A reason to throw in a no hang vote would be to prevent the mafia from having one of their members blitzing someone to hang at the last minute. Of course I doubt that was Rein's reasoning behind his vote.
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Isn't wifom incredible? Clearly, the doctor would protect the better players. But the mafia should know that, and target someone else. But the doctor should know that, and protect the less skilled players. But the mafia should know that as well, and then, target the more skilled players. Which is why the doctor should protect the more skilled players. And so on and so forth :)