Seems to me like it's a matter of gauging the other person's willingness to change their mind (an admittedly difficult task in itself). If there is no argument

Seems to me like it's a matter of gauging the other person's willingness to change their mind (an admittedly difficult task in itself). If there is no argument or evidence that you could present which would be good enough to do so, the entire conversation seems to be pointless and it's probably best to just walk away. If they are in fact open to being proven wrong, doing so politely may still not work (i.e. they may be able to counter any point you make), but the odds are better than if one does so rudely. All other things being equal, we're more likely to be persuaded by someone we like (or at least are neutral toward) than someone we dislike.