When you hear powerful evidence or arguments that should get you to revise your beliefs, not only do all sorts of cognitive biases fight the changes but so do the social factors of status and face saving. Perhaps I've long been a vocal proponent of X which implies Y, and you show me that Y isn't always true. It's very hard to just straight up admit "ok, I'm not a hardcore Xist anymore." There's a status loss in letting yourself be convinced.
For a long time I thought that I was stronger than this, that saving face only mattered as much as I let it matter. I wish I could freely admit when I've been convinced, but I often can't manage to. [1] Instead I'll finish a conversation defending my earlier beliefs and only later start acting on my new ones.
After a discussion where someone didn't admit to any change of mind, I'll often see them later having changed their behavior. So now if I'm trying to persuade someone I don't focus on securing verbal agreement. Instead I just try to be as convincing as possible, and notice if they come around later. [2]
(I also posted this on my blog)
[1] This is not a helpful trait: I'd like other people to let me know when I'm wrong or when they have evidence I'm not considering, but if they never get the satisfaction of knowing they've convinced me they may just feel like they've wasted their time, and not try in the future. So I'm working on it.
[2] Keeping people from feeling personally invested in one side or the other of an argument is probably also helpful: I understand discussions are much more likely to convince bystanders than participants.
Abandoning your previous position can also be a way of saving face, in at least two ways:
Being wrong is embarrassing, yes; but being wrong for a short period of time is less embarrassing than being wrong for an extended period of time. Best to stop the bleeding as quickly as possible.
You signal that you are a reasonable person who does not let emotional attachment to a position cloud his judgment. If you're dealing with someone of higher status, you show that your mistake doesn't matter that much because you corrected yourself quickly. If you are dealing with someone who is lower status than you, you come off appearing magnanimous.
In many cases this is true, but someone could also interpret this as you as being loose with your morals, one who betrays one's own ideals in a flash (and so are untrustworthy). Or maybe interpret it as you being a follower, who only thinks what people tell you to.