This isn't what Arrow's theorem says. It's not just that there is always someone whose vote happens to match the result (like a fixed-point theorem would). It s

This isn't what Arrow's theorem says. It's not just that there is always someone whose vote happens to match the result (like a fixed-point theorem would). It says that there is a ballot that determines the result. In other words, the quantifiers are reversed. Instead of "for every list of votes, there is a voter where the outcome is the same as his vote", it is "there is a voter such that for every list of votes, the outcome is the same as his vote." Now, this doesn't have to be the same person every time you take a vote. The voting systems described in Arrow's theorem just take an ordered list of votes and return a result, and the dictator corresponds to an index in this list. So you can order the votes differently each time if you want, but at the end of the day, it's just a process for deciding how to pick a dictator.