The Fastest Gun in the West problem has existed on SO since day one, and the SO team has put much effort into resolving it. I wonder how much the Enlightened badge encourages the problem. You get this badge for:
First answer was accepted with at least 10 up votes
So if I know the answer, I have an even greater incentive to fire off a quickie and be first, since only the first can get this badge (a silver one, no less!). AFAIK this is the only thing on SO that directly requires you to be first.
Some other incentives to be first were discussed in the latest podcast (episode 66), but they are all about the side-effects of being first. This badge actually rewards you for being first.
I don't have any suggestions as to what should be done with the badge, but if it does cause this problem, maybe it should be rethought.