treatment and control, i.e., $$\mu^d_\mathrm{ITR}(z) = \frac{1}{N} \sum_{i=1}^N {\mathrm{E}}^d[Y_i {\, | \,}Z_i = z_i].$$ We index these quantities by both the definition of effective treatments (ITR for “individualistic treatment response”, as in Section \[sec:treatment\_response\_assumptions\]) and the experimental design $d$, since the former determines the conditioning involved and the latter determines the distribution of $Z$ over which we take expectations.

When a vertex’s outcome depends on the treatment assignments of others, these quantities need not equal the quantities of interest. That is, they can suffer from some estimand bias, such that $
\tau^d_{\mathrm{ITR}}(1, 0) - \tau(1, 0)
$ is non-zero. Each vertex assigned to treatment contributes to this bias through the difference between its expected outcome when assigned to treatment (given the experimental design) and what would be observed under global treatment. More generally, for some global treatment vector $z$, vertex $i$ contributes to the bias of $\mu^d_\mathrm{ITR}(z)$ through $
{\mathrm{E}}^d[Y_i - Y_i(Z = z) {\, | \,}Z_i = z_i].
$ If the treatment assignment of other vertices do not affect vertex $i$’s behavior much, then this contribution might be quite small. Or this contribution could be more substantial.

### Bias reduction through design

We are now equipped to elaborate on 