Runoff

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A runoff is when two candidates face off in a separate (usually second) round of an election, with the winner of the runoff being the winner of the overall election. Usually choose-one FPTP voting is used for the runoff, though any voting method could be considered (Score voting being one of the more interesting to consider).

Voting honestly in an FPTP-based (or FPTP-equivslent runoff, if using something like STAR in the runoff on the assumption voters indicate all of their preferences) can never hurt a voter, because either the voter's preferred candidate wins with the added support of that voter where they otherwise wouldn't have, or they end up losing either way, with the voter's vote only reducing the margin of defeat between the two candidates in the runoff.

If FPTP is used in the runoff, then either one candidate is guaranteed to have a majority of votes, or both candidates are tied.

RCV (also sometimes called instant-runoff voting) and STAR voting both can be thought of as involving an automatic or instant runoff (meaning the runoff is done using the same ballots as were used to find the result of one of the previous rounds) which determines the winner.

Runoffs can be defined as a pairwise comparison between the two candidates in them. Because of this, some voting methods which involve automatic runoffs (i.e. STAR) can be counted by creating a pairwise comparison table to figure out the winner of any potential runoff, and when a voting method gives you the result of a runoff (i.e. if the IRV winner gets a majority against some other candidate), that can be used to partially fill out the pairwise comparison table.

Adding a runoff to a voting method often breaks several desirable properties of the underlying voting method, and adds potential for certain nonintuitive voting strategies such as pushover. For example, Score voting passes several generalized forms of Monotonicity and Favorite Betrayal, but neither its instant-runoff version, STAR voting, nor the delayed/separate-runoff version have those properties. Example:

9: A:5 B:1 C:0

12 B:5 C:1 A:0

8 C:5 A:1 B:0

The score totals are A 53, B 69, C 52. A and B go to the automatic runoff, and then A pairwise beats B 17 to 12 and wins. But if the 12 B:5 C:1 A:0 voters had instead voted B:5 C:4 A:0, they would've helped C enter the automatic runoff instead of A, where B pairwise beats C and wins instead. This is nonintuitive because by increasing their support for an irrelevant alternative, the B-top voters made their favorite win. [1] This is essentially because this is a Condorcet cycle situation, so any voting method passing the majority criterion in the two-candidate case, such as any voting method ending with a runoff, will shift its results depending on, when there are only two candidates in the election, who the two are.

  1. ""STAR voting"".