Runoff
This article combines three separate concepts: runoffs, elections with only two major candidates in them, and binary votes]. It really needs to be rewritten.
A runoff (or binary vote, or pairwise comparison) is when two candidates face off in a final (usually second round of an election using a multiple-round voting method, with the winner of the runoff being the winner of the overall election. Usually choose-one first-past-the-post (FPTP) voting is used for the runoff (i.e. the candidate preferred by more voters wins).
Voting honestly in an FPTP-based (or FPTP-equivalent runoff, if using a Category:Majority rule-based voting method) 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.
See Category:Runoff-based voting methods.
Automatic runoff
The normal version of a runoff (with a separate election) is often called a "delayed" runoff. 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. Also, 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.
Criticisms
Political faction violence
Delayed runoffs may actually encourage violence on the part of factions whose preferred candidates didn't make it into the runoff, because there is a period of time in which they are aware they will not get what they want.
Lack of voter choice
An abstract criticism of two-candidate elections in general is that they reduce voter choice; indeed, one major reason voting reform advocates wish to move away from FPTP for single-winner races is to encourage more candidates to run, and more popular candidates to win.
Runoffs are also subject to this criticism, albeit to a lesser degree, because voters can't support candidates who are not in the runoff.
Write-in option
The use of a write-in option in the runoff, combined with a voting method that allows for supporting multiple candidates (i.e. rated method or Condorcet methods), would allow alternative candidates to pick up support in the runoff. However, this breaks the strategyproofness of runoffs (see below).
Majoritarianism
By only offering two choices, it is often postulated that runoffs encourage candidates to campaign in a divisive manner to simply ensure more than half of the voters prefer them to their opponent, which may arbitrarily divide the electorate into two rival factions.
Strategic voting
While a runoff itself is strategyproof, the elections/rounds before the runoff are not. Because they are a type of Multiple-round voting method, runoffs are susceptible to some unusual forms of strategic voting, such as pushover; adding a runoff to a voting method often breaks several desirable properties of the underlying voting method.
Example for STAR voting
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.
Binary vote
Two-candidate case
A runoff can be thought of as the electoral (candidate-based) instance of a binary vote, which is simply any time voters choose between two options. For example, legislative voting often involves choosing between passing a bill or not passing it (i.e. staying with the status quo).
Strategyproofness
Binary votes are fundamental to voting theory, because they have a number of desirable properties (strategyproofness, or if using utilitarian voting, no incentive for coordinated strategic voting i.e. the only strategic voting required involves normalization) that are impossible when there are more than two candidates/options to vote on. See the Gibbard-Satterthwaite theorem for more information.