RRV trends majoritarian in party list?

[Lonán Dubh]

Recently, I was playing around with Gerrymandering, and whether certain voting methods were capable of mitigating the impacts thereof, and in my thought experiments I came across the logic that caused me to seek out my Multi-Seat Range voting method (described here https://groups.google.com/d/msg/electionscience/IHopnj49aP8/igPWq0tZCgAJ ) in the first place. 

It turns out that with Party List elections (potentially including the selection of US presidential electors), RRV actually trends more majoritarian than STV or even Single Non-Transferable vote.

Imagine the following scenario (based on the WashigntonPost Gerrymandering Graphic and Pew survey of D/R political opinions), with 500 voters and 5 seats:

94:      A3    B5    C6    D4    E2    F1
64:      A4    B6    C5    D3    E2    F1
42:      A6    B5    C4    D3    E2    F1
120:    A1    B2    C4    D6    E5    F3
99:      A1    B2    C3    D5    E6    F4
81:      A1    B2    C3    D4    E5    F6

STV returns the following sequence: D, E, C, F, B, as does SNTV.

The sum difference between perfect representation and STV or SNTV in this scenario is 24.8%, with the majority of that coming from the two smallest factions being over/under represented.

RRV, however, returns D, D, C, D, D.

24% of the population gets 80% of the representation.  The sumed difference from perfect representation under Party-List RRV is 114.4%, largely from the 54% over representation of the plurality faction.

Now, obviously, this only applies to scenarios where a single faction can win multiple seats, and not if each line on the ballot corresponds to an individual, but I am somewhat horrified to find that there is,in fact, a scenario where STV returns a more representative result than RRV.

This makes me think that the "Important Criteria" consideration that Parker had as #3 in his list of requirements for a Theoretically Perfect Multi-seat Voting Method should probably include proportionality, given the number of nations who use some form of Party List for their elections (US Presidential Electors, Party List elections, MMP, etc).

----

For completeness, I also ran this through my Apportioned Range method, and got the following: D, C, E, B, F (the same set as STV/SNTV, but in a different [more centrist biased] order).

[parker friedland]

> This makes me think that the "Important Criteria" consideration that Parker had as #3 in his list of requirements for a Theoretically Perfect Multi-seat Voting Method should probably include proportionality,
> given the number of nations who use some form of Party List for their elections (US Presidential Electors, Party List elections, MMP, etc).

See my 4th requirement for a Theoretically Perfect Multi-seat Voting Method

[parker friedland]

> RRV, however, returns D, D, C, D, D.

Perhaps that is because D is so broadly liked among all the voters. If D was more polarizing, he would get less representation, but because he is so broadly liked, when D voters are re-weighted, everybody is re-weighted because everybody likes D.

> 24% of the population gets 80% of the representation.

No, everybody likes D to an extent (his lowest rating is 3 stars which isn't that bad) so even the voters who gave him 4's and 3's are represented by him, just not as much as the voters who gave him a 6 are.

There are many ways you could try to get a measure of how proportional the results are in each of the outcomes. Here are two of them:

1. Find the total satisfaction each voter gets from the result (where the total satisfaction is equal to the sum of the scores they provided to each of the candidates). Give a higher priority to voters who got between 0 and 1 satisfaction units a higher priority then voters who got between 5 and 6 satisfaction units by weighting the different utility values according to the harmonic scale (where 1 satisfaction unit equals 1 utility unit, 2 satisfaction units equals 1 + 1/2 utility units, 3 satisfaction units equals 1 + 1/2 + 1/3 utility units, 4 satisfaction units equals 1 + 1/2 + 1/3 + 1/4 utility units, etc.) According to this measure, the outcome with the highest average utility should be the most proportional.

2. Find the most optimum way you can make connections between voters and candidates where each voter is only connected to a single candidate, each candidate is only connected to 100 voters, where the average score each of the voter's gives to the candidate they are connected to is the highest. That score should give you insight into how proportional the results are. According to this measure, the arrangement based off of the outcome with the highest average score each voter is giving to each candidate that they are connected with should be the most proportional.

On Sunday, January 7, 2018 at 1:22:06 PM UTC-8, Ciaran Dougherty wrote:

[Warren D Smith]

not sure how bad the example really is.

See, you made it sound bad by calling them "D" and "C" then saying
"look at all those Ds and Cs! Especially Ds!" But in fact, if you had
used other names
(all the Ds in fact are different people, and the voting method RRV,
also STV for that matter, have no idea that all the Ds happen to be members of
the same party) then maybe it would have looked good, not bad.

It is not so obvious how to judge this. Actually, it often is not so obvious
how to judge plenty of multiwinner elections, not just this one.




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[parker friedland]

After carefully looking at the results, it is UNDISPUTEABLE that RRV produced a better result then STV.

Lets look at each group of voters specifically.

Each voter's combined satisfaction

                    Under STV:    Under RRV:
94 voters:      18                   22
64 voters:      17                   17
42 voters:      15                   16
120 voters:     20                  28
99 voters:       20                  23
81 voters:       20                  19

The only voters who get just a slightly better outcome under STV are the 81 voters at the bottom. The rest of the voters would get a much better outcome under RRV. The 94 voters at the top having their satisfaction increase from 18 to 22 alone should be enough to more then cancel out the 81 voter's at the bottom having their utility slightly decrees from 20 to 19.

And if you weight each voter's utility using the harmonic series to gadget how proportional each result is:

Each voter's combined harmonically weighted utility

                    Under STV:    Under RRV:
94 voters:     3.495              3.691
64 voters:     3.440              3.440
42 voters:     3.318              3.381
120 voters:   3.598              3.927
99 voters:     3.598              3.734
81 voters:     3.598              3.548

Average:       3.53472          3.67476

On Sunday, January 7, 2018 at 1:22:06 PM UTC-8, Ciaran Dougherty wrote:

[Lonán Dubh]

> Also, you said that you were testing your method (Apportioned Range Voting) but yet you keep saying RRV. You do mean ARV, right? Because RRV stands for re-weighted range voting.

No, I meant what I wrote.

I didn't say I was testing ARV, I said I was testing the impact of various methods with respect to Gerrymandering, and the majoritarian results of RRV (for the "baseline," single-district results) reminded me why I sought a different solution in the first place, rather than being satisfied with the one that everybody already knows about (RRV).

On Sun, Jan 7, 2018 at 1:50 PM, parker friedland <parkerf...@gmail.com> wrote:

Also, you said that you were testing your method (Apportioned Range Voting) but yet you keep saying RRV. You do mean ARV, right? Because RRV stands for re-weighted range voting.

On Sunday, January 7, 2018 at 1:22:06 PM UTC-8, Ciaran Dougherty wrote:

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[Lonán Dubh]

>  "look at all those Ds and Cs! Especially Ds!"  But in fact, if you had used other names (all the Ds in fact are different people, and the voting method RRV, also STV for that matter, have no idea that all the Ds happen to be members of the same party) then maybe it would have looked good, not bad.

This is why I specifically pointed out that this is a problem with Party List scenarios: those D's aren't different names, they're multiple seats filled from the same party's list. 

That said, I wonder whether RRV isn't somewhat problematic for Individual Candidate races, too, when the range is narrower than the field of candidates.  Under such scenarios, you will end up with voting blocs aggregating to, somewhere around 5.5 rather than 6 for their faction's candidates, and lowering the scores for the other factions similarly; I worry that under such a scenario, you might still end up with similar results from a nearly party-line vote.

But let me present another example that may make it easier to see and understand my concern. Here is the real world vote count from California from the 2016 Presidential Election, converted to a score vote as plausibly as I could (maximum for the single mark candidate, minimum for the antithetical candidate, scoring others as low as reasonably possible while maintaining relative preferences):

        
8,753,788 Clinton   >>>   C6    T1    J3    S3
4,483,810 Trump    >>>   C1    T6    J3    S2
478,500    Johnson >>>   C2    T2    J6    S1
278,657    Stein      >>>   C3    T1    J2    S6

SNTV: Clinton 35, Trump 18, Johnson 1, Stein 1 (Droop quotas, final elector to Trump, by virtue of the largest remainder)
STV: Clinton 35, Trump 17, Johnson 2, Stein 1 (Droop quotas, final elector to Johnson, after votes transferred from Clinton, Stein)
ARV: Clinton, 34, Trump 18, Johnson 2, Stein 1 (Hare quotas, final elector to Johnson, based on Range of last Hare Quota)

....and RRV: Clinton 40, Trump 15, Johnson 0, Stein 0

Does that make my concern any clearer?

Stein and Johnson each have more than a full Hare quota's worth of voters that score their candidate maximum, and score only their candidate above the median, yet neither candidate receives a single one of the 55 electors under RRV.

Heck, Trump has a full 17.6 Hare Quotas of voters that scored him as the sole candidate above the median, yet under RRV, he only gets 15 electors?

Even if you adjust the scores so that the minimum score is 0 (so that, eg, a Clinton elector doesn't count at all against a Trump voter), the results are still 37 Clinton, 18 Trump, 0 Johnson, 0 Stein (SNTV, STV, and ARV would return the same results).

If that's what happens when Johnson and Stein voters put maximal space between their preferred candidates and the others (while maintaining relative preferences), what is the point of such voters voting at all under RRV?   It seems to me that honest votes would be more likely to yield actual results for minor parties/factions even under SNTV than they would under RRV.  Does that not horrify anyone else?

[parker friedland]

>> Also, you said that you were testing your method (Apportioned Range Voting) but yet you keep saying RRV. You do mean ARV, right? Because RRV stands for re-weighted range voting.

>

>No, I meant what I wrote.
>
>I didn't say I was testing ARV, I said I was testing the impact of various methods with respect to Gerrymandering, and the majoritarian results of RRV (for the "baseline," single-district results) reminded me >why I sought a different solution in the first place, rather than being satisfied with the one that everybody already knows about (RRV).

I realized that afterwards when I read this part of your post:

>For completeness, I also ran this through my Apportioned Range method, and got the following: D, C, E, B, F (the same set as STV/SNTV, but in a different [more centrist biased] order).

That is why I then deleted that comment I made after I realized my mistake, but unfortunately, deleting a comment on the google groups page is not the same thing as deleting it in the email thread.

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[parker friedland]

Let's break this down again.

Each voter's combined satisfaction

                                            Under SNTV:    Under STV:    Under ARG:    Under RRV:
8,753,788 Clinton voters:      234                    236                231                 255
4,483,810 Trump voters:       148                    145                150                 130
478,500 Johnson voters:       113                    117                117                 110
278,657 Stein voters:            131                    132                130                 135

Under this scenario, at first glance, it does seem like it makes more sense for Trump (or Trump's party) to win a higher portion of the legislature because Trump voters are not getting a high satisfaction from the result in comparison to Clinton voters, and proportional voting methods are suppose to benefit the individuals who are going to get a worse result (by giving higher precedence to increasing the satisfaction of an unhappy voter from 0 to 1 then increasing the satisfaction of a happy voter from 99 to 100). So while Clinton voter's satisfaction increases by 19 under RRV in comparison to STV, and Trump voter's satisfaction decreases by 15 under RRV in comparison to STV, the decrease of 15 should be worth the increase of 19 because Clinton voter are already a lot more happy. However, that is only at first glance. If you take a closer look, you will notice that there are roughly twice as many Clinton voters as Trump voters, which could make it a worthy trade off.

Since there are roughly 2 Clinton voters to every Trump voter, and there are very little 3rd party voters, Clinton should control a the portion of the legislature that is twice as big as the portion Trump controls. It is troubling that in RRV, Clinton voters are getting a much better deal, by getting a portion of the legislature that is 2.667 times the portion that Trump voters are getting. But perhaps that is because the voters in this election are not using the full range of their ballot. If Trump voters had rated Clinton as a 0 and Clinton had rated Trump as a 0, then perhaps, the results would be more proportional.

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[Warren D Smith]

well, RRV and STV are not intended for party list, they are intended for
named candidates.

[parker friedland]

But perhaps that is because the voters in this election are not using the full range of their ballot. If Trump voters had rated Clinton as a 0 and Clinton had rated Trump as a 0, then perhaps, the results would be more proportional.

However it is a bit unsettling that when the minimum score changes from 0 to 1, so much of RRV's proportionality is lost. That is because suppose that voters are voting honestly, and because they cannot compare their own satisfactions of each of the candidates with other voter's satisfactions of each of the candidates, voters are expected to normalize their scores so they give their favorite a max score and their least favorite a min score. Suppose that a really unpopular candidate inters the race, who doesn't win any seats themselves, but manages to become almost all of the voter's new least favorite candidate, and as a result, each voter's min score of 0 that they gave to their previous least favorite candidate becomes a min score of 1 or 2. As a result, a lot of proportionality in the legislature will be lost.

There should be a criterion about this, called the linear transformation criterion, which states that if linear transformations are applied to the scores of each of the voter's ballots, the election result should stay the same. However, After writing this, I now realize that it is impossible for a proportional voting method to pass this because proportional voting methods need to give a higher priority to pleasing unhappy voters.

Also, because the un-liked candidate does not win a seat, they are an irrelevant alternative and yet they still influenced the election, which means that if you assume that voters normalize their scores (which they should), RRV does not pass the Independence of Irrelevent Alternatives Criterion.

[parker friedland]

> well, RRV and STV are not intended for party list, they are intended for
> named candidates.

It is possible for voters to give candidates from the same party very similar scores, so that statement does not discredit any possibly flawed RRV results being discussed.

[parker friedland]

> There should be a criterion about this, called the linear transformation criterion, which states that if linear transformations are applied to the scores of each of the voter's ballots, the election result should
> stay the same. However, After writing this, I now realize that it is impossible for a proportional voting method to pass this because proportional voting methods need to give a higher priority to pleasing
> unhappy voters.

Actually, never-mind. It is possible for a proportional cardinal voting method to pass this criterion. Monroe's system passes it. Perhaps Monroe's voting system should replace RRV as the ideal multi-winner extension of score and approval voting. When there are only 3 candidates, Monroe's system is O(N^3) sum-able because of that quality function I came up with to describe it, so implementing 3 winner Monroe's system in actual elections might be plausible.

[Warren D Smith]

Also note RRV has a tuning parameter K.
In the examples with small parties not getting any seats,
the tuning parameter might be able to cure that?
http://rangevoting.org/RRV.html

[parker friedland]

> Also note RRV has a tuning parameter K.
> In the examples with small parties not getting any seats,
> the tuning parameter might be able to cure that?

But what the turning parameter is not able to fix is the fact that RRV fails my Linear Transformations Criterion, and as a result, the proportionality of RRV election results could be effected by irrelevant alternatives in the way that I just discribed far too often. The only value of K at which RRV passes LTC is when K equals infinity, which is when RRV is the least proportional, so that tells me that assigning a lower value of K to make it easier for third parties to win seats will only make the degree to which RRV fails LTC worse and the degree that irrelevant alternatives are able to influence the proportionality of the election even greater.

[Lonán Dubh]

> If Trump voters had rated Clinton as a 0 and Clinton had rated Trump as a 0, then perhaps, the results would be more proportional.

You're assuming that I was using 0-6, when I was in fact using 1-6.  But if we realign the scores for RRV calculation (1-6 --> 0-5), RRV does offer a more be more proportional result 37/18/0/0, rather than 40/15/0/0.

The trouble is that when we do, your calculation (if I understand it) asserts that it's a worse result. The Trump faction's satisfaction went up by 15 points, certainly, but the Clinton faction's went down by the same amount, and the Stein faction's satisfaction dropped, too.  And, as you already cited, there are twice as many Clinton voters as Trump, so their satisfaction has greater impact on the group's average satisfaction (recalculating the all the results, based on the minimum zero/off-by-one normalization)

                                        SNTV:     STV:     ARV:       RRV      Corrected RRV:
8,753,788 Clinton voters:     179         181       176          200            185
4,483,810 Trump voters:       93           90         95           75              90
478,500 Johnson voters:       58           62         62           55              55
278,657 Stein voters:           76           77         75           80              74

Average Satisfaction:         145.3      145.7    144.1       152.6           147.9

As we see, SNTV, STV, and ARV are all within 1.6 points of each other, while (offset) RRV is a full 6.9 points higher than the best of them... but you lose more than 2/3 of that "improvement" when we align the scores for proper calculation with RRV.

That means that, according to the "Undisputable" Voter Satisfaction calculation, in two different scenarios, the more majoritarian RRV results are "better" than the more proportional STV/SNTV/ARV results.   Further, when the offset is corrected to make the RRV math work correctly, we get a more proportional result, which the calculation calls "worse" than the uncorrected RRV result.

As such, I question the validity of that calculation for multi-seat races.

More specifically, I question the validity of calculating Satisfaction for the electorate at large; it makes sense when there is only a single seat, and therefore a single decision, but wouldn't it make sense to calculate it for every seating decision individually?

As proof of the problem with that calculation, I offer the calculation for the currently used method, single mark, winner take all:

                                        SNTV:     STV:     ARV:       RRV:      WTA:
8,753,788 Clinton voters:     179         181       176         185          275
4,483,810 Trump voters:       93           90         95           90             0
478,500 Johnson voters:       58           62         62           55           55
278,657 Stein voters:           76           77         75           74          110

Average Satisfaction:         145.3      145.7    144.1       147.9       176.1

If the calculation of Satisfaction at large is valid for multi-seat elections, then that would mean that the optimal solution is Range Winner Takes All; that increases the average satisfaction by more than 25 points over any of the other options.

This is why I kept referencing the idea of Hare Quotas, why my design of ARV attempts to link voters to seated candidates: I assert that any VS calculation should be done independently for each seat, and should only consider the voters that seat theoretically corresponds to.

After all, why should any voter have a say in a candidate that is supposed to represent someone else?

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[Lonán Dubh]

> > well, RRV and STV are not intended for party list, they are intended for
> > named candidates.

> It is possible for voters to give candidates from the same party very similar scores, so that statement does not discredit any possibly flawed RRV results being discussed

Further, all 50 states plus DC run a Closed Party List election at least once every four years for Presidential Electors (ME and NE calculating the results twice, once at the state level and once at the district level), and numerous nations use some form of Party List (Open PL, Closed PL, Mixed Member Proportional, etc) for their national elections.  If a method doesn't work properly in the most common forms of Proportional Representation elections, is that not a significant flaw?

On Mon, Jan 8, 2018 at 12:20 AM, parker friedland <parkerf...@gmail.com> wrote:

> well, RRV and STV are not intended for party list, they are intended for
> named candidates.

It is possible for voters to give candidates from the same party very similar scores, so that statement does not discredit any possibly flawed RRV results being discussed.

On Monday, January 8, 2018 at 12:05:56 AM UTC-8, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:

well, RRV and STV are not intended for party list, they are intended for
named candidates.

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[parker friedland]

The average satisfaction rating's themselves are not a good indication of proportionality. However comparing each voter group's average satisfaction can be useful for determining when one outcome is obviously better then another (such as when everyone gets an equivalent or happier result in one outcome then another, as was the case in my earlier post where I said that it was indisputable that RRV picked a better outcome). When some voting benefit more under one voting system and others benefit more then another voting system, then comparing group satisfactions alone is not enough to determine how proportional two outcomes are because average satisfactions themselves are biased in favor of majoritarian outcomes. That is why in a previous post, I also included each voter's harmonically weighted utility (which is biased in favor of proportionality) to judge each method's proportionality. I shall do that again with your second set of election results, however this time I will do it with the natural log function (but add 1 to the inputted satisfaction fist) instead because it is an estimate of the harmonic function and the easiest way of making the harmonic function on Desmos requires using factorial functions, and those blows up when you use numbers greater than 170.

Each voter's combined harmonically adjusted utility (with natural log estimate)

                                             SNTV:     STV:       ARV:       RRV:      Corrected RRV:      WTA:
8,753,788 Clinton voters:     5.19         182        5.18         5.30       5.23                        5.62
4,483,810 Trump voters:      4.54         5.20       4.56         4.33       4.51                        0.00
478,500 Johnson voters:      4.08         4.51       4.14         4.03       4.03                        4.03
278,657 Stein voters:           4.34         4.14       4.33         4.39       4.32                        4.71

Average:                              4.930       4.929     4.928       4.930     4.938                      3.747

[Lonán Dubh]

...but you're still not answering my question.

Why should the opinion of voters who are represented by Seat #1 be considered at all when it comes to choosing any other seat?

I'm not questioning your math, I'm questioning the usage of the algorithm in this case, since any child can see that a small plurality gaining a supermajority of the seats is not a proportional result.

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[parker friedland]

> Why should the opinion of voters who are represented by Seat #1 be considered at all when it comes to choosing any other seat?

Let's suppose that in a district, there are 50 republican voters and 50 democratic voters. Which set of representatives would do a better job at representing that district?  Outcome A where 1 democrat approved by all of the democrats and 1 republican approved by all of the republicans wins? Or outcome B where 1 democrat approved by all of the democrats and 1 independent approved by all of the democrats and 98% of the republicans wins. If you think that outcome A is more representative, because while the republicans are being represented a tiny bit less, but the democrats are being represented so much more that overall the population is more represented by outcome A, then I would agree. But in order to come to that conclusion, you had to take into consideration the opinions of the voters who were already represented by the democrat when deciding whether the independent or republican should win a seat.

>I'm questioning the usage of the algorithm in this case, since any child can see that a small plurality gaining a supermajority of the seats is not a proportional result.

Are you talking about my use of the harmonic function to determine which results do a better job at increasing voter representation? Because in those results, the winner take all result had the smallest proportionality index.

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[Lonán Dubh]

> If you think that outcome A is more representative, because while the republicans are being represented a tiny bit less, but the democrats are being represented so much more that overall the population is more represented by outcome A, then I would agree.

Not quite. A is more representative because the group of voters that does not yet have a representative is better represented by the candidate that all of them like rather than the one that most of them like. The people who are represented by the other candidate need not be considered at all. Indeed, they should not be considered, because to do so would be counting that group's vote twice.

The democrats' opinion of the candidate that is going to represent the republicans is no more relevant to their value as their (the r's) representative than the opinion of rural Texan voters' opinion of a representative for downtown Boston, for the same reason: they aren't who the candidate is supposed to represent.

So, no, it doesn't matter whether the independent candidate is liked by the democrat voters, nor even how much they like the republican candidate. All that matters is that the voters who will be represented by them (in this case, republicans) would be best represented by.

The fundamental problem that I have with the calculation you offered is that it is a single calculation attempting to measure the goodness of 55 independent decisions.  For a single decision, it's great. For multiple decisions, though?

If you were to apply it to your most recent scenario, would it not select for outcome B, just as it selected the clearly majoritarian D,D,D,D,C outcome over the unquestionably more proportional B,C,D,E,F outcome?

[Toby Pereira]

Does RRV work better in these cases with the KP transformation? For example:

2 voters: A=10, B=9

1 voter: A=9, B=10

So I take it you're saying that party A should get twice as many seats as party B? If we KP transform these ballots into approval, we get:

1.8 voters: AB

0.2 voters: A

0.9 voters: AB

0.1 voters: B

Equals:

2.7 voters: AB

0.2 voters: A

0.1 voters: B

Under RRV/Thiele, the 2.7 all-approving ballots are effectively ignored, leaving:

0.2 voters: A

0.1 voters: B

So this should give the result you want.

I would always use the KP transformation when using RRV. It gives what I would consider to be undesirable results otherwise.

On Monday, 8 January 2018 08:50:29 UTC, parker friedland wrote:

[Toby Pereira]

On Monday, 8 January 2018 08:34:05 UTC, parker friedland wrote:

> There should be a criterion about this, called the linear transformation criterion, which states that if linear transformations are applied to the scores of each of the voter's ballots, the election result should
> stay the same. However, After writing this, I now realize that it is impossible for a proportional voting method to pass this because proportional voting methods need to give a higher priority to pleasing
> unhappy voters.

Actually, never-mind. It is possible for a proportional cardinal voting method to pass this criterion. Monroe's system passes it. Perhaps Monroe's voting system should replace RRV as the ideal multi-winner extension of score and approval voting. When there are only 3 candidates, Monroe's system is O(N^3) sum-able because of that quality function I came up with to describe it, so implementing 3 winner Monroe's system in actual elections might be plausible.

I'm not a fan of Monroe's method. See http://rangevoting.org/MonroeMW.html and the heading "Pereira's complaints about Monroe".

[Toby Pereira]

I'll look at it again with standard RRV.

2 voters: A=10, B=9

1 voter: A=9, B=10

A total = 29

B total = 28

A is elected.

Using the 1/(1+SUM/MAX) weighting, the ballots are now:

2 voters: A=5, B=4.5

1 voter: A=4.74, B=5.26

A total = 14.74

B total = 14.26

A is elected again. Under the KP transformation, the second seat would be a tie (using as we are D'Hondt divisors). Now the ballots become:

2 voters: A=3.33, B=3

1 voter: A=3.21, B=3.57

A total = 9.88

B total = 9.57

A is elected again!

Maybe I'll try to work out how many seats A would win before the first B seat a bit later, but for now I think this shows that using the KP transformation makes a big difference, so it would be worth working through the previous examples again with that in mind.

Toby