Proportionality for Solid Coalitions

Proportionality for Solid Coalitions (PSC) is a voting system criterion relating to ranked voting systems. It's the essential requirement[1] to guarantee a proportional representation of voters in multiple winner ranked voting systems.

Solid coalitions

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

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

${\displaystyle k}$–PSC

${\displaystyle k}$–PSC is defined with respect to 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).[2] This criterion was proposed by Michael Dummett.[3]

${\displaystyle k+1}$–PSC

${\displaystyle k+1}$–PSC is defined like ${\displaystyle k}$–PSC, but with respect to the Hagenbach-Bischoff quota ${\displaystyle n/(k+1)}$ instead of the Hare quota: the number of voters in ${\displaystyle V}$ must exceed ${\displaystyle j}$ Hagenbach-Bischoff quotas.[2] It is a generalization of the majority criterion in the sense that it relates to groups of supported candidates (solid coalitions) instead of just one candidate, and there may be more than one seat to be filled. Because some authors call the fraction ${\displaystyle n/(k+1)}$ Droop quota, ${\displaystyle k+1}$–PSC is also known as Droop proportionality criterion.[1]

One major implication of Droop proportionality is that a majority 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, which is equivalent to a number of voters exceeding half of the Hagwnbach-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).

References

1. D. R. Woodall: Monotonicity of single-seat preferential election rules. Discrete Applied Mathematics 77 (1997), p. 83–84.
2. Tideman N.: Collective Decisions and Voting. Ashgate Publishing Ltd, Aldershot, 2006, p. 268–269.
3. Dummett, M.: Voting procedures. Oxford Clarendon Press (1984).