Down voting a question gives -2 rep to the asker, but up voting gives +5.
Since the up votes are halved on questions, why the down votes are not halved as well? Specially since they gives no penalty to the down voter.
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You're talking about two different things. The +5 for a question upvote is a reward for asking a good question, but the -2 for a downvote is a penalty for asking a bad one. We want to encourage good answers more than good questions, so answer upvotes are worth more. But we want to discourage bad questions just as much, so the penalty for downvotes are the same.