LS@parker @kroner @Hoss @thendrix also an interesting paragraph from a proof linked in that article:
https://www.math.ucla.edu/~tao/arrow.pdf
Voting theory has a vast literature. A sample result (due to Gil Kalai): If there are three candidates, we assume all of the above axioms except for voting system rationality, and all voters vote randomly, then a rational outcome can only be ob- tained at most 91.92...% of the time. (By comparison, simple majority vote yields a rational outcome 91.22..% of the time, when the number of voters approaches infinity; this seems to be due to Gulibaud). So it seems that voting paradoxes are not isolated to a pathological set of scenarios of infinitesimal probability, but are a genuine and non-trivial difficulty inherent in voting systems.
Of course people don't vote 'randomly', but still interesting.
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