Question

Is it possible to have a vote for three candidates using a ranked ballot that produces a tie in each of the following systems, all at the same time: plurality, top-two runoff, elimination run-off, and Borda count?

Answer 1

This situation is very unlikely, to have a tie in all the systems above at the same time.

Under ranked ballot, people use ranks like on the ordinal from first, second and third.

A plurality vote describes the situation when a candidate polls more votes than any other but does not receive more than half of all votes cast.

Top two run off is a type of ranked preferential voting method used in single-seat elections with more than two candidates. Instead of indicating support for only one candidate, voters can rank the candidates in order of preference. Ballots are initially counted for each voter's top choice. If a candidate has more than half of the vote based on first-choices, that candidate wins. If not, then the candidate with the least votes is eliminated. The voters who selected the defeated candidate as a first choice then have their votes added to the totals of their next choice. This process continues until a candidate has more than half of the votes. When the field is reduced to two, it has become an "instant runoff" that allows a comparison of the top two candidates head-to-head. Compared to plurality voting, this method can reduce the effect of vote-splitting when multiple candidates earn support from like-minded voters.

Elimination run-offs is a voting system used to elect a single winner, whereby only two candidates from the first round continue to the second round, where one candidate will win.

The Borda count decides the winner of an election by giving each candidate, for each ballot, a number of points corresponding to the number of candidates ranked lower. Once all votes have been counted the candidate with the most points is the winner.

So, it is not possible to have a vote for three candidates using a ranked ballot that produces a tie in each of the systems discussed above at the same time.