The argument

You dont need to win an argument to make a point.

No matter how the session went, afterwards everyone will hindsight review and take the point. Might not agree with it, but it will be reitareated.

I think this is usually common in negotiations. Especially after the first iterations. Claims are set, conflict emerges, no solution in sight. Yet, everyone will hindsight the session and the next iteration will factor in some of the demands from the other party - in one form or another.

Sometimes its the ego that pushes for a formal "win" whereas you need to leave people an exit to save face.

So once you made the point, let it stew. See where they try to meet you in the middle. You dont need to "win" the argument.

I mean, really, you're making a point not fighting the egos.