How would you prevent voters being coerced into showing their ballots to their landlords?
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Or coerced by abusive partners. Or to get paid for their vote. It's a key but obscure aspect of anonymity. If you can prove you are person X, someone can seek to coerce that you are person X. If you can distinguish one set of anonymous from another, members can be coerced.
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There's a CAP theorem for voting. I forget the exact set, but iirc, they are Provable (that the vote I wanted was counted), anonymity (that a third party can't figure out my vote), and deniability (that I can't be forced to prove who I voted for, think gangs of New York).
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You can only have two in any voting system as getting those two requires forfeiting the third. I can't find it but there was a site from back in 2008 time frame that examined each voting system and what the provided vs sacrificed.
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The story goes that the inability to prove how they voted helps to protect voters from coercion.
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