By the time you're in an argument, you've already failed to change the other persons mind - their defenses are up, and you're not getting in unless you have some combination of unusually good debating skills, unusually high weight of evidence on your side, and an unusual willingness of the other participant to change their mind mid-tack.

An argument is a fight, and people struggle harder against losing fights than possibly anything else. On the other hand, if you just walk away, let their emotions cool down, give them time to consider your points when they don't feel threatened, you might be surprised as to how often they'll come around - especially if they can do it without losing face (ie: you don't gloat and rub it in their face). The only arguments I've ever "won", I've won like this.