The most important reason to argue with someone is that someone should change their mind.

Not sure why Marshall is being voted down here; I agree with him completely. The main reason I "argue" with someone is to seek truth. Perfectly rational agents with the same information should never disagree. So if I disagree with someone, either one (or more) of us is not rational, or one (or more) of us has information that the other one doesn't.

If I argue with someone, I am doing them a favor by expending effort to provide them with more information or helping them see their irrationalities. If someone argues with me, they are doing me a favor by expending effort to provide me with more information or helping me to see my irrationalities.

When I argue with people who are relatively rational (e.g. most of my friends), this works well. Usually one of us learns something new.

When I argue with people who are less rational (e.g. most people in general), this does not work very well, and I run into the problems described by Yvain here.