reply* model [@blume_statistical_1995], when vertices anticipate neighbors taking the same action in the present round as they did in the previous round. In particular, we can interpret $Y_{i, t}^*$ as the payoff for vertex $i$ to adopt behavior 1 at time $t$. When $\gamma > 0$, then this is a semi-anonymous graphical game with strategic complements [@jackson_social_2008 Ch. 9].

Analysis and estimation {#estimation}
-----------------------

We focus on the ATE (the average treatment effect; $\tau$ in Equation \[tau\_is\_ate\]), which is naturally of interest when considering whether a new treatment would be beneficial if applied to all units.

There are many options available for estimating the ATE. For example, if the relevant network is completely unknown or if peer effects are not expected, then one might use estimators for experiments without interference, such as a simple difference-in-means between the outcomes of vertices assigned to treatment and control. To clarify the sources of error in estimation, we begin with the population analogs of these quantities — i.e., the associated estimands — and return to the estimators themselves in Section \[sec:estimators\]. Consider the simple difference-in-means estimand $$\label{eq:tau_itr}
\tau^d_{\mathrm{ITR}}(1, 0) = \mu^d_\mathrm{ITR}(1) - \mu^d_\mathrm{ITR}(0)$$ where the $\mu^d_\mathrm{ITR}$ are mean outcomes when a vertex is in 