There are some people for whom this comes naturally. They sit with their hands folded and wait for you to convince them. You can talk and talk, wave your hands, raise your voice, and after you're finished they say very quietly, "I'm not sure." Or they just repeat the silly or hurtful thing they said at the outset. There's no way of knowing whether they were any paying attention at all while you carefully explained your point of view.
There were people in my family growing up who were like that. They annoyed the hell out of me. Because we had the same genes, and I've seen evidence of their intelligence in other contexts, I believe they knew they were being annoyingly illogical. I think that was the point. They got some kind of pleasure out of winding me up and spinning me around.
I've learned that sometimes the best thing is to fold my hands and listen and see what comes back, and see if it changes my mind. I know that I am really listening, and am open to changing my mind. That way there's a point in having the conversation.
Ideally the balance, between every pair of people, should be right smack in the middle. We care just as much about what they think as they care about what we think. Or approximately in the middle. If it appears out of whack, or getting out of whack, that's a good moment to take a deep breath, and step outside the immediate situation, and make a decision -- is this conversation worth continuing?
All this assumes the conversation isn't personal. Those conversations should never be allowed to continue. If someone is saying they see a flaw in you, or for some reason you feel compelled to tell someone about their supposed flaws, stop that conversation in its tracks. No one, not even a parent or a teacher, not even a master, is entitled to an opinion about you. It's fair to have an opinion about the weather, or the Mets, or a teacher you both have. But it's not fair to expect someone to listen to or participate in a discussion about themselves as a person. About something they did? That's fair if it's handled well. But it's hard to do that without offending.
Update: Paul Krugman is like me, that's why I like him. He's always trying to convince everyone of the errors of their ways. Appealing to their better nature. "Wake up!" says Krugman, "opportunity is slipping away from you!" He believes people are smart, but doesn't understand why they act so stupidly. I think most people (but not me) are chuckling. Someone needs to say to Krugman -- you're a Nobel Laureate. You're off-the-charts smart. You teach at Princeton and have a column in the NY Times. Let them convince you for a change.
Update #2: Glenn Greenwald is another tireless convincer. He pays attention to things people say and do, and remembers them, and can play them back in different contexts. If you appreciate this kind of rhetoric (I do!!) he's capable of masterpieces. But he never convinces the people he's talking about, most of whom are smart enough to understand what he's saying, to care enough to do anything about it.
Update #3: People like me make shitty investors. We always see things from the other guy's point of view. But if you're a smart entrepreneur you want to have a few of us around. Paradox. But we make excellent marketers and evangelists. Not sales people though, their magic is different.