Satisfaction of the Condorcet criterion implies that of the majority criterion, but not vice versa. With the Condorcet criterion there may be different majorities depending on who X and Y are, but the majority criterion only applies when there is a single majority which has X as their first choice, preferred over every other candidate.
Proportionality for Solid Coalitions
Proportionality for Solid Coalitions (PSC) is a voting system criterion for ranked voting systems. It is important for giving proportional representation for voters in multiple-winner ranked voting systems.
A group of voters formula_1 is a "solid coalition" for a group of candidates formula_2, if every voter in formula_1 ranks every candidate in formula_2 ahead of every candidate that is not in formula_2.
Let formula_6 be the number of voters, formula_7 be the number of seats to be filled and formula_8 be some positive number.
formula_7–PSC is based on the Hare quota formula_11. If formula_1 is a solid coalition for formula_2 and the number of voters in formula_1 is at least formula_8 Hare quotas, then at least formula_8 candidates from formula_2 must be elected (if formula_2 has less than formula_8 candidates at all, then all of them have to be elected). This criterion was created by Michael Dummett.
formula_20–PSC is defined like formula_7–PSC, but is based on the Hagenbach-Bischoff quota formula_23 instead of the Hare quota, and the number of voters in formula_1 must exceed formula_8 Hagenbach-Bischoff quotas. It is a more general form of the majority criterion because it allows for groups of supported candidates (solid coalitions) instead of just one candidate, and there may be more than one candidate that will be elected. Because some authors call the fraction formula_23 a Droop quota, formula_20–PSC is also known as the Droop proportionality criterion.
Droop proportionality means that a majority-size solid coalition will always be able to elect at least half of the seats. This is because a majority is always over n/2 voters, and that will contain half of the Hagenbach-Bischoff quotas (There are (k+1) Hagenbach-Bischoff quotas in an election, since (n/(k+1)) * (k+1) = n, so (k+1)/2, which is half of the quotas * n/(k+1), which is the quota, = n/2).
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