I don't know whether forums would qualify as a game, but I don't reckon it's a stretch to think that most people are playing a role. What I post on here is (drunk postings aside) thought out and reread before I click "post". I think you get a lot of that on the net - in many cases you get the choice of a screen name and avatar, and it's a chance to start fresh in a way. I've heard the same thing said about people who use online dating sites. It can be one way to meet people, but it's only when you meet them in person that their real, spontaneous personalities really come out.
I bet there are a fair few studies that have looked into it, but I don't know any.

May have to see if my old online Uni library login still works (when I'm sober).
Edit: I'm not saying that the majority of people are acting completely out of character, just that you get a chance to edit yourself online in a way that I don't think is quite as easy IRL.