Majority favorite criterion

The majority favorite criterion is a voting system criterion that says that, if a candidate would win more than half the vote in a first-preference plurality election, that candidate should win. Equivalently, if only one candidate is ranked first by a over 50% of voters, that candidate must win. It is occasionally referred to simply as the "majority criterion", but this term is more often used to refer to Condorcet's majority-rule principle.

Some methods that comply with this criterion include any Condorcet method, instant-runoff voting, Bucklin voting, plurality voting, and approval voting.

The criterion was originally defined only for methods based on ranked ballots, so while ranked systems such as Borda fail the criterion under any definition, its application to methods that give weight to preference strength is disputed, as is the desirability of satisfying such a criterion (see tyranny of the majority).

The mutual majority criterion is a generalized form of the criterion meant to account for when the majority prefers multiple candidates above all others; voting methods which pass majority but fail mutual majority can encourage all but one of the majority's preferred candidates to drop out in order to ensure one of the majority-preferred candidates wins, creating a spoiler effect.

Difference from the Condorcet criterion
By the majority favorite criterion, a candidate C should win if a majority of voters answers affirmatively to the question "Do you (strictly) prefer C to every other candidate?"

The Condorcet criterion gives a stronger and more intuitive notion of majoritarianism (and as such is sometimes referred to as majority rule). According to it, a candidate C should win if for every other candidate Y there is a majority of voters that answers affirmatively to the question "Do you prefer C to Y?" A Condorcet system necessarily satisfies the majority favorite criterion, but not vice versa.

A Condorcet winner C only has to defeat every other candidate "one-on-one"—in other words, when comparing C to any specific alternative. To be the majority choice of the electorate, a candidate C must be able to defeat every other candidate simultaneously—i.e. voters who are asked to choose between C and "anyone else" must pick "C" instead of any other candidate.

Equivalently, a Condorcet winner can have several different majority coalitions supporting them in each one-on-one matchup. A majority winner must instead have a single (consistent) majority that supports them across all one-on-one matchups.

Application to cardinal voting methods
The majority favorite criterion was initially defined with respect to voting systems based only on preference order. In systems with absolute rating categories such as score and highest median methods, it is not clear how the majority favorite criterion should be defined. There are three notable definitions of for a candidate A:


 * 1) If a majority of voters have (only) A receiving a higher score than any other candidate (even if this is not the highest possible score), this candidate will be elected.
 * 2) If (only) A receives a perfect score from more than half of all voters, this candidate will be elected.
 * 3) If a majority of voters prefer (only) A to any other candidate, they can choose to elect candidate A by strategizing.

The first criterion is not satisfied by any common cardinal voting method, but arguably lacks the persuasive force it has when comparing ordinal systems. Ordinal ballots can only tell us whether A is preferred to B (not by how much A is preferred to B), and so if we only know most voters prefer A to B, it is reasonable to say the majority should win. However, with cardinal voting systems, there is more information available, as voters also state the strength of their preferences. Thus cardinal voting systems do not disregard minority voices when making decisions; a sufficiently-motivated minority can sometimes outweigh the voices of a majority, if they would be strongly harmed by a policy or candidate.

Approval voting
Approval voting trivially satisfies the majority favorite criterion: if a majority of voters approve of A, but a majority do not approve of any other candidate, then A will have an average approval above 50%, while all other candidates will have an average approval below 50%, and A will be elected.

Plurality voting
Any candidate receiving more than 50% of the vote will be elected by plurality.

Instant runoff
Instant-runoff voting satisfies majority--if a candidate is rated first by 50% of the electorate, they will win in the first round.

Borda count
For example 100 voters cast the following votes:

A has 110 Borda points (55 × 2 + 35 × 0 + 10 × 0). B has 135 Borda points (55 × 1 + 35 × 2 + 10 × 1). C has 55 Borda points (55 × 0 + 35 × 1 + 10 × 2).

Candidate A is the first choice of a majority of voters but candidate B wins the election.

Condorcet methods
Any Condorcet method will automatically satisfy the majority favorite criterion, as a majority choice of the electorate is always a Condorcet winner.

Score voting
For example 100 voters cast the following votes:

Candidate B would win with a total of 80 × 9 + 20 × 10 = 720 + 200 = 920 rating points, versus 800 for candidate A.

Because candidate A is rated higher than candidate B by a (substantial) majority of the voters, but B is declared winner, this voting system fails to satisfy the criterion due to using additional information about the voters' opinion. Conversely, if the bloc of voters who rate A highest know they are in the majority, such as from pre-election polls, they can strategically give a maximal rating to A, a minimal rating to all others, and thereby guarantee the election of their favorite candidate. In this regard, score voting gives a majority the power to elect their favorite, but just as with approval voting, it does not force them to.

STAR voting
STAR voting fails majority, but satisfies the majority loser criterion.

Highest medians
It is controversial how to interpret the term "prefer" in the definition of the criterion. If majority support is interpreted in a relative sense, with a majority rating a preferred candidate above any other, the method does not pass, even with only two candidates. If the word "prefer" is interpreted in an absolute sense, as rating the preferred candidate with the highest available rating, then it does.

Criterion 1
If "A is preferred" means that the voter gives a better grade to A than to every other candidate, majority judgment can fail catastrophically. Consider the case below when $n$ is large: A is preferred by a majority, but B 's median is Good and A 's median is only Fair, so B would win. In fact, A can be preferred by up to (but not including) 100% of all voters, an exceptionally severe violation of the criterion.

Criterion 2
If we define the majority favorite criterion as requiring a voter to uniquely top-rate candidate A, then this system passes the criterion; any candidate who receives the highest grade from a majority of voters receives the highest grade (and so can only be defeated by another candidate who has majority support).