Redistributing Votes when Candidates Drop Out of an Election

The U.S. Presidential Primary is a series of statewide primaries and caucuses used by the Democratic and Republican parties to winnow a pool of candidates to arrive at their nominees. As unsuccessful candidates leave the race, there is a question as to how voters who planned to vote for the candidate reassign their votes in the upcoming primaries/caucuses. When a profile of voters’ preferences is independent of elimination under plurality, then voters whose first-place candidate has dropped out of the race have their votes distributed to the remaining candidates in the same proportion of the remaining candidates’ first-place votes. For any number of candidates, we show that this notion of independence can be reframed in terms of a probabilistic view of the election. We relate the repeated dropping out of candidates to Harville’s equations from probability theory. To motivate our analysis, we look at data from the 2024 Republican Presidential Primary.