Tie

A tie is when two or more options have the same amount of something, usually the same number of votes. They appear in many different contexts in voting methods, and are usually broken randomly (i.e. every option in the tie has an equal probability of being selected, whether it's being selected for winning or something else).

Some ties can be broken without randomness; for example, STAR voting can have ties in the automatic runoff broken by electing the tied candidate with a higher score, if there is one.

Sometimes, certain criteria compliances can only be achieved with more complex tie-breaking procedures. For example, Sequential Monroe voting only retains compliance with Pareto when the tiebreaker for multiple candidates having a maximal quota score is to choose the candidate among the tied who has the highest overall score.

A type of tie is a symmetrical pairwise or ranked tie. For example:

1 A>B>C 1 B>C>A 1 C>A>B

All three candidates are in a symmetrical situation, that is, one could swap their name labels and they would still be in the exact same situation as before: ranked 1st on 1 ballot, ranked 2nd on 1 ballot, and ranked 3rd on 1 ballot. These types of ties are interesting in the context of evaluating Condorcet cycles. They also feature when discussing an Equally Weighted Vote; some voting methods which pass the "test of balance" can fail to produce an equally weighted vote when there are symmetrical ties; this includes many voting methods which pass the majority criterion in the two-candidate case; see Equally Weighted Vote for an example.

Probability
Some voting method criteria are/must be defined with respect to candidates having or not having positive probability rather than winning or not winning. For example, the plurality criterion can be defined either as "B must not win" or "A's probability of winning must be at least as good as B's" (see also Plurality criterion).

Tiebreaker procedures
Usually each tied candidate has an equal probability of winning. This can lead to technical (but likely very rare) failures of various criteria, however, so some more complex procedures have been created to avoid that.

Random Voter Hierarchy
See Maximize Affirmed Majorities.