If the base rates (prevalence of outcomes) differ between groups, it is mathematically impossible to satisfy demographic parity, predictive rate parity, and equal opportunity simultaneously.

If the base rates (prevalence of outcomes) differ between groups, it is mathematically impossible to satisfy demographic parity, predictive rate parity, and equal opportunity simultaneously.

• A. Equal Opportunity
• B. Mathematical Impossibility
• C. Problem Specification
• D. Transparency

Answer: (B)

When the base rates differ between groups, you can’t satisfy those three fairness criteria at the same time. Demographic parity requires the same rate of positive predictions for every group, regardless of actual outcomes. Equal opportunity requires the true positive rate to be the same across groups. Predictive rate parity (precision) requires the positive predictive value to be the same across groups. If you try to enforce equal positive prediction rates while also keeping equal true positive rates, the differing base rates force you to tune thresholds differently for each group. Those different thresholds then typically cause the precision to diverge across groups. Conversely, making the precision equal across groups while keeping the same true positive rate and the same predicted-positives rate leads to a contradiction unless the base rates are equal. So with differing base rates, you can’t have all three hold simultaneously. That’s why the statement is described as a mathematical impossibility.

When the base rates differ between groups, you can’t satisfy those three fairness criteria at the same time. Demographic parity requires the same rate of positive predictions for every group, regardless of actual outcomes. Equal opportunity requires the true positive rate to be the same across groups. Predictive rate parity (precision) requires the positive predictive value to be the same across groups.

If you try to enforce equal positive prediction rates while also keeping equal true positive rates, the differing base rates force you to tune thresholds differently for each group. Those different thresholds then typically cause the precision to diverge across groups. Conversely, making the precision equal across groups while keeping the same true positive rate and the same predicted-positives rate leads to a contradiction unless the base rates are equal.

So with differing base rates, you can’t have all three hold simultaneously. That’s why the statement is described as a mathematical impossibility.