I am reading the paper "Structural Causal Bandits: Where to intervene?" by Lee & Bareinboim (2018). Here is a link: https://papers.nips.cc/paper_files/paper/2018/file/c0a271bc0ecb776a094786474322cb82-Paper.pdf
I am trying to expand their results to the multi-objective case, that is, there are multiple target variables $\mathbf{Y} = \{Y_1,\dots,Y_m \}$. Here is a reformulation the definition for possibly-optimal minimal intervention sets:
Definition. Given $\langle \mathcal{G}, \mathbf{Y}, \mathbf{X} \rangle$, let $\mathbf{X}_s$ be a minimal intervention set and $\mathbf{x}_s \in \mathcal{D}(\mathbf{X}_s)$. Then, $\mathbf{X}_s$ is called possibly-optimal if there is no other minimal intervention set $\mathbf{X}_s'$ and $\mathbf{x}_s' \in \mathcal{D}(\mathbf{X}_s')$ with $\mathbb{E}[Y_i | \text{do}(\mathbf{X}_s = \mathbf{x}_s)] \geq \mathbb{E}[Y_i | \text{do}(\mathbf{X}_s'=\mathbf{x}_s')]$ for all $1 \leq i \leq m$ and $\mathbb{E}[Y_i | \text{do}(\mathbf{X}_s = \mathbf{x}_s)] > \mathbb{E}[Y_i | \text{do}(\mathbf{X}_s'=\mathbf{x}_s')]$ for at least one $1 \leq i \leq m$, for some SCM conforming to $\mathcal{G}$.
I want to prove proposition 2 for the multi objective case. Here is a reformulation of the proposition in the multi-objective case and the proof I have come up with:
Proposition. If no $Y_i$ is confounded with $\text{an}(Y_i)_{\mathcal{G}}$ via unobserved confounders, then $\text{pa}(\mathbf{Y})_{\mathcal{G}}$ is the only possibly-optimal minimal intervention set.
Proof. Let $\mathbf{X}_s$ be a minimal intervention set and $\mathbf{x}_s \in \mathcal{D}(\mathbf{X}_s)$. Let $\mathbf{X}_s' = \text{pa}(\mathbf{Y})_{\mathcal{G}}$ and $\mathbf{Z} = \mathbf{X}_s' \backslash (\mathbf{X}_s \cap \mathbf{X}_s')$. Then, for all $i=1,\dots,m$ it holds \begin{align} \mathbb{E}[Y_i | \text{do}(\mathbf{X}_s=\mathbf{x}_s)] &= \int_{\mathcal{D}(\mathbf{Z})} \mathbb{E}[Y_i | \text{do}(\mathbf{X}_s=\mathbf{x}_s), \mathbf{Z} = \mathbf{z}] P(\mathbf{Z} = \mathbf{z} | \text{do}(\mathbf{X}_s=\mathbf{x}_s)) d\mathbf{z} \\ &= \int_{\mathcal{D}(\mathbf{Z})} \mathbb{E}[Y_i | \text{do}(\mathbf{X}_s=\mathbf{x}_s, \mathbf{Z} = \mathbf{z})] P(\mathbf{Z} = \mathbf{z} | \text{do}(\mathbf{X}_s=\mathbf{x}_s)) d\mathbf{z}\\ &= \int_{\mathcal{D}(\mathbf{Z})} \mathbb{E}[Y_i | \text{do}(\mathbf{X}_s \cap \mathbf{X}_s'=\mathbf{x}_s[\mathbf{X}_s'], \mathbf{Z} = \mathbf{z})] P(\mathbf{Z} = \mathbf{z} | \text{do}(\mathbf{X}_s=\mathbf{x}_s)) d\mathbf{z} \\ & \geq \int_{\mathcal{D}(\mathbf{Z})} \mathbb{E}[Y_i | \text{do}(\mathbf{X}_s'=\mathbf{x}_s')] P(\mathbf{Z} = \mathbf{z} | \text{do}(\mathbf{X}_s=\mathbf{x}_s)) d\mathbf{z}\\ & = \mathbb{E}[Y_i | \text{do}(\mathbf{X}_s'=\mathbf{x}_s')] \end{align} where $\mathbf{x}_s' \in \mathcal{D}(\mathbf{X}_s')$ is a non-dominated intervention value. Note that the second equality is derived through the second rule of do-calculus and the third equality assumes that $Y_i$ is not confounded with $\text{an}(Y_i)_{\mathcal{G}}$ via unobserved confounders. If $\mathbf{X}_s \neq \text{pa}(\mathbf{Y})_{\mathcal{G}}$, it is possible to construct an \textsc{scm}, conforming to $\mathcal{G}$, such that strict inequality holds for at least one $Y_i$. This shows that $\mathbf{X}_s'$ can be the only possibly-optimal minimal intervention set.
Unfortunately, I am pretty sure that this proof is incorrect due to the fact that the inequality does not need to hold for all $\mathbf {z} \in \mathbf{Z}$. Can someone help me? That would be highly appreciated!
