For example, one answer explains my problem in details and the other explains a short solution. Which do I pick? Do I write my own?
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I believe the approach most in keeping with the spirit of the site is to accept the most helpful/correct answer, and edit it to be complete (even if that means coming back when you have enough rep to do so).
Usually I accept what helped me the most.
If both did in exactly the same way, consider answering yourself as community wiki. I don't like doing this because I feel I don't give enough credit to the people helping me, but sometime it's what makes more sense.
If one more than the other, maybe you could ask the user to incorporate the other's answer and mark it as cw.
Expanding on Kara's answer, don't forget that both answers will be there no mather which one you accept. The community allows both of them to bubble up. In any case, they will be near each other until someone edits the accepted answer and adds the info from the other one.
Accepted answer means that it worked for you. Votes talk about concensus among the community. I always read all answers when researching a problem.