On Stack Exchange sites I have always felt answering in the right time can result in a good score. The first few answers always get some advantage. If the scores are hidden for some specific time (maybe 6-12 hours) and all the answers are shown randomly, then I think the process will be more unbiased.
Edit:
Considering what Gamecat has told, the score may be kept visible only for the respective answerers. May be for the questioner also.
This has been discussed several times before. And I think it's not going to work.
The problem is, people like to know how they are doing. So they frequently get back to see the number of up (or down) votes on their answer. They edit the answer if things were not clear and if other answers provide new views on the subject.
The fact is that the earliest answers have the biggest chance to get the most upvotes. So some people give a short answer first and elaborate on this later. And this is not wrong, in the end we like to have a site with high quality answers and questions.
This idea is not as outrageous as it sounds. Sometimes I do get the vibe both that upvotes are a question of timing and that "rich get richer", sometimes independent of answer quality.
For example, a slight variation of the OP's suggestion is the standard procedure to review a scientific paper. You (and 2-4 other reviewers) get the paper (question) and are asked to write a review (an answer). Then the editors (or area chairs) collect all reviews and make a decision. You don't know the rep or who wrote the paper (question) and you do not know the rep or who the other reviewers (answerers) are. The reviewers are not allowed to work together, and do not get instant feed back on the factual accuracy of their review.
The system seems to be working reasonably well for science.
I do think that a timeout period, to "cool off" the popularity contest and during which you're forced to vote only evaluating the merits of the question and answer could be very helpful.
EDIT: This is one example after looking through one tag, for yesterday's activity:
Concatenating two strings iteratively in C++
In this case, I consider both answers at least equally good, and if pressed, I'd say the answer that came in second, is better, more thorough. BTW, as this will change during the discussion, when I linked it the first answer has 5 upvotes and the second 3 and the OP chose the first answer as correct.
I believe I have seen exactly what the OP is talking about. A question is asked that has a "quick but incomplete" answer, or a "slow answer that includes an explanation". Two people start answering - one writes the fast answer, the other the slower one. A third person comes along just after the first answer is posted - thinks "hey, that's kinda what I would have said", and upvotes the answer as helpful. By the time the second answer appears, the first has two upvotes. A short while later the second answer appears, the first answerer decided to elaborate based on the other answer; in the end the two answers become "interchangeable" but the OP accepts the "first" answer in an attempt to be fair.
A system in which no voting is possible for the first 10 minutes would encourage everyone to "give a good answer first time", rather than rewarding a "get there first then edit" approach.
