Finite-population potential outcomes under partial interference

Let $\mathcal{I}$ be a finite set of treatment-assignment units partitioned into $N\ge 1$ nonempty groups ${\mathcal{I}}_1,\dots ,{\mathcal{I}}_N$, with $n_i=\mathrm{\mid }{\mathcal{I}}_i\mathrm{\mid }$. Label the members of each group ${\mathcal{I}}_i$ by $j=1,\dots ,n_i$. Let $\mathcal{Z}\subseteq \{0,1{\}}^{\mathcal{I}}$ be the nonempty finite set of allowed global assignments in a finite randomized treatment-assignment model (Finite randomized treatment-assignment models, Finite probability spaces, outcome weights, events, and event probabilities). Write ${\mathbf{z}}_i$ for the restriction of $\mathbf{z}\in \mathcal{Z}$ to group $i$, and $Z_{ij}$ for unit $j$'s coordinate in its group.

A fixed finite-population potential-outcome schedule is a family of real-valued functions $Y_{ij}:\mathcal{Z}\to \mathbb{R}$, one for every unit $(i,j)$. The observed outcome at assignment $\mathbf{Z}$ is $Y_{ij}^{\mathrm{o}\mathrm{b}\mathrm{s}}=Y_{ij}(\mathbf{Z})$. The schedule is fixed; only the declared assignment is random. This notation permits arbitrary dependence of $Y_{ij}(\mathbf{z})$ on every component of $\mathbf{z}$.

The schedule satisfies partial interference when, for every unit $(i,j)$ and all allowed $\mathbf{z},{\mathbf{z}}^{\mathrm{\prime }}\in \mathcal{Z}$, equality of their within-group assignments ${\mathbf{z}}_i={\mathbf{z}}_i^{\mathrm{\prime }}$ implies $Y_{ij}(\mathbf{z})=Y_{ij}({\mathbf{z}}^{\mathrm{\prime }})$. In that case the common value is well-defined as $Y_{ij}({\mathbf{z}}_i)$: outcomes may depend arbitrarily on assignments within group $i$, but not on assignments outside that group. Partial interference is a substantive model/design restriction, not a consequence of randomization.