# Hare-PSC's desirability, variants, and various implementations in cardinal PR

The 3 main cardinal PR proposals (SSS, SMV, RRV) all fail Hare-PSC, a criterion for PR methods stating that a group of voters comprising k Hare quotas who prefer a set of candidates above all others should always elect at least k of those candidates. Hare-PSC seems like an elephant in the room when discussing which cardinal PR proposals are best.

Something to consider is that if the Hare quota is 10 and some voters vote:

9 A
1 D>A

This is not technically a solid coalition, yet arguably A deserves a guarantee of winning just as much as they would if D wasnâ€™t running.

Some ideas for implementing Hare-PSC:

• Before running a cardinal PR method, first identify all solid coalitions, then use combinatorics to decide which candidates must win. Use the cardinal PR methodâ€™s selection method to decide which candidate within the PSC-guaranteed set is best, elect them, reweight their supportersâ€™ ballots, then recalculate the PSC set of candidates and repeat. Once all PSC guarantees are fulfilled, only then run the method as normal.

• When running a cardinal PR method, at each stage eliminate all candidates whose election would force violation of PSC. (I think this doesnâ€™t cause problems with cardinal methods since they pass IIA with unchanged ballots, but not sure.)

It must be a really quiet elephant because I have never heard of it. It seems like Proportional Justified Representation but I do not know without a definition.

I have made an Electowiki page with some discussion on it relating to cardinal methods: https://electowiki.org/wiki/Proportionality_for_Solid_Coalitions

Its definition is

5-winner example:

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.)

Note that PSC applies whether or not the set of candidates for a group can shrink i.e. 2-winner example:

10 A:10 B:1
10 C:10 D:1

Suppose B and D win. This satisfies the solid coalitions of 10 voters each who prefer (A, B) > all others and (C, D) > all others, but it does not also satisfy the Hare coalitions who prefer A > all others and C > all others.

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A 3-winner example, but this time there is technically no Hare solid coalition for the final seat:

2 voters most prefer c1-3, but prefer e1-4 over d1
1 voter most prefers c1-3 and d1
6 voters most prefer e1-3

By Hare-PSC, 2 e candidates must win, but there are no guarantees for the c candidates, despite a Hare quota loosely preferring them. I think by PSC logic, one of (c1-3, e1-4, d1) should be guaranteed to win the final seat.

2 to elect, max score 10:

n voters: A=10, B=0, C=9, D=9
n voters: A=0, B=10, C=9, D=9

If I understand you correctly, then your criterion demands the election of A and B, whereas most cardinal methods would go for C and D. But if thatâ€™s what you want, are cardinal ballots even desirable? You might as well go ordinal.

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Youâ€™ve got it. Cardinal Hare-PSC compliant methods may be preferable for those who want a method which is cardinal in the single-winner case but also maximally represents Hare quotas in the single-winner case (when looking at their favorite representatives, not their utility from the whole winner set). Also, itâ€™s worth considering that the cardinal information is still useful for deciding which PSC-compliant outcome is best. Itâ€™s simply one more option that ought to exist in case anyone wants it.

So you think that the 10, 0 result is better than the 9, 9 result above? I think it stems from the (in my opinion incorrect) view that each voter just has one representative that is effectively assigned to them. In a multi-winner election, I as a voter would want to have the set of candidates that overall best represents my views, not just any set that contains my single favourite candidate. And thatâ€™s also why I donâ€™t really agree with the definition of:

### Fully Satisfied Voters

The Fraction of voters who got candidates with a total score of MAX or more. In the single winner case getting somebody who you scored MAX would leave you satisfied. This translates to the multiwinner case if the one can assume that the mapping of score to Utility obeys Cauchyâ€™s functional equation which essentially means that it is linear. Higher values are better with a max of 1.

From here.

Obviously itâ€™s just a name though, and Iâ€™m not necessarily saying itâ€™s a bad metric to test for in itself.

Iâ€™m gonna pass on the question. Iâ€™m not trying to argue that PSC is a great thing, simply that there may be a lot of people who would be easier to swing around to cardinal PR if they knew there are good PSC-compliant variants.

I would simply note that Iâ€™m unsure whether this is what most people want out of multi-winner methods. At the very least, I think that when we take, for example, a 5-winner election, a strictly utilitarian view might say that a voter getting 5 candidates who are each 1/5 is equal to getting 1 candidate who is a 5/5, whereas I doubt this would be seen as equivalent by most people. As we discussed briefly earlier

thereâ€™s always ways to mix and match different properties.

Would this system be any good for your needs?

The following PR approval voting procedure is an approval limit of Schulze STV

A score for each candidate set is determined in the following way: The vote of each ballot is distributed amongst the ballotâ€™s approved candidates in the candidate set. The score for each candidate set is the largest possible vote for the candidate in the set with the smallest vote. The candidate set with the highest score wins the election.

example: 2 seats
approval voting profile
10 a
6 a b
2 b
5 a b c
4 c
The possible candidate sets are: {a b}, {a c}, and {b c}.

score for {a b} determined from
10 a
11 a b
2 b
score for {a b} = 11.5

score for {a c} determined from
16 a
5 a c
4 c
score for {a c} = 9

score for {b c} determined from
8 b
5 b c
4 c
score for {b c} = 8.5

set {a b} wins.

First off, thank you for being helping out and entertaining a discussion into something you donâ€™t prefer.

That method you mention is very interesting (Marylander discussed something similar but for Score: Monroe PR doesn't work properly
but PSC is about preferences, which Approval ballots donâ€™t really indicate. Take a look at this post An example of maximal divergence between SMV and Hare-PSC

Which I think actually shows a PSC failure for all the main cardinal PR methods. The only ways I see to get the compliance are to either start by electing someone in the solid coalitions for the first seat(s) until a non-compliant outcome is impossible, then running the voting method as usual, or eliminate candidates at each stage in such a way that a non-PSC-compliant outcome becomes impossible (but this has some weird issues as far as I can tell, such as voters in solid coalitions possibly getting double the representation by electing who they want when all candidates are uneliminated, and then getting a second opportunity to choose when everyone other than their coalitionâ€™s candidates is eliminated.)

I do want to point out that SMV passes a stronger PSC-related criterion than other cardinal PR methods: a solid coalition of k Hare quotas guarantees at least k of their preferred candidates win simply by max-rating them. Most other cardinal PR methods also require less-preferred candidates to be min-scored.

I meant to add that you could use e.g. the KP-transformation to make it a score method. I think it would be indifferent between AB and CD in the following case though:

1 voter: A=10, B=10, C=10, D=0
1 voter: A=10, B=10, C=0, D=10

However, you can brush that under the carpet by electing sequentially and it would prefer the 10, 0 to 9, 9 scenarios. But I think most methods that do would also be indifferent between the 10, 10 and 10, 0 scenarios.