# Proportionality for Solid Coalitions

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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 (i.e. they are solidly committed to/solidly support these candidates), 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

### 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 is the same as Hare-PSC but holds for Hagenbach-Bischoff quotas instead (many authors call the Hagenbach-Bischoff quota/Hb quota a Droop quota), and requires the solid coalition's preferred candidates 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, and more generally guarantees that a majority will always win at least half of the seats.

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

Most methods that pass weak forms of PSC (i.e. any method that passes a weak form of Droop-PSC) allow a majority to strategically vote to get at least half of the seats. The more candidates there are that are preferred by a solid coalition, however, the less likely this may be; for example, Approval voting will likely pass the majority criterion in practice more often than the mutual majority criterion, because things like the chicken dilemma may encourage voters to not approve all solidly supported candidates.

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):

(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 in the set 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.)

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 a quota of voters, with each quota of voters giving no support to any other candidates, and the solid coalition as a whole giving no support to any candidates other one of the k candidates, then at least k of their preferred candidates must win.

SNTV and cumulative voting pass 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]

The reason Droop-PSC guarantees a majority wins at least half of the seats is that majority solid coalitions always constitute more voters than the number of Hagenbach-Bischoff quotas corresponding to half of 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 left 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 HB quota here is ~16.667, and thus by Droop-PSC the majority has (51/16.667 rounded down to the nearest number)=3 seat or 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 STV using Hare quotas, which passes Hare-PSC, yields:

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]

### Solid coalition overlaps

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.

## Complying methods

STV, Party list, and Expanding Approvals Rule pass forms of PSC. Specifically, the choice of quota in those methods determines which type of PSC they pass.

D'Hondt passes Droop-PSC, so many methods that reduce to it in the party list case (such as Schulze STV), with the notable exception of most cardinal PR methods, pass it too.

## 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.

## Factions

Inherent to the discussion of PSC are factions and subfactions, which are defined here as solid coalitions (i.e. they prefer some candidates above all others). PSC not only guarantees a certain number of seats to a political faction or party, but also guarantees that the wings or subfactions of that group are also proportionally represented i.e. if 40% of the voters are Democrats, and 50% of the Democrats are more liberal Democrats, then 40% of the seats will go to the Dems and 20% of all seats will go to liberal Dems. There is some discussion over how valid the PSC guarantees for subfactions are, however; for example, in Party list, the ordering of the candidates on the list may not accurately represent the subfactions (i.e. if there are 5 spots on the list, with the party expected to get 4 seats, and there is a subfaction that constitutes 25% of the party, then that subfaction proportionally deserves 1 of the 4 seats, but the party leaders might put other subfactions' candidates in the top 4 spots of the list, effectively preventing that subfaction from winning a seat).[4]

## Notes

With rated ballots, it is possible for a voter to express less-than-full support for any candidate. Because of this, one way to apply PSC to rated ballots and thus cardinal PR methods would be to require that every voter in each solid coalition give at least one candidate in their coalition the maximum score. An even stricter requirement could be to require them to give every candidate in the coalition a perfect score, though this could instead be thought of as a genuinely slightly weaker form of PSC; it is already satisfied by SMV. This is modeled off of the majority criterion for rated ballots.

### Single-winner case

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.

### Free-riding

Though Droop-PSC guarantees a majority half of the seats, it doesn't guarantee a plurality group half the seats when it could take them using vote management. 5-winner example:

51 A>C 49 B 10 C

The HB quota is (110/(5)+1)=~18.333, so A is guaranteed only (51/18.333 rounded down) = 2 out of the 5 seats, less than half. With vote management, Party A could split into 3 groups of 17 votes each, and most PR methods would then give them 3 seats. So it may be worth considering a stronger type of Droop-PSC based on giving solid coalitions at least as many seats as they'd get in D'Hondt. Schulze STV is an example of a method that does so.

### Semi-solid coalitions

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.

### Other multiwinner properties analogous to PSC

Properties analagous to PSC can be considered for multi-winner voting methods that aren't proportional. For example, Bloc Score voting guarantees that a majority solid coalition can elect all of its preferred candidates if they set their approval threshold between their preferred candidates and all other candidates.

### Finding solid coalitions from ballots

Because a solid coalition is defined as a group of voters who all agree on their k highest-ranked candidates, it can be found by looking at the first choice(s) of a ballot, then the second choices, etc. until some group of voters are found to be in agreement on the candidates examined so far. Note that if some voters equally rank candidates, then it is necessary to avoid counting more candidates on their ballots than the number of candidates that have already been examined on other ballots.