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It's not mandatory to gift someone else. You're given the option to do so.
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Eh you could make an argument that, by participating in the festivity of Christmas, you provide tacit consent to agree to a social contract of sorts that requires you to play by the common rules of said holiday, i.e. giving gifts to others in exchange for receiving gifts from yourself. Giving yourself gifts and still expecting gifts from others would be a violation of this. Also it makes you seem like an asshole.