Majority criterion

The majority criterion is a criterion for evaluating voting systems. It can be most simply thought of as "if a majority prefers a candidate as their unique 1st choice (i.e. they prefer this candidate above all other candidates), then the majority's 1st choice must win."

It can be stated as follows:

Or in plain English as

Example
51 A

25 B>C

24 C>B 51 voters out of 100 prefer A over all others (B and C), therefore A must win by the majority criterion.

Complying methods
Practically every serious ranked voting method passes the majority criterion, with the notable exception of Borda.

Stronger forms
The mutual majority criterion, which is sometimes simply called the majority criterion, generalizes the constraint to sets of candidates.

The Condorcet criterion implies the majority criterion.

Weaker forms of the criterion
Some voting methods (most rated voting methods) pass a weaker form of the majority criterion, which only requires that a majority be able to force their 1st choice to win by coordinating and voting strategically. Note that it is not always the case that the majority will have the ability to safely vote strategically I.e. if they're unsure as to whether there is or who their collective 1st choice is.

Majority criterion for rated ballots
There are some Cardinal systems which are designed to fulfil Majoritarianism not Utilitarianism. The majority criterion for rated ballots is a weaker, separate criterion which says that a candidate given a perfect (maximal) rating by a majority of voters must win if no other candidate received a perfect rating from that majority.

The difference between the two versions can be seen with this example: 51 A:1

49 B:5 If the highest score is a 5, then the majority criterion for rated ballots allows either A or B to win. This is in contrast to the regular majority criterion, which requires A to win. Arguably, the majority criterion for rated ballots is more appropriate in the context of rated ballots, since a voter who doesn't give their 1st choice a perfect score is essentially choosing not to use all of their voting power, and thus their preference need not be (or even perhaps, shouldn't) be maximally respected or enforced.

Independence of irrelevant alternatives
The majority criterion implies failure of the Independence of irrelevant alternatives criterion; see the Condorcet paradox for an example.

Majority rule as an approximation of utilitarianism
It is important to emphasize that majoritarianism and cardinal utilitarianism are not opposing principles. Majoritarianism can be understood as an approximation of utilitarian principles under certain conservative assumptions.

Within a theoretical framework using strictly ranked preferences (ordinal utilities), as in many models in modern neoclassical economics, all one can hope to achieve from a collection of social preferences is what is referred to as a Pareto equilibrium: a situation where no individual can be better off without making at least one individual worse off. This concept is used, for example, to establish the Pareto equilibrium within free markets and their usage of available resources. For a given set of individual preferences many such Pareto equilibria may exist, forming what it is called a Pareto frontier.

However, Pareto equilibria by themselves can be arbitrarily anti-democratic. As an extreme example, an authoritarian dictatorship where the dictator holds all the power and wealth, and the rest of the population has none, is a perfectly legitimate Pareto equilibrium. In order to improve the lot of everyone else with the exception of the dictator, the social choice function has to violate the preferences the dictator has to remain wealthy and in power. That is, the social choice function must necessarily use some additional criterion to navigate the Pareto frontier (violating at least one individual's preferences) in order to reach an equilibrium that is perceived as "socially better". Under strict ordinal utilities there is no way to distinguish between any equilibrium in the frontier, all are equally "good", so there is no way to determine a "better" one.

Majority rule is introduced in order to make this additional distinction between social states. It is used to justify the violation of preferences of a minority (like the sole dictator) in order to pursue a "better" equilibrium (the majority of the population).

However, the notion of "counting" preferences does not exist under a strict ranked preference mathematical framework. "Counting", be it with integers or real numbers, is inherently a cardinal procedure.

In order to invoke majority rule an assumption must be made that is inherently cardinally utilitarian: that satisfying each individual's preference has the same cardinal utility gain for every person, and that these utilities can be aggregated and totals compared. Framed differently, every individual's A>B preference can exactly cancel (it's commensurable) to any other individual's B>A preference, that is, every individual has exactly the same right to violate every other individual's preferences.

This is fundamentally a cardinal utility procedure, and in the case of two options immediately produces majority rule as a result of maximization of utility: if between any two options, A and B, one has 60% of people preferring A>B and 40% preferring B>A, then the net utility of A will be 60 - 40 = +20 against the net utility of B, 40 - 60 = -20. So a maximization of social utility chooses A, favoring the majority.

Thus, all ranked systems can be seen as approximations of cardinal utilitarianism to various extents, and operate under the same core assumption of democracy as cardinal voting methods: that every individual has some fundamentally commensurable value that may be counted, and that the preferences of one individual can override the preferences of any other. (Whether there are different strengths of preference is irrelevant.)

Condorcet voting systems, by applying majority rule to all pairwise comparisons, are effectively looking for the most consistently approximately utilitarian candidate. This intuitively explains the better utilitarian performance of Condorcet systems under various numerical simulations.