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

Let $(X, S, Y) \in \mathbb{R}^p \times \{1, 2\} \times \mathbb{R}$ be a triplet following some joint distribution $\mathbb{P}$ 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 $f^*(x) = \mathbb{E}[Y | X = x]$. We provide a non-trivial example of a prediction $x \to f(x)$ which satisfies two common group-fairness notions: Demographic Parity \begin{align} (f(X) | S = 1) &\stackrel{d}{=} (f(X) | S = 2) \end{align} and Equal Group-Wise Risks \begin{align} \mathbb{E}[(f^*(X) - f(X))^2 | S = 1] = \mathbb{E}[(f^*(X) - f(X))^2 | S = 2]. \end{align} 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.