# Proportionality for Solid Coalitions

Proportionality for Solid Coalitions (PSC) is a criterion for proportional methods requiring that sufficiently-sized groups of voters (solid coalitions) always elect a proportional number of candidates from their set of mutually most-preferred candidates. In general, any time any group of voters prefers any set of candidates over all others, a certain minimum number of candidates from that set must win to pass the criterion, and the same must hold if the preferred set of candidates for a group can be shrunk or enlargened. It is the main conceptualization of Proportional Representation generally used throughout the world (Party List and STV pass versions of it.)

Any voting method that collects enough information to distinguish solid coalitions (generally scored or ranked methods, since preferences can be inferred from their ballots) can be forced to be PSC-compliant by first electing the proportionally correct number of candidates from each solid coalition before doing anything else.

## Types of PSC

For the purposes of PSC, maximal support to a candidate or set of candidates is generally determined as:

- with ranked ballots, all of the candidates are ranked above all candidates not in the set.

- with rated, approval, and choose-one ballots, all of the candidates in the set are given the maximum score or are marked.

### Hare-PSC

k-PSC or Hare-PSC is a condition requiring a solid coalition of c candidates supported by k Hare quotas to be always elect at least ${\displaystyle \min(c, k)}$ most-preferred candidates (i.e. k candidates whenever c is not less than k.)

### Droop-PSC

k+1-PSC or Droop-PSC (since many authors call the Hagenbach-Bischoff quota a Droop quota) is the same as Hare-PSC but holds for Hagenbach-Bischoff quotas instead, and requires the solid coalition to be supported by more than k HB quotas, rather than at least that amount. The Droop-PSC criterion is also called the Droop proportionality criterion. Note that Droop proportionality implies the mutual majority criterion.

The main difference between Hare-PSC and Droop-PSC can be seen with an example: Suppose you can buy two boxes of pizza, with over 2/3rds of voters wanting Cheese pizza, and under 1/3rds of the voters wanting Pepperoni pizza. Hare-PSC would say that you should buy at least one box of Cheese pizza, but has no opinion on what you should buy for the second box, whereas Droop-PSC would say that you should buy two boxes of Cheese pizza. This can be explained as happening partially because if the 2/3rds group of cheese-preferring voters split themselves into two equally sized groups of over 1/3rd of voters each, then these "two" groups that want Cheese would each outnumber the group of under 1/3rds of voters that want Pepperoni.

### Weak forms of PSC

PSC is a requirement that holds for honest voters. Many voting methods pass weaker requirements that hold only for strategic voters, with the difficulty of the strategy depending on the method. In general, any method that passes such weaker versions of PSC is considered to be at least semi-proportional. Note that PSC implies all of these weaker forms of PSC. Here are some of these weaker requirements (note that the requirements vary slightly depending on whether you're using the Hare quota, HB quota, or other quota):
If a solid coalition of k quotas evenly distributes its support among k of their preferred candidates such that each of the k candidates receives maximal support from at least a quota of voters,then at least k of their preferred candidates must win.
SNTV passes this with Droop quotas.
If a solid coalition of k quotas gives maximal support to k of their preferred candidates, and no support to all other candidates, then at least k of their preferred candidates must win.
Most cardinal PR methods pass this for Hare quotas.
If a solid coalition of k quotas gives maximal support to at least k of their preferred candidates, and less-than-maximal support to all other candidates, at least k of those preferred candidates must win.
Sequential Monroe voting passes this, making it the best cardinal PR method from the perspective of PSC. This is probably the strongest PSC-like requirement that a natural voting method can pass without actually passing PSC.

## Examples

5-winner example, Hare quota 50 (example done using scored ballots):

Number Ballots
10 A1:10 A2:7 A3:7 A4:7 A5:7 B1:1 C1:0 D1:0 E1:0 F1:0
10 A1:7 A2:10 A3:7 A4:7 A5:7 B1:0 C1:1 D1:0 E1:0 F1:0
10 A1:7 A2:7 A3:10 A4:7 A5:7 B1:0 C1:0 D1:1 E1:0 F1:0
10 A1:7 A2:7 A3:7 A4:10 A5:7 B1:0 C1:0 D1:0 E1:1 F1:0
10 A1:7 A2:7 A3:7 A4:7 A5:10 B1:0 C1:0 D1:0 E1:0 F1:0
40 A1:2 A2:0 A3:0 A4:0 A5:1 B1:10 C1:0 D1:0 E1:0 F1:0
40 A1:0 A2:2 A3:0 A4:0 A5:1 B1:0 C1:10 D1:0 E1:0 F1:0
40 A1:0 A2:0 A3:2 A4:0 A5:1 B1:0 C1:0 D1:10 E1:0 F1:0
40 A1:0 A2:0 A3:0 A4:2 A5:1 B1:0 C1:0 D1:0 E1:10 F1:0
40 A1:0 A2:0 A3:0 A4:0 A5:0 B1:0 C1:0 D1:0 E1:0 F1:10

Looking at the top 5 lines, 50 voters, a Hare quota, mutually most prefer the set of candidates (A1-5) over all other candidates, so Hare-PSC requires at least one of (A1-5) must win. (Note that Sequential Monroe voting fails Hare-PSC in this example. However, one could forcibly make SMV do so by declaring that the candidate with the highest Monroe score within the set (A1-5) must win the first seat, for example.) [1]

Generally, Droop-PSC makes it more likely that a majority will win at least half the seats than only Hare-PSC. The reason for this is that majority solid coalitions almost always constitute enough Droop quotas (and always constitute enough Hagenbach-Bischoff quotas) to always win at least half the seats, while with Hare quotas they can only guarantee they will win just under half the seats and have over half a Hare quota to win the additional seat required to get at least half the seats. 5-winner example using STV with Hare quotas:

Number Ballots
26 A2>A1>A3>B1>B2>B3
25 A1>A3>A2>B1>B2>B3
17 B1>B2>B3>A1>A3>A2
16 B2>B1>B3>A1>A3>A2
16 B3>B2>B1>A1>A3>A2

Note that 51 voters, a majority, prefer (A1-3) over all other candidates, and thus electing all 3 of them would mean 3 out of 5 seats, a majority, would belong to the majority. Also, the Droop quota here is 17, and thus by Droop-PSC the majority has 51/17=3 PR guarantees, which would give them a majority of seats. However, by Hare-PSC, they only have 51/20 rounded down = 2 PR guarantees, and thus by Hare-STV:

So, the Hare quota here is 20. A1 and A2 are immediately elected, but post-transfer A3 only has 11 votes, and is thus eliminated first. B1, B2, B3 take the remaining 3 seats.[2]

There can be quota overlaps when assigning PSC claims; suppose a group constituting 80% of a quota of voters vote A>B>C=D, another group of 80% of a quota vote B>A>C=D, and another group of 50% of a quota vote C>A=B=D. Then, 2 candidates must be elected from the set (A, B, C, D), since in total 2.1 quotas mutually most prefer that set, but a further constraint is that 1 candidate must win from within (A, B), since 1.6 quotas mutually most prefer them. It would not satisfy PSC if the final winner set had neither A or B in it in other words, even if it had C and D.

## Generalised solid coalitions

The Expanding Approvals Rule passes a stricter PR axiom than PSC:

Definition 5 (Generalised solid coalition) A set of voters N′ is a generalised solid coalition for a set of candidates C′ if every voter in N′ weakly prefers every candidate in C′ at least as high as every candidate in C\C′. That is, for all i ∈ N′ and for any c′ ∈ C′
${\displaystyle \forall c \in C \setminus C^{\prime}\quad c^{\prime} \succsim_{i} c}$.

We note that under strict preferences, a generalised solid coalition is equivalent to solid coalition. Let c(i, j) denotes voter i’s j-th most preferred candidate. In case the voter’s preference has indifferences, we use lexicographic tie-breaking to identify the candidate in the j-th position.

Definition 6 (Generalised q-PSC) Let q ∈ (n/(k + 1), n/k]. A committee W satisfies generalised q-PSC if for all generalised solid coalitions N′ supporting candidate subset C′ with size |N′| ≥ ℓq, there exists a set C′′ ⊆ W with size at least min{ℓ, |C′|} such that for all c′′ ∈ C′′

∃i ∈ N′ : c′′ i c(i,|C′ |).

The idea behind generalised q-PSC is identical to that of q-PSC and in fact generalised q-PSC is equivalent to q-PSC under linear preferences. Note that in the definition above, a voter i in the solid coalition of voters N′ does not demand membership of candidates from the solidly supported subset C′ but of any candidate that is at least as preferred as a least preferred candidate in C′. Generalised weak q-PSC is a natural weakening of generalised q-PSC in which we require that C′ is of size at most ℓ.

Definition 7 (Generalised weak q-PSC) Let q ∈ (n/(k + 1), n/k]. A committee W satisfies weak generalised q-PSC if for every positive integer ℓ, and every generalised solid coalition N′ supporting a candidate subset C′

|C′| ≤ ℓ with size |N′| ≥ ℓq, there

exists a set C′′ ⊆ W with size at least min{ℓ, |C′|} such that for all c′′ ∈ C′′

∃i ∈ N′ : c′′ i c(i,|C′ |).[3]

By analogy to Descending Acquiescing Coalitions, the generalized PSC could also be called proportionality for acquiescing coalitions.

## Notes

Droop-PSC implies Hare-PSC, since a Hare quota is simply a large Droop quota, but the same doesn't hold the other way around. Hare-PSC is equivalent to the unanimity criterion and Droop-PSC to the mutual majority criterion in the single-winner case. Note that this means cardinal PR methods can only pass Hare-PSC and not Droop-PSC in order to reduce to cardinal methods that fail the mutual majority criterion in the single-winner case, which is most of them.

Note that PSC doesn't hold if some voters in a coalition back out-of-coalition candidates i.e. 1-winner example with Droop quota of 51:

26 A>B

25 B

49 C

STV would elect C here. Yet if A hadn't run, then B would've been 51 voter's 1st choice, so a Droop solid coalition would've been backing them, and since STV passes Droop-PSC, B would've guaranteeably won.

One could also consider various extensions of the solid coalition idea, in part to address examples like the above one; 5-winner example:

9 A>F>G>H>I>J

9 B>F>G>H>I>J

9 C>F>G>H>I>J

9 D>F>G>H>I>J

9 E>F>G>H>I>J

8 K

7 L

Arguably there is some kind of coalition of 45 voters backing candidates A through J here, and since the largest opposing coalition is 8 voters, D'Hondt would say that the 45-voter coalition ought to win all 5 seats. At that point, one could eliminate all candidates outside the 45-voter coalition (K and L) at which point A through E all are a Hare quota's 1st choice and must all win. This sort of thinking is generally what Condorcet PR methods such as Schulze STV do.