Aren't these badges supposed to symbolize and showcase distinguished achievements (such as asking good questions, reviewing stacks, etc.)? You should be proud of having these badges, and I think the rest of the community thinks so too, but this "peer pressure" badge seems to have a negative impact. I personally don't think this badge is worth trying to get (and whether or not I try to get my posts downvoted play a big factor in that), and I vote for its removal. Does anybody else think so, and why? If not, then why not?
I agree the badge is somewhat pointless, but I don't think it's detrimental to the site.
I think it's fair to say that most people here not only care about their "reputation" in terms of the site's points, but also their intellectual reputation they convey when asking and answering questions. This provides an incentive for users to not write poor posts in the first place- they don't want their contributions to be seen negatively.
However, consider the user who has made a mistake or unintentionally wrote a post he/she realized was poorly received. Under the same conscientiousness mentioned in the previous paragraph, this user probably doesn't need a badge to know he/she should delete their comment.
Of course, every now and then, you will stumble across users who have forgotten to delete their scores of $-7$ or $-8$. These badges encourage users to "try new features" and delete their poor responses.
If there are a select few writing poor posts just to delete them instantly, then I highly doubt they're a burden to the system much. Usually there is something wrong with their posts, and the site's moderators do a fine job of closing flagged posts that accompany poorly-received posts.
Bottom line? Having the badge isn't much of an issue. Furthermore, if it was an issue, its removal should probably be discussed on meta.stackexchange.com instead of Math Meta.
I recently learned about the existence of this badge ... after it was "awarded" to me. I just shrugged it off. I had posted a correct and complete answer (I know when I do) and it kept being downvoted, along with several other correct answers to that same problem. It seemed strange, I flagged it, this didn't seem to change things much, I got tired of it (seeing new downvotes for that same answer for several days, and not just my answer, but also as I said other correct answers to that question), and I deleted my answer, after I decided I didn't want to have anything to do with that question.
Now, I tried to look at that same question, so I could provide a link. I got a "Page not found", with the explanation This question was removed from Mathematics Stack Exchange for reasons of moderation. https://math.stackexchange.com/questions/1022883/let-xn-be-a-bounded-but-not-convergent-sequence-prove-that-xn-has-two-subse/1022892#1022892
But, I seem to keep the badge ... (and, for that matter, I do not seem to really mind it ... the more badges, the better :)