Is it allowed to win a game that you had given away previously?

It is allowed only if it's not the same copy you received.
If you won a game and activated it, then you are safe. If after a while you decide to buy the same game to give away, then again you are safe.

http://www.steamgifts.com/about/faq
http://www.steamgifts.com/about/guidelines

You are safe, yes. As long as you do not regift won copies, there is no problem. A lot of people might give away a game and then realize they'd like it for themselves too, since you discover the game and look at its info to make a giveaway.

I hope you win the games you want. ^_^

Like others said, "yes you can".

you can always enter GA for games that are not in your library.
that's the only rule. and don't regift ;)

Oh, and if you got the answer, better close the topic, cause people seem to think you want to regift stuff. and they might blacklist you for thinking heresy :(

Most of these comments seem to have the question arse-about...OP is asking if they're allowed to win a game they previously gave away, not give away a game they previously won...the answer is still yes though