# 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.

## Solid coalitions

A group of voters ${\displaystyle V}$  is a solid coalition for a group of candidates ${\displaystyle C}$ , if every voter in ${\displaystyle V}$  ranks every candidate in ${\displaystyle C}$  ahead of every candidate that is not in ${\displaystyle C}$ .

Let ${\displaystyle n}$  be the number of voters, ${\displaystyle k}$  be the number of seats to be filled and ${\displaystyle j}$  be some positive number.

## ${\displaystyle k}$–PSC

${\displaystyle k}$ –PSC is based on the Hare quota ${\displaystyle n/k}$ . If ${\displaystyle V}$  is a solid coalition for ${\displaystyle C}$  and the number of voters in ${\displaystyle V}$  is at least ${\displaystyle j}$  Hare quotas, then at least ${\displaystyle j}$  candidates from ${\displaystyle C}$  must be elected (if ${\displaystyle C}$  has less than ${\displaystyle j}$  candidates at all, then all of them have to be elected). This criterion was created by Michael Dummett.

## ${\displaystyle k+1}$–PSC

${\displaystyle k+1}$ –PSC is defined like ${\displaystyle k}$ –PSC, but is based on the Hagenbach-Bischoff quota ${\displaystyle n/(k+1)}$  instead of the Hare quota, and the number of voters in ${\displaystyle V}$  must exceed ${\displaystyle j}$  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 ${\displaystyle n/(k+1)}$  a Droop quota, ${\displaystyle k+1}$ –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).