SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Then any decision policy  $D$  satisfying benefit fairness also satisfies demographic parity.
$$
\begin{array}{l} P (d \mid x _ {1}) = \sum_ {\delta} P (d \mid x _ {1}, \Delta = \delta) P (\Delta \mid x _ {1}) (101) \\ = \sum_ {\delta} P (d \mid x _ {0}, \Delta = \delta) P (\Delta = \delta \mid x _ {1}) \quad \text {u s i n g B F} (102) \\ = \sum_ {\delta} P (d \mid x _ {0}, \Delta = \delta) P (\Delta = \delta \mid x _ {0}) \quad \text {u s i n g} \Delta \mid x _ {1} \stackrel {d} {=} \Delta \mid x _ {0} (103) \\ = P \left(d \mid x _ {0}\right), (104) \\ \end{array}
$$
implying demographic parity.
G Practical Considerations for Decompositions in Alg. 2
In this appendix, we discuss some practical details about how to perform the decompositions of the treatment disparity and benefit disparity as described in Alg. 2. In particular, we begin by showing that the benefit  $\Delta$  can be considered as variable in the causal diagram.
Proposition 11 (Connection of Benefit with Structural Causal Models). Let  $\mathcal{M}$  be an SCM compatible with the SFM in Fig. 1. Let the benefit  $\Delta$  be defined as in Eq. 14. Let  $f_{Y}$  be the structural mechanism of the outcome  $Y$ , taking  $X, Z, W, D$  as inputs, together with the noise variable  $U_{Y}$ . The structural mechanism for the benefit  $\Delta$  is then given by:
$$
f _ {\Delta} (x, z, w) = \mathbb {E} _ {u _ {y}} \left[ f _ {Y} (x, z, w, d _ {1}, u _ {y}) \right] - \mathbb {E} _ {u _ {y}} \left[ f _ {Y} (x, z, w, d _ {0}, u _ {y}) \right], \tag {105}
$$
where  $\mathbb{E}_{u_y}$  integrates over the randomness in  $U_{Y}$ . Therefore, the benefit  $\Delta$  is a deterministic function of  $X$ ,  $Z$ ,  $W$ , and we can add it to the causal diagram as follows:
Therefore, after adding  $\Delta$  to the causal diagram as in the above proposition, we can view it as another outcome variable. In particular, it also allows us to define various counterfactuals with respect to it, via the standard definitions for counterfactuals used in structural causal models. For instance, the potential outcome  $\Delta_{x_0}$  (which is a random variable) is simply given by  $f_{\Delta}(x_0,Z,W)$ , where  $Z,W$  are random variables but  $X = x_0$  is fixed. Other, possibly nested counterfactuals are also defined analogously.
Equipped with a structural understanding of how benefit relates to the original causal model, we can now explain in more detail how the decompositions in Alg. 2 are performed. In particular, the decompositions follow the previously proposed approaches of [44, 31]:
Proposition 12 (Computing Decompositions in Alg. 2). Let  $\mathcal{M}$  be an SCM compatible with the SFM in Fig. 1. Let the benefit  $\Delta$  be defined as in Eq. 14 and suppose it is added to the causal diagram as in Prop. 11. The decompositions of the disparity in treatment allocation, and the disparity in expected
benefit, can be computed as follows:
$$
\begin{array}{l} P (d \mid x _ {1}) - P (d \mid x _ {0}) = \left(\underbrace {P \left(d _ {x _ {1} , W _ {x _ {0}}} \mid x _ {0}\right) - P \left(d _ {x _ {0}} \mid x _ {0}\right)} _ {\text {d i r e c t e f f e c t D E} (X \rightarrow D)}\right) (106) \\ - \left(\underbrace {P \left(d _ {x _ {1} , W _ {x _ {0}}} \mid x _ {0}\right) - P \left(d _ {x _ {1}} \mid x _ {0}\right)} _ {\text {i n d i r e c t e f f e c t I E} (X \rightarrow D)}\right) (107) \\ - \left(\underbrace {P \left(d _ {x _ {1}} \mid x _ {0}\right) - P \left(d _ {x _ {1}} \mid x _ {1}\right)} _ {\text {s p u r i o u s e f f e c t S E} (X \leftrightarrow - \rightarrow D)}\right). (108) \\ \end{array}
$$
$$
\begin{array}{l} \mathbb {E} \left(\Delta \mid x _ {1}\right) - \mathbb {E} \left(\Delta \mid x _ {0}\right) = \left(\underbrace {\mathbb {E} \left(\Delta_ {x _ {1} , W _ {x _ {0}}} \mid x _ {0}\right) - \mathbb {E} \left(\Delta_ {x _ {0}} \mid x _ {0}\right)} _ {\text {d i r e c t e f f e c t D E} (X \rightarrow \Delta)}\right) (109) \\ - \left(\underbrace {\mathbb {E} \left(\Delta_ {x _ {1} , W _ {x _ {0}}} \mid x _ {0}\right) - \mathbb {E} \left(\Delta_ {x _ {1}} \mid x _ {0}\right)} _ {\text {i n d i r e c t e f f e c t I E} (X \rightarrow \Delta)}\right) (110) \\ - \left(\underbrace {\mathbb {E} \left(\Delta_ {x _ {1}} \mid x _ {0}\right) - \mathbb {E} \left(\Delta_ {x _ {1}} \mid x _ {1}\right)} _ {\text {s p u r i o u s e f f e c t S E (X \leftrightarrow \rightarrow \Delta)}}\right). (111) \\ \end{array}
$$
Furthermore, under the assumptions of the SFM in Fig. 1 both of the decompositions are identifiable and their identification expressions are given by
$$
D E (X \rightarrow F) = \sum_ {z, w} \left[ \mathbb {E} (F \mid x _ {1}, z, w) - \mathbb {E} (F \mid x _ {0}, z, w) \right] P (w \mid x _ {0}, z) P (z \mid x _ {0}), \tag {112}
$$
$$
I E (X \rightarrow F) = \sum_ {z, w} \mathbb {E} (F \mid x _ {1}, z, w) \left[ P (w \mid x _ {0}, z) - P (w \mid x _ {1}, z) \right] P (z \mid x _ {0}), \tag {113}
$$
$$
S E (X \leftrightarrow F) = \sum_ {z} \mathbb {E} (F \mid x _ {1}, z) \left[ P (z \mid x _ {0}) - P (z \mid x _ {1}) \right], \tag {114}
$$
where the random variable  $F$  is either replaced by the treatment decision  $D$  to obtain the direct, indirect, and spurious terms in Eqs. 106-108 or by the benefit  $\Delta$  to obtain the terms in Eqs. 109-111.