Causal inference where potential outcome is somehow "violated"? The fundamental problem of causal inference says that only one potential outcome is observed for each unit.
What happens if both outcomes from control and treatment can be observed? Can we still make use of analysis tools like causal trees to understand heterogeneous treatment effects?
As a concrete example, suppose we are an online search engine and want to better understand how to serve ads. Each time a user enters a search query (request), we pick an ad from a collection of $N$ ads and show it to the user. For each of these $N$ ads, there are 2 versions, one with an image and one without. We randomize users into a control group and a treatment group (same distribution of users in each group), where users in the control group will see ads without an image and users in the treatment group will see ads with an image. By the end of the experiment, for each ad we record the total number of clicks by users in the control group as well as in the treatment group.
In this particular case, each ad is an experimental unit, and we are able to observe outcomes from both the treatment group and the control group. We want to understand if we should add an image to the ads or not.
In addition, each ad has its feature (e.g., associated company, promoted product, etc.), and we want to understand which group of ads will benefit the most from the addition of images. My question is, in such a case, can we follow the routines in treatment effect analysis? If not, what's a more suitable framework?

I would set up your data as an ad-level crosssection, where:

*

*Each row is a distinct ad

*Ad characteristics are the other columns.

*The outcome column is the treatment-control difference in the two clicks per impression rates.

*I would also include the number of impressions that the difference is calculated from as a column, in case some ads appear more frequently than others.

You can fit a model of the effect as a function of the ad-level variables, possibly using the number of impressions to weight the data. If you have few ad characteristics relative to ads, linear regression or a non-parametric model will work. If you have many covariates relative to the number of ads, the lasso modified for inference is a good first start:
Belloni, Alexandre, Victor Chernozhukov, and Christian Hansen. 2014. "High-Dimensional Methods and Inference on Structural and Treatment Effects." Journal of Economic Perspectives, 28 (2): 29-50. https://www.aeaweb.org/articles?id=10.1257/jep.28.2.29
This modification is similar in spirit to the causal trees approach you mentioned.
