Equal Opportunity

Equal opportunity requires that among the people who should receive the positive outcome, every group is identified at the same rate:

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

In plain terms: equal recall per group. If 80% of qualified applicants in one group are approved and 60% in another, qualified people in the second group are being failed by the system at twice the rate, and equal opportunity is the criterion that names it.

It comes from Hardt, Price and Srebro, who proposed it alongside the stronger equalized odds and showed how to adjust an already-trained predictor to satisfy either. The relaxation is deliberate — it concentrates on the error that denies someone something they qualified for.

Follow the qualified population

In the formula, Y is the recorded outcome, Ŷ is the decision, and A identifies the group. The calculation includes only people labeled positive: true positives divided by true positives plus false negatives. It therefore compares the chance of being selected conditional on that label. It does not require equal selection rates among all applicants, or equal numbers selected.

Consider a constructed example with 100 qualified applicants in each group. Selecting 80 qualified applicants from each gives an 80% true positive rate and a 20% false negative rate. Both groups lose the same share of qualified candidates. If one group instead contains 200 qualified applicants, selecting 160 of them would preserve the same rate; requiring exactly 80 selections would not.

Equal recall can leave unequal wrongful selections

Now suppose each group also has 100 applicants labeled unqualified. The system selects 40 of these in A and 10 in B. Equal opportunity still holds: both groups’ true positive rates remain 80%. But the false positive rates are 40% and 10%, and precision is 80/120 = 66.7% in A versus 80/90 = 88.9% in B. The constraint never examines the negative-label population, so it cannot detect this difference.

The consequences depend on the decision. Extra interviews use limited capacity; wrongful investigation flags can directly harm the people selected. Even a grant of an apparent benefit can be costly or unsuitable. Equal opportunity identifies a particular missed-opportunity disparity, not every relevant harm. A policy that selects everyone has recall of 100% in every group and still provides no discrimination between outcomes.

Reaching a target recall

When higher scores indicate a more likely positive outcome, lowering a cutoff generally increases recall by admitting more cases. The additional cases can include both true and false positives. Choose an operating point using validation data, compare the accompanying false positive rate and precision, and evaluate on data not used to select the cutoff. Group-specific score distributions may require different cutoffs for similar recall; ties or coarse scores may require a tolerance or an explicit randomized rule.

A lower numerical threshold does not by itself diagnose a worse model. Rescaling scores monotonically can move the cutoff without changing their ranking. Compare error rates at matched recall and inspect calibration before attributing a difference to poor measurement, limited data or other causes. A simulated noise difference is an illustration of a possible mechanism, not proof that underrepresentation caused a real disparity.

It inherits whatever the label inherited

Equal opportunity is defined against \(Y\), the recorded outcome, and this is usually presented as its advantage over demographic parity — it accounts for who actually qualified rather than counting heads.

The same property is its dependency. “Who actually qualified” is a record, produced by a process. If promotions were granted unequally, the promotion label encodes that. If one group was policed more heavily, the arrest label does. Equalizing true positive rates against such a label equalizes performance at reproducing a decision that was already unequal, and the criterion has no way to detect this — it is what Hardt and colleagues call an oblivious measure, depending only on the joint statistics of prediction, target and group.

Before using it, examine how Y was produced and whose outcomes are observable. For example, performance may be recorded only for applicants previously hired. Equal recall within that selected sample need not describe all applicants. Demographic parity provides a label-independent comparison of selection shares, but does not repair missing or biased labels.

Using it

• Report the false positive rate beside it, always. The two numbers together describe the error profile; either alone describes half of it.
• Check the denominators. True positive rate is computed over positives only, so in a small group with a rare outcome it may rest on a few dozen cases and move several points on noise.
• Investigate different thresholds using comparable performance measures. A lower numerical cut-off need not mean a less informative score: a strictly increasing change of scale preserves ranking while moving the threshold. Compare false-positive rates at matched recall, calibration and the score scales before diagnosing the difference.
• Check how recall parity interacts with precision. Equal recall and equal precision can coexist at different prevalences if false-positive rates differ. The stricter combination of equalized odds and predictive parity has additional feasibility conditions; the arithmetic in fairness parity must be read with those conditions.

Equal opportunity is a reasonable default when false negatives are the harm that matters and the label can be trusted. It is a claim about one of the two errors, and it should be reported as one.

References

• Hardt, Price & Srebro, Equality of Opportunity in Supervised Learning, NeurIPS 2016
• Chouldechova, Fair prediction with disparate impact, 2017