This doesn't work. If a third candidate comes along and places themself anywhere along the beach that the two candidates are in the middle of, they'll lose to t

This doesn't work. If a third candidate comes along and places themself anywhere along the beach that the two candidates are in the middle of, they'll lose to the candidate opposite. (I think the single-dimensional political axis is also just a simplification, and the theorem works in higher dimensions.) For a distribution of voters without statistic lumpiness or cross-correlation, I think the only way for a third candidate to win is to identify some neglected political axis. (Maybe the theorem holds if the voter population is convex?) I think the point you are trying to make is that voters don't just always choose the politician exactly closest to them -- they pick one with some likelihood based on its relative closeness. In that case the two identical centrist candidates will split the vote. Other circumstances the theorem has difficulty with are primary votes. If only left-leaning voters vote in left-party primaries, "centrist for lefty-party" candidates will win lefty-party primaries -- not "centrist for the electorate" candidates. The theorem only really holds for "strong parties" that can choose their candidates and their positions, not the weird US free-for-all where essentially every candidate is an independent.