The term 'melodrama' is a somewhat loaded term - ask a few people if X media counts as melodrama, and you're likely to get a variety of answers. Michael Abbott discusses melodrama in one accepted context (a definition that I would quibble with based on my own background dealing with 'melodramatic representation') in reference to games. Yes, we do do melodrama - everything from GTA to Metal Gear to Final Fantasy plays with at least one interpretation of melodrama:
Lest you blanch at the notion of Solid Snake lumped in with Days of Our Lives or Waiting to Exhale, I would suggest to fans of Braveheart, Lost, CSI, and virtually every sports movie ever made that you are also fans of melodrama. The Call of Duty series, the Final Fantasy series, Bioshock - even significant portions of GTA IV - all rely on melodrama to deliver their experiences.
And at the center of these tales is the classic Melodrama Hero - a man (sometimes, but rarely a woman) of strength and courage who must do great deeds in an environment of heightened emotional intensity; a hero who operates within a clearly defined world of good and evil, charged with restoring order and stability from chaos. Solid Snake and Dudley Do-Right are cut from the same cloth. One may be a conflicted hero with lots more backstory (and, okay, Dudley is a cartoon caricature), but dramaturgically they function in remarkably similar ways.
I have to say I would think most people would blanch at the idea of Solid Snake lumped in with soap operas ... but he's got a point. Melodrama is a hugely effective narrative style — and the reasonably clear dichotomies we see in many narrative-driven games is one critical part in labeling them as 'melodramas,' or at least as media possessing melodramatic elements. However, I don't think the world is quite ready for the Days of Our Lives RPG. At least, I certainly hope not.
We do melodrama [The Brainy Gamer]