You should say something like "Now that we can agree on x, let's discuss y." (Given that agreeingt on y is dependent on a prior agreement about x.) Getting someone to agree with you is not the end of the conversation, it is the beginning. Thanking them for agreeing with you makes it seem like all matters are settled and now you may peacefully part ways.
Also, I disagree that it is intrinsically rational or polite or part of some gentlemen's agreement to thank someone when they concede your point.
Ok, sure. I am following your criticism, I think. In a conversation I expect to be temporarily conceding points and suspending disbelief, etc. I don't plan on thanking them every step of the way while we build toward a fully formed argument. I plan on thanking them at the end, when they feel like the ultimate outcome of our discussion has led them to change their mind on the topic under debate. The feeling of changing your mind on something is pretty uncomfortable, and I sometimes feel like the guy who talked me into it owes me big. This is purely an...
Would it be supercilious to thank someone for updating? I know I would feel uncomfortable doing it, but I often feel the urge to do so anyway.
There seems to be something vicious about thanking them. My own estimation of my own belief has not changed in these situations. I feel fairly satisfied that the other has considered my view and has shakily come to agree with it. I worry it would be a little like saying 'screw your opinion, now you see I'm right.'
However:
It is both rational and polite to thank them. The gentlemen's agreement of rationality allows for one person to be wrong and not lose face at all. When someone concedes something to me (a point) I typically feel the need to thank them.
What do you think?