Why can I vote to close my own question? Under what circumstances would I want to do that, or should I consider doing that? Do I still get to vote to reopen even if I voted to close my own question?
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To answer your questions in more detail:
Why can I vote to close my own question?
You can delete your own question until someone responds and/or gets upvoted. At that point if you delete your post, then you also delete their posts, which is not acceptable to the community.
Once others have contributed to your question, you can vote on closing/deleting it, but you may not be able to close or delete it yourself under most circumstances.
Under what circumstances would I want to do that, or should I consider doing that?
If you feel your question does not add anything valuable to the community, or the question itself is confusing, malformed, or may result in other people making a mistake, then you might want to remove it.
Some people are also "pruning" their questions so they look better for their Careers.Stackoverflow CV.
The ability to delete your own post is a reminder that though SO is largely community owned and to some degree wiki-style, you are still a owner for your question, responsible for it, and thus you are granted this additional power over it.
Do I still get to vote to reopen even if I voted to close my own question?
No, it works like other open and close votes - you can vote once, and that's it. This is mainly due to implementation (it's easier to extend 3k+ close/open voting power than to create a special power) but also as a direct result of the open/close and edit wars that happened early on when 3k+ moderation powers were first created. By allowing only one vote, wars have a finite lifetime.
If you post a dupe on accident, and someone points it out, and you realize your mistake, you can help correct your mistake by voting to close your question. You wouldn't want to have a duplicate question hanging around just because there weren't enough people who could vote to close it, would you?
Because this is ... a democratic site? :-)
Maybe you changed your mind or the question is no longer relevant or for some other reason you want to delete it but you don't anymore have the right to do that.