For the Average Treatment Effect (ATE) in causal inference, is it usually defined as
$$ E(Y(1) - Y(0)) $$
I am wondering what the most commonly referred to name for $E(Y(1))$ is? Is it not the average treatment effect on the treated (ATT), but calling it the expected response for treatment doesn't seem quite there. Is there a formal name for it in the literature?
ATE is the difference in potential outcomes if an individual is treated and if they are not treated, averaged over all individuals. So you need to add subscripts:
$$ ATE = E[Y_{1i} - Y_{0i}] $$
where 0 and 1 denote whether an individual is treated (1) or not treated (0). The expected value operator then gives us the average of the difference in potential outcomes across all individuals in the population. In other words, it gives the average of individual-level causal effects.
So to answer your question, $E[Y_{1i}]$ then just gives us the average of potential outcomes for individuals if they are all treated. These are potential outcomes, not necessarily observed for all individuals. If everyone was in fact treated, then this would give you average outcome for the population. Instead, this is the average outcome had everyone been treated.
This is a great question, because the important book by Judea Pearl ("Causality", 2nd ed.) actually calls
$P(Y|do(X = x)) = P(Y(x))$
the "causal effect" of $X$ on $Y$ (Definition 3.2.1), so that
$E[Y(x)] = \sum_y y P(Y(x) = y)$
would be the "average causal effect".
Although the general focus on the ATE as a difference makes sense, there are a few good reasons to focus on $E[Y(x)]$. First, the ATE can always be constructed from $E[Y(x)]$ for varying $x$, but not the other way around. Second, $E[Y(x)]$ is indeed the average outcome after one has implemented $X = x$ in the whole population, which in itself is interesting to know.
I personally would call $E[Y(x)]$ the "post-intervention mean", just like Pearl also sometimes calls $P(Y(x))$ the "post-intervention distribution".
