictions. Mathematically: P(Ŷ=1|Y=0, A=a) = P(Ŷ=1|Y=0, A=b).

The formal statistical term for the condition satisfied by Equalized Odds. A classifier satisfies separation if its prediction score (Ŷ) is conditionally independent of the protected attribute (A) given the true outcome (Y). This means all the information about group membership is contained within the ground-truth label. It is a stronger condition than sufficiency but weaker than independence.

A fairness criterion that is mutually exclusive with Equalized Odds in most real-world scenarios (except when base rates are equal). A classifier satisfies sufficiency if the true outcome (Y) is conditionally independent of the protected attribute (A) given the prediction score (Ŷ). This ensures that a given score means the same thing for all groups—i.e., the Positive Predictive Value (PPV) is equalized. It is often preferred in risk assessment contexts.

A fairness definition closely related to sufficiency. A classifier is well-calibrated if, for every predicted probability score, the proportion of true positives is identical across all groups. For example, if a model assigns a 70% risk score, exactly 70% of individuals in every group receiving that score should experience the event. This is critical for decision-making but is mathematically incompatible with Equalized Odds when base rates differ.

A foundational result in algorithmic fairness proving that Equalized Odds, Calibration, and non-trivial predictions cannot coexist when base rates differ between groups. Formally proven by Kleinberg et al. (2016) and Chouldechova (2017), this theorem forces practitioners to choose which fairness criteria to prioritize. It demonstrates that no single metric can satisfy all intuitive notions of fairness simultaneously in a realistic setting.