Equalized Odds (Fairness)

Equalized odds requires that both error rates match across groups — within the positive class, the flagging rate is equal across groups; within the negative class, it is also equal across groups:

\[ P(\hat{Y}=1 \mid Y=y, A=a) = P(\hat{Y}=1 \mid Y=y, A=b) \quad \forall \, y \in \{0,1\}, \; \forall \, a, b \]

It is the two-sided version of equal opportunity, which asks only for the \(y=1\) half, and both come from Hardt, Price and Srebro. Equalized odds does imply equal opportunity — the second is a relaxation of the first.

That containment is real, and it is often extended into a ranking that is not: demographic parity weakest, equal opportunity stronger, equalized odds strongest. Demographic parity is not part of that implication chain.

Two comparisons, each with its own denominator

The true positive rate is TP/(TP + FN), measured among actual positives. The false positive rate is FP/(FP + TN), measured among actual negatives. Equalized odds requires each of these rates to match across groups. It does not require TPR to equal FPR within a group. An informative system can have a high TPR and a low FPR while satisfying the criterion.

For a constructed example, give each group 1,000 people. A contains 200 actual positives and 800 negatives; B contains 500 positives and 500 negatives. At TPR = 80% and FPR = 10% in both groups, A has 160 true positives and 80 false positives, while B has 400 true positives and 50 false positives. The system misses 20% of actual positives and wrongly flags 10% of actual negatives in either group, so equalized odds holds exactly in these counts.

Selection rates are nevertheless 24% and 45%. Precision is 160/240 = 66.7% in A and 400/450 = 88.9% in B. The difference comes from mixing positives and negatives in different proportions. If p is a group’s positive-label prevalence, its selection rate is p × TPR + (1 − p) × FPR. Matching the two conditional rates does not generally match this prevalence-weighted average.

Conversely, equal selection rates can conceal unequal error rates. Among 100 positives and 100 negatives in each group, a policy selecting 80 positives and 20 negatives in A but 60 positives and 40 negatives in B selects half of each group. It satisfies demographic parity, but its TPRs are 80% and 60%, and its FPRs are 20% and 40%. Neither criterion implies the other.

Different prevalences also constrain predictive parity. If equalized odds holds at a positive TPR and PPV lies strictly between zero and one, equal PPV cannot also hold at different prevalences. Boundary cases matter: FPR = 0 with the same positive TPR gives PPV = 1 in both groups, even when some positives are missed. Approximate parity requires analysis of the actual tolerances rather than a blanket impossibility claim.

Why one threshold may not match both rates

Changing a threshold moves both TPR and FPR. Plotting FPR horizontally and TPR vertically gives a group’s ROC curve, showing the operating points available as its threshold changes. Equalized odds needs a common point for both groups. Two deterministic threshold rules need not offer a useful common point; matching recall alone can leave their false positive rates different.

Randomization can create additional points. If one rule has (FPR, TPR) = (0.10, 0.60) and another has (0.30, 0.90), choosing either rule with probability one-half independently of the outcome gives expected rates (0.20, 0.75). Such mixtures expand the feasible operating points. Hardt and colleagues use this idea in post-processing learned predictors. Randomized population rates do not guarantee exact finite-sample counts, and the acceptability of randomizing consequential decisions must be assessed in the application.

Parity alone permits unhelpful policies, including rejecting everyone or selecting everyone. Choose among feasible policies using the costs of both errors, required performance and operational constraints. A tolerance may allow a more useful policy than exact matching, but its justification must concern the resulting harms and uncertainty. It should not be chosen after seeing which value makes a report pass.

When it is the right criterion

Equalized odds is relevant when differences in both kinds of error deserve attention, such as missed disease and unnecessary follow-up in screening. Equal rates are still not identical harms: the severity of an error can vary by individual, and a group average can conceal differences within the group. Define the action and the outcome label before interpreting the two rates.

Constraining both rates can restrict the available policies when the main concern is missed positives; equal opportunity is the targeted version. And like every criterion defined against \(Y\), it is only as meaningful as the recorded outcome — if the label came from an unequal process, equalizing error rates against it equalizes fidelity to that process.

Whichever is chosen, report the other three at the same operating point. Include the counts behind each rate and uncertainty. A single satisfied criterion leaves other aspects of the decision unexamined. The arithmetic behind the incompatibility is set out in fairness parity.

References

• Hardt, Price & Srebro, Equality of Opportunity in Supervised Learning, NeurIPS 2016
• Chouldechova, Fair prediction with disparate impact, 2017
• Kleinberg, Mullainathan & Raghavan, Inherent Trade-Offs in the Fair Determination of Risk Scores, ITCS 2017