PAGO PAGO, American Samoa—As the 351 votes cast in the presidential caucus were pulled from a wooden box and read aloud inside the old Oasis bowling alley on Super Tuesday, it became clear that an upset was under way on this lush tropical archipelago in the South Pacific.

With 175 votes, Mike Bloomberg was finally seeing a payoff from his roughly $620 million campaign to persuade Americans to elect him president. He was winning in American Samoa, a tiny territory closer to New Zealand than the continental U.S.

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