terms of estimators’ bias and error, under varied (although simple) decision mechanisms. Before considering these processes themselves, we consider outcomes as a function of treatment assignment.

### Treatment response assumptions {#sec:treatment_response_assumptions}

In the following presentation, we use the language of “treatment response” assumptions developed by @manski_identification_2013 to organize our discussion of outcome generation. Consider vertices’ outcomes as determined by a function from the global treatment assignment $Z \in \mathbb{Z}^N$ and an independent stochastic component $U \in \mathbb{U}^N$ to an outcome vector $Y \in \mathbb{Y}^N$: $$f(\cdot): \mathbb{Z}^N \times \mathbb{U}^N \rightarrow \mathbb{Y}^N.$$ We then observe $Y = f(Z, U)$. We can decompose this function into a function for each vertex $$f_i(\cdot): \mathbb{Z}^N \times \mathbb{U}^N \rightarrow \mathbb{Y}.$$ We can, as we have done above, continue to write $Y_i(Z = z)$ to refer to the outcome for vertex $i$ that would be observed under assignment $z$; by suppressing dependence on $U$, this treats $Y_i(\cdot)$ as a stochastic function.

If vertices’ outcomes are not affected by others’ treatment assignment, then SUTVA is true. Perhaps more felicitously, @manski_identification_2013 calls this assumption *individualistic treatment response* (ITR). Under ITR we could then consider vertices as having a function from only their own assignment to their outcome: $$f_i(\cdot): \mathbb{Z} 