This question was deleted by Community♦ and undeleted by a moderator, but to avoid that it gets deleted again, I will answer it now.
The first problem is, as I already said in my comment: it's not fair because of different time zones. I could be sleeping if someone posts a challenge about a topic I'm very interested in. I would never win.
It would be unfair if you have a great answer, but you post it 5 seconds after a bad answer, as Victor mentioned in his comment. Or even worse: you could have a great answer, and you are about to post it, but then you lose your Internet connection for some time. Or your browser crashes. Then you probably won't be the first one who answers the question.
Another problem: someone could easily abuse the 5 minutes editing grace period. Someone could post an answer that actually doesn't work, you post a working answer 2 minutes later, but then the previous answerer fixes his answer. Actually, you are the winner, but it will look like the previous answerer is the winner because he posted his answer first, without an "Edited at this time" link is shown.