Well watching people argue, ways to "win an argument" - to give everyone the impression that your points are better - include

a) not listening to the other person b) intimidating them by claiming or implying they are stupid for not agreeing c) making better points and counterarguments

c) usually gets lost in the messy nature of arguments. But you can make the better points, and because of the limits of human knowledge, they still tend to be 'the best guess we have is that X is true', so could still be wrong

In debating technique, I spotted a while ago that it was clever to sidestep the confrontational style of argument sometimes, and turn the conversation to things you can both - apparently harmlessly - agree on. That way when you deliver your logic - even if it, in reality, contradicts what your opponent is saying, their stubbornness hasn't been tweaked in the same way. If they disagree they may well do it more sensibly

Doesn't always work that way. Some people - even if they are very capable of dispassionate analysis - are politically passionate by nature, and prefer to have a cause, and a whopping great shouting-match :)