Table 1. Overview of exact fairness definitions.

Overview of our exact derived necessary, sufficient and easy-to-test fairness conditions (corollary 2.6 and 2.13) based on the corresponding definitions of Barocas, Hardt & Narayanan (2019) (definition 2.5 and 2.12).

	Definition according to Barocas et al.	Derived necessary and sufficient condition
Independence	$\hat{Y}⫫S$	$\mathbb{P}(\hat{Y}=y|S=s_{k_1})$
		=
		$\mathbb{P}(\hat{Y}=y|S=s_{k_2})$
		$\mathrm{\forall }y\in \mathcal{Y},s_{k_1},s_{k_2}\in \mathcal{S}$
Separation	$\hat{Y}⫫S|Y$	$\mathbb{P}(\hat{Y}=\hat{y}|S=s_{k_1},Y=y)$
		=
		$\mathbb{P}(\hat{Y}=\hat{y}|S=s_{k_2},Y=y)$
		$\mathrm{\forall }y,\hat{y}\in \mathcal{Y},s_{k_1},s_{k_2}\in \mathcal{S}$