The Ballot Theorem

Suppose in an election with two candidates, we are certain of the final tally. However, as the votes are counted sequentially, the lead may switch between the candidates during the process. This was certainly the case in the 2020 U.S. presidential election, where in several swing states, different types of votes (mail, early-voting, in-person) were counted at different stages. While the final tally does not change in a recount, the lead change may. One may ask: How likely will there be lead changes in a recount? Can the winning candidate always lead from start to finish? The Ballot Theorem provides a quantitative answer to this question. (Informal) Suppose candidates and each have […]

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