I'll usually admit that I'm wrong when the other person has explained, sufficiently, why their answer is correct. This means I know why I was incorrect and I have less of a chance of being correct. For instance, I got a 64 on a Physics test recently (I don't care, I'll still get an A) and I had forgotten to divide by 2 on my calculator and added instead of subtracted for another thing. However, I don't think that I should have lost 30% for those two things (which my teacher said she'd look into). But, anyways, this has made me less likely to make a mistake like that, because my subconscious will make sure that I'm a little more careful with those problems.
Of course, admitting I'm wrong usually only deals with things like Math and Physics, or other fact-based subjects, like History. It's essentially impossible to be wrong on opinions that aren't disproved by facts, especially if you're solipsistic (you don't trust any of your senses), which I'm not, though I'll try to be at least somewhat when I see someone's opinion that I don't agree with, unless facts can refute it (which is where the solipsism ends, since I'm trusting facts, sense-gathered data).
As for the arguments you referred to, it really depends on what the argument is over, and I'd use the above process of deciding whether it could be argued for/against. I also tend to not argue things that I'm not well-versed in (like Politics).