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Let $(X, S, Y) \in \mathbb{R}^p \times \{1, 2\} \times \mathbb{R}$ be atriplet 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 $x \tof(x)$ which satisfies two common group-fairness notions: Demographic Parity\begin{align} (f(X) | S = 1) &\stackrel{d}{=} (f(X) | S = 2) \end{align} andEqual 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 explicitconstruction of a non-constant predictor satisfying the above. We discussseveral implications of this result on better understanding of mathematicalnotions of algorithmic fairness.

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