An example of prediction which complies with Demographic Parity and equalizes group-wise risks in the context of regression

Let (X, S, Y) ∈ ℝp × 1, 2 × ℝ be a triplet following some joint distribution mathbbP with feature vector X, sensitive attribute S , and target variable Y. The Bayes optimal prediction f^* which does not produce Disparate Treatment is defined as f^*(x) = mathbbE[Y | X = x]. We provide a non-trivial example of a prediction x → f(x) which satisfies two common group-fairness notions: Demographic Parity beginalign (f(X) | S = 1) &stackreld= (f(X) | S = 2) endalign and Equal Group-Wise Risks beginalign mathbbE[(f^*(X) - f(X))2 | S = 1] = mathbbE[(f^*(X) - f(X))2 | S = 2]. endalign To the best of our knowledge this is the first explicit construction of a non-constant predictor satisfying the above. We discuss several implications of this result on better understanding of mathematical notions of algorithmic fairness.