# 3 Unitary Proportional Variations on Sequentially Spent Score

Premise: After 2 years of thinking though the myrid options for 5 star PR methods I am tentatively convinced that Keith’s ‘Vote Unitarity’ concept -that ballot weight can be split between winners but never created or destroyed during the voting system’s calculation of winners- is key for fairness and equality. I love that his method of subtraction is much simpler to explain and count than re-weighting. But, I still feel like preference order should be taken into account more.

My gut is that considering the preference order of as many voters as possible is important for preventing the election of highly polarizing candidates when a faction could be represented by someone less antagonistic. I think it’s also the key for preventing strategic voting and incentivizing honest, expressive voting.

In Keith Edmonds’ Sequentially Spent Score method, (SSS), if a candidate you gave 3 stars to is elected, the max score you could give to any other candidate is 2. You only have 2 stars left.

But here’s the issue… Candidates who you originally gave low scores, 1s and 2s, now have the same score as your favorites. You are essentially subtracting different amounts from different candidates, and in subsequent rounds your ballot may no longer show your preference order or degree of support between a candidate who only had 2 stars originally and one who originally had 5. In subsequent rounds you might use up your last stars on a candidate you didn’t really like even though your favorite is still in the race.

Breakthrough? I think I’ve come up with two new and distinct ways to address this, and either would have a side effect of combating strategic voting too. Below I’ll outline each of those two solutions, and also a 3rd variation that includes both at the same time.

Please let me know if these new systems pass the various definitions of PR, if any of them satisfy the quota criterion, which of these Unitary methods your like best, and any other thoughts you might have.

ENTER A VARIATION: Sequentially Spent STAR 2.0
(SSS + Runoffs for each seat with voters still in play)
In this variation if a candidate you gave 3 stars to is elected, your max score for all other candidates is reduced to 2. You have spent 3 out of your total 5 stars, and you only have 2 left. (So far this is just like Keith’s SSS method.)

Your ballot no longer shows a preference order or degree of support between a candidate who only had 2 stars originally and one who originally had 5… but, this is corrected by a runoff in each round using voters’ real original preference order.

Description: For each seat up for election sum all scores. The two highest scoring candidates overall advance to a runoff. The finalist you prefer gets your vote. Your original preference order is fixed and never changes throughout the election. Once a voter has spent their full 5 stars their ballot is not included in future scoring or runoff rounds.

*Note: Previous proposals talking about runoffs didn’t specify that only voters who’s ballot is not spent get to participate in the runoffs. This is a new concept as far as I know.

Sequentially Spent STAR 2.0 Example Ballot:
Candidate A: 5
Candidate B: 3
Candidate C: 3
Candidate D: 2
Candidate E: 0
Candidate F: 3
Candidate G: 1

Sequentially Spent STAR 2.0 Example Election:

• In the first round Candidate B and C are the highest scoring, and B is preferred by more voters. B is elected and I’ve spent 3 stars.
• I still have 2 stars left to spend. In the second round candidates C and D are the high scorers. My vote would be 0 stars for either, I am still pulling for my favorite, candidate A.
• Then there’s a runoff between C and D. My vote goes to C in this round but since I didn’t give any stars to C in the last scoring round I still have my 2 stars left.
• In the next round E and D are the highest scoring. My ballot shows 0 for both. My vote goes to D in the runoff, they are elected, and I still have 2 stars.
• In the next round A and E are the highest scoring. My ballot shows 2 for A (my favorite) and 0 for E (my worst case scenario.) My vote goes to A in the runoff, they are elected, and I spend my last 2 stars.
• If there are any more seats up for election my vote is no longer in play. I’ve achieved full representation.

This variation could be easily acted out by people in a room holding their ballots and holding cards to represent their remaining ballot weight. Voters would go stand next to the candidates they’re voting for in each runoff. Voters whose ballot is spent would sit down.

Takeaways for this voter: My preference order and degree of support for the candidates made a difference, and even though my favorite didn’t win in the early rounds I still got to support them in the end. I feel good about that and in future elections I’ll feel comfortable showing my honest preference order and degree of support.

ANOTHER VARIATION: Sequentially Subtracted Score 2.0
(SSS with spent score subtracted from all remaining candidates. Voters in play till all 5 stars are spent.)
Rather than lowering the max score remaining for each voter, (essentially subtracting from your favorites while leaving your less preferred candidates alone) what if you subtracted the score spent from each of your remaining candidates? So, if you have spent 3 stars already (because a decent but not great candidate was elected in the first round) then a candidate who you had originally given 5 stars to would have 2, and a candidate who you had originally given 2 stars would have -1.

Description: For each seat, the highest scoring candidate is elected. The amount you spent on a winning candidate is subtracted from all your remaining candidates. Scores, including negative numbers, are summed. Once your 5 stars are all spent your ballot is not included in following rounds.

Sequentially Subtracted Score 2.0 Example Ballot:
Candidate A: 5
Candidate B: 3
Candidate C: 3
Candidate D: 2
Candidate E: 0
Candidate F: 3
Candidate G: 1

Sequentially Subtracted Score 2.0 Example election:

• In the first round Candidate B is elected. I gave B 3 stars, so we subtract 3 from all remaining candidates.
• I still have 2 stars left to spend. In the second round my ballot shows 2 for my favorite A, 0 for C, and -1 for D, etc. Candidate C is the highest scoring candidates overall so they are elected. My ballot showed 0 for C so nothing is subtracted from my remaining candidates.
• I still have 2 stars left to spend. My ballot shows 2 for A (my favorite,) 0 for C, and -1 for D, etc. D wins the next seat. My ballot showed -1 for D so nothing is subtracted from my remaining candidates.
• I still have 2 stars left to spend. My ballot shows 2 for A, 3 for E, 0 for F, and -2 for G. A wins the next seat. I spend my last 2 stars and my vote is done. I no longer have any impact on remaining seats.
• The process repeats with remaining voters until all seats are filled.

Takeaways for this voter: My preference order and degree of support for the candidates made a difference, and even though my favorite didn’t win in the early rounds I still got to support them in the end. I feel good about that and in future elections I’ll feel comfortable showing my honest preference order and degree of support.

BOTH ABOVE VARIATIONS COMBINED: Sequentially Subtracted STAR 2.0
(Spent score subtracted from all candidates and runoffs with all voters still in play.)

Description: Elect the STAR winner to the first seat. Subtract the score spent from each of your remaining candidates. For each seat up for election sum scores (including negative numbers.) The two highest scoring candidates are finalists and the finalist who is preferred wins.

Notes: Your original preference order is used in all runoffs and remains fixed. Once you have spent all 5 stars your ballot is done and your ballot is no longer included in the scoring round or in runoffs.

Example Ballot:
Candidate A: 5
Candidate B: 3
Candidate C: 3
Candidate D: 2
Candidate E: 0
Candidate F: 3
Candidate G: 1

• In the first round Candidate B is elected. I gave B 3 stars, so we subtract 3 from all remaining candidates. Voters who had given B 5 stars are done and their ballots are not included at all for future seats up for election.
• I still have 2 stars left to spend. For the second seat candidates C and D are the high scorers. My ballot shows 0 for C, and -1 for D. My vote goes to C in the runoff and since my ballot shows a 0 for them nothing is subtracted. (If D had won I would not have spent anything on them either. Spent score can not be regained.) Candidate C is preferred by more voters in play and is elected.
• I still have 2 stars left to spend. Candidates A and D are the highest scoring for the 3rd seat, and my ballot shows 2 for A and -1 for D. D is preferred overall and wins.
• I still have 2 stars left to spend. Candidates A and E are the highest scoring for the 4th seat, and my ballot shows 2 for A and -3 for E. A is preferred and wins. I spend my last 2 stars, and my vote no longer has an impact on remaining seats.
• The process repeats with remaining voters until all seats are filled.

This variation could also be acted out by people in a room holding their ballots and holding cards to represent their remaining ballot weight. Voters would go stand next to the candidates they’re voting for in each runoff. Voters whose ballot is spent would sit down. (See illustration of what a hand count ballot might look like. For the top variation a hand count ballot would be the same but with 0s instead of negative numbers.)

Takeaways for this voter: My preference order and degree of support for the candidates made a difference, and even though my favorite didn’t win in the early rounds I still got to support them in the end. I feel good about that and in future elections I’ll feel comfortable showing my honest preference order and degree of support.

Compare these results to the original Sequentially Spent Score proposal from Keith Edmonds.

Description: In Sequentially Spent Score the highest scoring candidate for each seat wins. Every time a candidate you support is elected, your max score available is lessened by the amount you spent so far. If you scored a winner 3 then 3 is subtracted from your max score remaining. Your 5 star candidate would have a score of 2, and anyone scored 2 or lower then they would still have the same score they started with… until someone else you support is elected. Your ballot remains in play until you spend all 5 of your stars.

Original Sequentially Spent Score Example Ballot:
Candidate A: 5
Candidate B: 3
Candidate C: 3
Candidate D: 2
Candidate E: 0
Candidate F: 3
Candidate G: 1

Original Sequentially Spent Score Example Election:

• In the first round Candidate B is the highest scoring. B is elected, and I spend 3 stars.
• I still have 2 stars left to spend. In the second round my ballot shows 2 for my favorite A, 2 stars for C, 2 for D, 0 for E, 2 for F, and 1 for G. C is elected and I spend my last two stars on them. My vote is not counted in remaining rounds. My favorite doesn’t get elected, but I am somewhat represented by B and C.

Takeaways for this voter If I knew that B and C were front-runners maybe I shouldn’t have voted for them in order to make sure my vote went to my favorite, A. (Free-riding strategy.)

2 Likes

This is why I prefer weighting to capping.

If you subtract the same amount of points from each candidate in the reweight step, then it all cancels out, so the method won’t be proportional.

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This is a bit of an extreme example I got from @parker_friedland but it shows why STAR at each round is not PR. It relies on group B only running one candidate but they can never get them elected. I’ll also exaggerate the numbers to make it more clear. Lets say we want to elect 10 winners in a score 1000 system with votes

51% = A:1 , B:0
49% = A:0 , B:1000

So the first runoff is A and B as

51% = A:1 , B:0
49% = A:0 , B:1

The A supporters then elect a candidate at the cost of 1 of their 1000 points so that effectively in the next round the runoff is identical with a new A candidate since they have 999 score points left. And they win again. Then again and again till all seats are used up.

I find this to be not very intuitive and only running one candidate would be a weird thing for B to do but I am sure there are more complicated examples based on this.

I think this is not PR either. Consider multiple clones A, B and C in a 3 winner race

34% = A:5 , B:0 , C:0
33% = A:0 , B:5, C:0
33% = A:0 , B:0, C:5

A is elected first but since the quota is 33 % the A voters are still in it after this

34% = A:0 , B:-5 , C:-5
33% = A:0 , B:5, C:0
33% = A:0 , B:0, C:5

So the A group wins the next round too making it not PR. I think the issue with this is the edge effect of going from full influence to none when you have spent all your score. Maybe there is a way to do something like this while also lowering ballot weight to account for the influence. When thinking about SSS, the subtraction is the mechanism but it is about removing influence. There is nothing fancy about the subtraction it is the subtraction combined with the lowering of influence.

The reason the why after spending 3 you cannot distinguish between somebody you gave a 2 and a 5 is because either one will give you full satisfaction. It is up to the others to decide which is better. You cannot break that without breaking vote unitarity. Of course you want to be able to take the person you gave 5 if everybody else views them as the same but this is not going to happen without there being a tie. In the case of a tie then sure lets take the winner with the higher sum before subtraction. If others have a preference then they get to decide because either will leave you satisfied.

If you scale instead of cap then you are just lowering your support for intermediate candidates in order to preserve preference. This does not really help you in the end because you are just going to lose both in a race where you could have won the intermediate candidate.

In your first example I’m confused. Is this a 2 candidate two winner example? Is this a 0-5 star ballot?
Why did a majority of the electorate not give almost any support to their favorite? 1000 what?
This looks like a single-winner example in a voting method which is not what we’re talking about.

More importantly, did @parker_friedland’s concern look at removing voters from the runoff after they achieve representation? I think his concern stemmed from allowing all voters to vote in all the runoffs, which would obviously not be PR.

In the second example group A is done after the first round because they have spent all 5 stars. So B wins the next seat and then C wins the 3rd seat.

Reworked Example of Sequentially spent STAR 2.0
Two winners. 4 candidates.
51 voters = A:5 , B:0, C:1, D:0
49 voters = A:0 , B:5: C:1, D:0

Total Scores Round 1:
A:5 255
B:0 = 245
C:1 = 100
D:0 = 0

A wins first seat. A block’s votes are fully spent. B wins next seat.

If the A block had scored A 4 stars and given nobody a max score then B would have won the first seat and A would win the next one.)

Elect 10 winners. A has many clones and B is only one person.

no 0-1000

It is exaggerated to make the effect clear.

Yes the influence is lowered. I think the issue is fundamentally that the selection is majoritarian but the influence lowering is utilitarian so there is a mismatch.

No they would not have spent it all. the quota is 1/3 and they are 34% so they have a little left. I don’t want to do the math to tell you how much. This is what I was getting at when I said that there also needed to be a way to lower influence.

You are missing surplus handling. Maybe that is the key here. You can have the scores go negative but you need to also use that for the surplus handling. I do not see exactly how to do it but there may be a way.

I think that you could do any selection you wanted with an allocation system. The basic allocation system is Allocated Score. Unless I am missing something an “Allocated STAR” or “Allocated STLR” can be invented.

No my concern was about having a runoff every single round while using a re-weighting algorithm that reduces the weight of voters supporting the winning candidate by different amounts depending on how high of a score they gave that winning candidate. Even when the amount of weight voters have in each runoff is weighted just the same as their votes are weighted each time the top 2 score winners are selected, if they are able to take advantage of the fact that you only need to make the scores you give to the two candidates in the runoff differ by 1 they can avoid having to have their ballot re-weighted as much as their opponents by giving a 1 to their candidate and a 0 to their opponent. Whether that runoff is weighted doesn’t fix that.

I think that’s the case. RRV/SDV style re-weighting are taylor made for score and approval methods and as a result they don’t really play nice with other selection methods. And as Keith pointed out, his method suffers from the same incompatibility issues as RRV and SDV.

If you want to have a method where the STAR winner is selected each round (rather then just the last round) then combining that with allocation is probably the simplest way to yield a PR system out of that.

Though as somebody who is a fan of that type of re-weighting and whom doesn’t have a huge preference between STAR and score/approval, I would sacrifice using STAR every round before sacrificing which type of re-weighting algorithm I think is superior.

The other easy way to work STAR into a PR re-weighting algorithm is to just only use STAR in the last round (the round that matters the most anyways) and just use score for all previous rounds (which is what the re-weighted STAR proposal does).

2 Likes

I prefer Sequential Monroe Voting to Allocated score because it has preference to the same people it would exhaust in the selection. I would think there is a elegant way to combine the SMV selection with STAR.

Similarly, in the surplus case. It would be best to allocate those who scored higher first but I am not sure this does not cause an issue with STAR.

@parker_friedland do you have a recommendation for a STAR variant of SMV. @Sara_Wolf is very keen to have a PR extension of STAR and I think all the Unitary and Thiele type systems are not possible.

Not really, unless you’re satisfied with just running STAR on the last round like with the re-weighted STAR proposal.

The thing about SMV is that it not only defines the re-weighting algorithm but also the selection method. If you didn’t use the top hare quotas of voters then it wouldn’t be SMV. And if you tried to run STAR on the weighted ballots from the hare selection, then you would be running STAR voting on the weighted ballots in the hare quota, you would be running STAR voting on the ballots that most support a given candidate, making the use of STAR voting kind of redundant. Even if the hare quota of voters that most support X would prefer Y under STAR voting, then the quota that you are exhausting would differ from the quota of voters you initially compared. The best way to resolve this might be to find the hare quota of voters who most support X and the hare quota that most support Y (where X and Y are the candidates with the most support among their hare quota) and then try to preform a runoff between those two. But then who should get a say in that runoff? All voters? The voters who are in at-least one of the two quotas (weighted by how much ballot weight they have in that quota or the weight of the which of the two they have more weight in if their ballot is in both)? The voters who are in both quotas (weighted by which quota they have less ballot weight in)? Add the weights from each quota together (if their ballot fully appears in 1 quota and isn’t included in the other, they have 1.0 votes in the runoff, if their vote is fully in both, 2.0 votes, if it is partially in 1 or both of the quotas, then just add the weight they have in either together)?

Doing something like this would both ruin the simplicity and elegance of SMV and it could also possibly have unintended undesirable consequences that we are not able to foresee. I also don’t see any benefit you would get out of it aside from being able to say that you are using STAR voting each round, and I don’t recommend adding all this extra complexity for no other reason besides the sake of being able to say that STAR voting is used every round.

This is sort of the lines I was thinking along

I would think that you keep the intersection of the two quotas and do the runoff. It would be one candidates best supporters vs the other candidates best supporters. In theory many would have scored them equally so it is really the people who have preference among the supporters.

Exactly, this is sort of where I got to when I made the last post. Coding SMV is already much harder than RRV or SSS so adding complexity is not a good idea. I hoped that you would see something I missed.

OK so it seems the best way to do a STAR PR is to have it be Allocated Score but with STAR. “Allocated STAR.” Its not a bad system. I would still take it over STV any day.

It’s worth noting that it does have the downside that if you use it to elect more then 2 winners it could potentially be non-monotonic in some extremely rare instances.

Possible scenario where this could happen:

When you give C a higher score (but everything else remains unchanged):

1st round: A and C enter the runoff, A wins. The voters who gave A the highest scores get exhausted.

2nd round: B enters the runoff against another candidate (doesn’t really matter who as long as B wins), wins, and then the voters who gave B the highest score whom were not already exhausted by A get exhausted.

3rd round: C and D enter the runoff, D wins because D has more support among A voters and A supporters were de-weighted less then B.

Result: A, B, and D win.

When you give C a lower score (but everything else remains unchanged):

1st round: A and B enter the runoff, B wins. The voters who gave B the highest scores get exhausted.

2nd round: A enters the runoff against another candidate (doesn’t really matter who as long as A wins), wins, and then the voters who gave A the highest score whom were not already exhausted by B get exhausted.

3rd round: C and D enter the runoff, C wins because C has more support among B voters and B supporters were de-weighted less then A.

Result: A, B, and C win.

The reason why this scenario might be possible is because the votes are exhausted differently based on whether A or B wins first. If there is a lot of overlap between A and B supporters, a lot of the voters who gave A the highest scores might also of given B the highest scores, meaning that the 2nd candidate to get elected is going to have their remaining scores also exhausted. This problem of the supporters of candidates that win latter getting de-weighed more exists in a lot of systems (including your SSS), not just allocation (this is one of the nice things about the divisor methods: the order by which previous candidates were elected doesn’t change the weight voters will have in fallowing rounds), however this problem can’t cause monotonicity violations when the single winner selection method passes the independence of irrelevant clones. The problem is STAR doesn’t pass IIA, which means that combining it with an allocation method that has this problem could potentially result in non-monotonicity violations (though if those violations could occur, they should be extremely rare and only be possible when electing at-least 3 winners).

Yes, I think you are right here. That’s a veto on this system for me. If I was going to entertain the idea of having a nonmontonic system I would do a sequential exhaustion system like IRNR. Well nobody can say we did not try.