I think this ultimately comes down to the problem of attribution in marketing -- how do you determine if an ad or story is effective in actually influencing somebody to buy a product or vote for a candidate? We know millions of people engaged with content from Russian trolls masquerading as Americans, but (like any marketing campaign with an offline action) it's difficult to quantitatively measure the ultimate impact they had.
Yeah, but even a simple survey would at least start to unravel this. "Did you either fail to vote or change your vote based upon paid advertising you saw on Facebook?" would at least be a good start. Even anecdotal stories of people being swayed by a paid Facebook ad would be a start. I haven't seen a single one, and I've looked.
The whole point of using the data like this is to change people's opinion without them knowing why, so I doubt anyone can answer a survey like this accurately.
Perhaps if there were two similar candidates, this would be true. However, that wasn't the case here. These candidates and their supporters were polar opposites. If they were swayed at all, it wouldn't have been through subtlety. The stories would be "I was going to vote for Hillary, but then I saw [X] on Facebook and was so horrified that I decided to [not vote at all or vote for Trump]".
I'm pretty sure that polls like that are ineffective for discerning the impact of any type of marketing. The best evidence I can think of for whether somebody found something influential is whether they liked or re-shared a post, and there's plenty of evidence for that. Those are, after all, the sorts of metrics typically used for measuring the success of a social media campaign: https://www.socialmediaexaminer.com/10-metrics-to-track-for-...
https://www.theatlantic.com/technology/archive/2018/02/the-r...