I'm not sure there's an "official" policy, per se, but there does seem to be a general "netiquette" on this topic, which roughly matches with my personal practices:
This is the age old Fastest Gun In The West problem.
Those who get the answer first should obviously be rewarded for their speed, but if there are instances where users post the same thing within seconds within one another, I will tend to upvote all of them. If it's within a few minutes, I will not be so lenient because that should have been enough time to review the already posted answers or load any new answers that were written during the construction of your own.
Seconds apart = close enough to reward both
Minutes apart = not close enough, fastest should get the upvote.
-- from TheTXI's answer
Of course, there is a third scenario not mentioned here, which I've seen pop up from time to time here on scifi.se: an answer duplicates a previous answer, providing absolutely no new information, and is posted hours, days, or even months after the previous answer it replicates.
For these answers, I sometimes (I admit I am not terribly consistent on this; it depends on a number of factors that are probably too subjective to merit mentioning) downvote and comment.
Comments on answers like this is a touchy subject, though. It is easy for comments to be discouraging, especially to new users, particularly if they honestly didn't read the other answers and were making a genuine effort at contribution (it happens). The best phrasing I could come up with would be something along the lines of
While this answer is right [assuming it is], it seems to be a duplicate of a previous answer. This may result in you getting some downvotes. Is there any additional information you could add to distinguish your answer from the others?"
Deleting, however, does not seem right, given what I understand on SE's policy for when/why high rep users should vote to delete.