Exposure mappings and exposure consistency

In the finite-population model of Finite-population potential outcomes under partial interference, an exposure mapping for unit $(i,j)$ is a function $f_{ij}:{\mathcal{Z}}_i\to {\mathcal{D}}_{ij}$, where ${\mathcal{Z}}_i$ is the finite set of allowed assignments to group $i$ and ${\mathcal{D}}_{ij}$ is a finite set of exposure labels. At assignment $\mathbf{Z}$, the unit's exposure is the random label $D_{ij}=f_{ij}({\mathbf{Z}}_i)$. Its design probability is ${\pi }_{ij}(d)=\mathbb{P}\{f_{ij}({\mathbf{Z}}_i)=d\}={\sum }_{\mathbf{z}\in \mathcal{Z}:f_{ij}({\mathbf{z}}_i)=d}p(\mathbf{z}),$ where $p(\mathbf{z})=\mathbb{P}(\mathbf{Z}=\mathbf{z})$ is the declared assignment law.

The mapping is sufficient for the potential outcome of unit $(i,j)$ on a set of labels if, whenever $f_{ij}({\mathbf{z}}_i)=f_{ij}({\mathbf{z}}_i^{\mathrm{\prime }})=d$, the fixed outcomes satisfy $Y_{ij}({\mathbf{z}}_i)=Y_{ij}({\mathbf{z}}_i^{\mathrm{\prime }})$. For each label in the image of $f_{ij}$ this condition defines a unique value $Y_{ij}(d)$ equal to the common outcome over its preimage. The observed-outcome consistency identity is then $Y_{ij}^{\mathrm{o}\mathrm{b}\mathrm{s}}=Y_{ij}(D_{ij})$. An exposure-specific finite-population target may include unit $(i,j)$ at label $d$ only when ${\pi }_{ij}(d)>0$; otherwise its inverse-probability estimator is undefined. Sufficiency is an assumption about the schedule and map, not a consequence of knowing the assignment probabilities.