Greatest possible consensus criterion

Any voting system that guarantees the election of a greatest possible consensus winner satisfies the greatest possible consensus criterion and unanimous consensus criterion. Approval voting and disapproval voting satisfy the greatest possible consensus criterion. Condorcet methods, the Borda count, IRV, and plurality voting do not.

Greatest possible consensus winner
The alternative with the greatest number of consents (or tied with one or more alternatives for the greatest number of consents) in a given election is a greatest possible consensus winner. Any election that has total preference orders submitted by all voters guarantees the existence of such a winner. There may be multiple greatest possible consensus winners. Any unanimous consensus winner is necessarily a greatest possible consensus winner.

Approval voting guarantees the election of greatest possible consensus winners, when it ask voters "which alternatives do you consent to?" Disapproval voting guarantees the election of greatest possible consensus winners, when it ask voters "which alternatives do you not consent to?"

Voting systems that do not guarantee the election of greatest possible consensus winners include: Condorcet methods, Borda Count, IRV, and plurality voting.

If an assumption is made that a voter maximally supports all candidates they consent to, and maximally opposes all candidates they don't consent to (for voting systems with equal-ranking allowed, this means equally ranking all consented-to candidates first, and equally ranking all other candidates last. For rating systems, this means giving the maximal rating to consented-to candidates, and giving the minimal rating to all other candidates), then the following voting systems guarantee the election of greatest possible consensus winners: Condorcet methods with equal-ranking allowed, and Score Voting.

Unanimous consensus criterion
A voting system that guarantees the election of an alternative that is consented to by all voters (the unanimous consensus winner) in the voting population satisfies the unanimous consensus criterion.

Details
An alternative that is consented by all voters is called a unanimous consensus winner. It is possible that no such winner exists in a given election. Also, in a given election with two or more alternatives, there can exist more than one unanimous consensus winner. Voting systems that guarantee the election of a unanimous consensus winner must have a tie-breaking mechanism among multiple winners in order to be decisive.

Any voting system that satisfies the greatest possible consensus criterion necessarily satisfies this criterion.

Approval voting and Llull Voting satisfy the unanimous consensus criterion. Condorcet methods, Plurality voting, IRV, and the Borda count do not satisfy this criterion.