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Except, Facebook doesn't like you doing this. In fact, it's against their TOS, and of late, they've made it considerably harder and harder to be able to do this. But since most of us want to keep our ACTUAL friends by NOT bombarding them with game requests, you can either friend people you don't know who are already playing the game, or you can create a second account and use that for games, which Facebook ALSO doesn't want you to do.
Why should they care who you friend?
Two big reasons.