EA has put a "Mass Effect 3 Mission Command" app on Facebook, through which fans can unlock some Avatar items and possibly get chosen for early access to the game's upcoming beta. There's also a big sweepstakes opportunity to win one of two custom Xbox 360s, painted in the game's Paragon and Renegade themes. EA sent us these pictures earlier today.
[UPDATE] Big props to commenter LeetEmu for noticing that "Renegade" is in fact misspelled on the image supplied (drive slot). Wow. Hope they fix that, pronto.
[UPDATE] We've received word that Microsoft has definitely noticed the misspelling, which appears to be on promotional material only, not a physical console itself, and it will not award any misspelled consoles to any contest winner.
The Mission Command app isn't really a game per se; you're given a short paragraph, asked to respond to it in either paragon or renegade character, and then type up a reply corresponding to that. No dialogue wheels here! That's why I think my response to the "mission" of flirting with Liara was most appropriate. Head on over and check it out if that seems like something fun to you.