@jack-waugh The cost is that the forum is difficult to navigate, and it is hard to stumble upon interesting material. For example, say I'm doing research into a novel proportional representation method for a legislature that gives substantial control to parties for candidate nomination (e.g. closed list). I've both developed new mathematical models as well as designed some statistical analysis on simulated data and created some visualizations.

Do I put that thread in Research and Projects, in the subcategory Research, or in the subsubcategory Simulations. Actually wait no, I want this to be actually used in my home country, so I should put the thread in Election Policy and Reform, but that one too has a seemingly relevant subcategory Forms of Government which looks like it might be right since that's what my method is: a new form of government.

Don't even get me started if I should put it in the top level category of Voting Method Discussion, its seemingly synonymous subcategory Voting Methods, or that subsubcategory Multi-winner, or the subsubSUBcategory Proportional Representation. Any of these could be the correct category, unless of course I want to just stick my thread under New Voting Methods and Variations like everybody else seems to be doing

There are 10 different places that I could reasonably put my thread, many of which are redundant, and many of which are barren. Personally, I think this is a huge and unnecessary barrier to engagement for new users of the forum.

@marylander I suppose it depends on how you define 'efficient.' Rule X satisfies EJR, and I would not say it is much more complicated than any other quota-spending method! Computationally it is the same amount of work.

no excuse to use a method that doesn't pass PJR

Completely agree. This is why welfarist rules like RRV are basically a non-starter in my eyes.

@jack-waugh

How about

• Single Winner
• Proportional Representation
• Other Reform Discussion
• News / Advocacy / Projects
• Meta / Forum Business

And with no subcategories. Just those 5 as top-level.

@jack-waugh That would be identical to Borda.

@jack-waugh The existence of this kind of construction is precisely why I am so obnoxiously pedantic to everybody about how they are defining the terms they are using; great example.

I'm not sure if anybody else feels the same way, but I find this forum very confusing to navigate. There are like 30+ categories and many of them are either entirely barren or somewhat redundant.

Given how low-traffic this forum is already, probably would be good to be as low-friction as possible to use. Could we possibly get rid of like, the majority of categories and only keep maybe 4 or 5?

Hi,

Recently I have been interested in the method "Rule X," introduced in this paper as well as section 3.5 of this paper. There are rigorous definitions there, so in this post I'll try just to give intuition and informal descriptions.

I will describe how it works on approval ballots, then how I have attempted to extend it to score.

The basic idea is: each voter starts with $1 (or just voting power 1, but money is a more fun metaphor). A candidate costs a Hare quota $q to elect. We can interpret an approval from a voter as willingness to spend their remaining budget to elect that candidate. Rule X sequentially chooses the candidate who can be purchased for $q at the lowest uniform price (not exactly uniform since some voters may exhaust their entire budget).

It's quite similar to, for example, Sequentially Spent Score, so let's compare the differences (using approval ballots!). Say the quota is $20. There are 35 supporters of candidate A who indicated a willingness and ability to pay $28 at $0.8 each. There are 30 supporters of candidate B who indicated a willingness and ability to pay $30 at $1 each.

SSS would choose B on this round, since the indicated demand is higher. Rule X would choose A on this round, since the price per supporter is lower. For each method they remove $q from the budgets of the supporters of the winner, but they also do this in different ways.

SSS treats a voter's remaining budget more like a multiplicative deweighting (yes I know this is handwavey), and subtracts a fraction of the $q in proportion to that voter's indicated willingness and ability to pay. On the other hand, Rule X treats a voter's remaining budget more like a subtractive deweighting, and chooses to subtract uniformly the smallest price possible such that if all the supporters pay that (up to their remaining budget) it will equal $q.

The reason I am interested in this rule is its excellent axiomatic characteristics. It satisfies Extended Justified Representation (EJR) and is a logarithmic approximation to the core.

I have extended Rule X to scored ballots with the following principle: when a voter gives a candidate a score of s < 1, then all payments made by that voter to elect that candidate should be made with efficiency s, so if a voter has scored a candidate 0.3 then for price set at 0 < p <= 1 they will spend 0.3p. I'll illustrate this by way of example, but I can draw up a formal definition if the example does not make it clear enough what I'm doing.

• There is a coalition of 22 voters for candidate A with $0.6 remaining who scored A as 0.7
• There is a coalition of 31 voters for candidate A with $0.35 remaining who scored A as 1.

Say the quota is again $20. Then the price p is the solution to 22*0.7*min(p, 0.6) + 31*1*min(p, 0.35) = 20 so the price will be set at p = $0.458. The coalition of 31 voters will be exhausted and the coalition of 22 voters will spend 0.7*p = 0.32 each to be left with $0.28 remaining.

If there is no other candidate that can be purchased for a price of lower than $0.458 then A will be selected and the ballots will be spent as above.

You may ask "what if no candidate can get a full quota of demand?"
This is a good question. What the authors of the method suggest is just to give every voter a little money uniformly until some candidate can be purchased (this is seq-Phragmen). Alternatively, it could just iteratively selected the highest-demand candidate and completely exhaust the budgets of its supporters. Note that this is exactly what SSS and Allocated Score do, so they run into the same problem of 'unaffordable' candidates, just it is a little harder to see.

There is already significant discussion in those papers about Rule X in general. Of course, if there are glaring issues with the approval variant that would be important to hear, but what I am more interested in is an evaluation of my attempt to extend it to score ballots.

• Does the scored variant have any holes in terms of quality or strategic behavior?
• Is this the most natural way to extend Rule X to scored ballots or is there another that makes more sense?

Looking forward to hearing thoughts, and again if my informal definition-by-example is not sufficient I am happy to provide pseudocode or a real formula.

@rob I'm just going off of the definitions, no assumptions needed

Agreed that voters' ballots will likely change depending on the candidates in the race in Approval more with a higher probability than in most other voting rules.

@Toby-Pereira to answer your question I'm looking into it. Arrow's Theorem actually has quite a few (slightly different) formalizations, and it looks like what I said is technically not true for the version defined on Wikipedia since that one only allows (strict) linear orders, but I feel quite sure I saw a formalization where the domain was all (weak) linear orders. I will try to find it.

@cfrank I don't mean for this to sound too harsh but I figure you wouldn't have posted this if you did not want honest feedback: I think you are ascribing significantly too much philosophical meaning to scores.

Ultimately a score a voter gives to a candidate is just a number they happened to choose within the rules of the game set before them. (The "game" being the election). I would not go anywhere near as far to say that score represents something as meaningful as "consent." I mean, just consider the case where there are only two viable and popular candidates but you (personally) hate both of them. I would fully expect a voter to grit their teeth and strategically score one of them highly, but this hardly means they "consent" to that candidate in the way I suspect you mean it.

That being said you touch on some useful ideas here. You bring up the topic of positional dominance around the 11:24 mark and this is something people study in the topic of generalized scoring rules, and there is also sort of a one-off paper I found once on a certain kind of generalized Condorcet winner according to positional dominance (this paper I think).

Also purely as a matter of engineering, the metric you propose around minute 18 (bucket into quantiles and then take some uniform mean) might likely work well in some senses, and actually if I am understanding correctly it is pretty similar to something @Ted-Stern has been trying to sell me on with "Top Biased Hare Quota" .

However, I think you do not give enough attention to transparency and understandability of a democratic process. If you are doing something very engineering-y e.g. trying to optimize a load balancer and you want to use some kind of voting mechanism to decide where to send your item, possibly you could use a rule like this as some kind of noise-robustness mechanism.

However, as it stands this is incredibly complicated to understand, especially if you are determined to include the part where the parameters are self-updating from one election to the next. And I do not think it is conducive to a political environment where voters feel like they can trust and understand the system and think that it is fair.

@rob

Just to expand once more:

https://twitter.com/mcpli/status/1574795032967712773

Check out the gerrymandering shenanigans happening down south *right now*

Due to incredibly racially polarized voting, Black voters can get a majority in only a single district. This is not an effect of FPTP vs Condorcet vs Approval vs whatnot---since we are talking about true majorities here the results will all be the same.

Proportional representation means that gerrymandering like this is impossible, or at least significantly more difficult and for less potential impact.

@rob

You have many good points there and I'm sorry if I sound like I am brushing them off, but I cannot help but feel like you are ducking the question.

Let's put aside toy examples, mathematical models, and speculation. The facts of real life are

1. Alabama voters are highly polarized along racial lines, and this is mostly for historical & social reasons and has little to do with the choice of voting method.
2. Districts are being drawn such that Black voters are a majority in only a single district, despite being over a quarter of the population
3. Any method satisfying the Majority criterion, including Condorcet, will not give Black voters adequate representation in the legislature. Because of point 1., we do not have to think too hard about what constitutes 'adequate' representation since most voters are simply polarized on racial lines.

And the question is: how can we elect more Black representatives commensurate with the Black percentage of the population without PR? I agree with you that Condorcet is excellent for single-winner elections. But Condorcet + gerrymandering is not excellent for multi-winner elections.

I have also lived in Alabama by the way, about two years in Ft. Rucker, but this was before I was of voting age.

what is "proportional" about PR if it doesn't refer to parties -- or at least distinct, hard-edged clusters. Can that be expressed in a simple sentence?

In layman's terms, these definitions boil down to the idea that any x% of the electorate should control x% of the seats. Basically any collection of ballots, if they constitute at least some number X quotas, and if that collection is sufficiently 'cohesive' (meaning they agree on some candidates), then they deserve at least X elected winners.

Here, 'cohesiveness' does not have to be on party lines, and parties do not even have to exist. The coalitions are determined post-hoc based on how the submitted ballots overlap in support.

@rob

I do notice that, on electowiki, it says "In the case of non-partisan voting, the definition of proportional representation is undefined".

Electowiki is (largely) written by people who tend not to read academic research, and I would take anything you read there with a grain of salt. PR emphatically does have rigorous definitions---in fact there are multiple. My favorite is Proportional Justified Representation.

Black people are an obvious distinction, because in most cases, you are either Black or not Black, and it is seen as an important part of your identity. Under a PR system, would there be a "Black party"? If not, what good does PR do in this case?

Yes, in the case of Alabama, voters are incredibly polarized across racial lines and we should almost certainly expect to see a Black coalition. I also wish you would not equate PR with party-list PR, because while I do think party-list PR is fine, there are plenty of PR schemes which do not require parties.

That district map you sent is not the one I am referring to, which is being drawn this year and is heavily gerrymandered. It doesn't matter if you use the fanciest Condorcet method in the world when the districts are drawn like this---nonblack voters have a true majority in nearly every district.

Software developers are also a minority. I was a dog owner. Dog owners are also a minority. I was over 6 feet tall. People over 6 feet tall are also a minority. Everybody is in tons of minority groups.

Exactly! Yes yes yes. Everybody is in tons of minority groups---why not let people choose which of these minority affiliations matter most to them, rather than artificially slicing along geographic lines?

Generally speaking, gerrymandering is a form of Simpson's paradox

It does not matter if you find the "median in each district," since depending on how you draw the districts this can give you an overall outcome very very very far from the median of the entire electorate.

In your thermostat example, the median of [30, 31, 32, 70, 70, 70, 70, 70, 90, 91, 91, 92, 92, 93, 94] is 70, but the median of the medians of
[30, 90, 91], [31, 91, 92], [32, 92, 93], [70, 70, 70], [70, 70, 94] is 90.

@rob said in Are Equal-ranking Condorcet Systems susceptible to Duverger’s law?:

My argument is that a good, median seeking method (Condorcet compliant, blah blah blah) would solve the problem just as well as using PR.

Let's return to the example of what's happening in Alabama right now. How would using Condorcet solve the fact that Black voters are a minority in nearly every district, yet are a substantial percentage of the population?

@rob said in Are Equal-ranking Condorcet Systems susceptible to Duverger’s law?:

They just seem very vague and hand-wavy, and they don't generalize well to situations with nuance and real-world messiness.

I don't think it has to be hand-wavy whatsoever---it's just how we tend to talk about it here. Try looking at definitions like Proportional Justified Representation. Very formal.

@rob

Just to expand once more:

https://twitter.com/mcpli/status/1574795032967712773

Check out the gerrymandering shenanigans happening down south *right now*

Due to incredibly racially polarized voting, Black voters can get a majority in only a single district. This is not an effect of FPTP vs Condorcet vs Approval vs whatnot---since we are talking about true majorities here the results will all be the same.

Proportional representation means that gerrymandering like this is impossible, or at least significantly more difficult and for less potential impact.

@rob

Voter candidate choice tends to be driven by demographic factors and social pressure significantly more strongly than it is driven by any romanticized idea of policy or ideology.

If you pick basically any minority group based on ethnicity, income, ability, age, profession, religion, nationality, etc. etc. you will find that not only are they underrepresented in the legislature, but that it can be difficult to draw single-winner districts at all such that these communities could get a seat, even if one were to intentionally set out to do so.

Splitting seats up by geography is one way to guarantee some amount of diversity, but doesn't it seem to you like sort of an arbitrary axis along which to slice people? Why not split seats up by income level? Or gender? PR lets voters choose which of these axes they lie on matter most to them.

@rob said in Are Equal-ranking Condorcet Systems susceptible to Duverger’s law?:

Maybe someone can tell me what PR would give us that single winner -- but selected with a good method such as a Condorcet compliant one -- doesn't.

The ability to represent in government minority groups who are not a majority in any particular geographic location. This is commonly known as the "Massachusetts Problem," named after the difficulty of drawing districts in MA to give Republicans adequate representation.

@rob said in Are Equal-ranking Condorcet Systems susceptible to Duverger’s law?:

To someone who considers themselves an independent

Do you think that if there were, say, 10 parties, that none of them would be likely to closely match your desired platform?

@rob said in Are Equal-ranking Condorcet Systems susceptible to Duverger’s law?:

Regardless I don't see PR/multi-winner having much impact in the US,

I definitely don't agree here but I think Matthew Shugart among other preeminent political scientists can probably defend the idea more eloquently than I can

Open letter to congress to end single-member congressional districts and adopt proportional representation

Emergency electoral reform OLPR for the US House

Wait, Nader the Condorcet winner? I've never heard that.

Sorry, brain fart, you're right. Gore was likely Condorcet winner. I'll use the "I couldn't read yet when that election happened" defense

@rob said in Are Equal-ranking Condorcet Systems susceptible to Duverger’s law?:

Are you saying that the number of social cleavages exists independently of the voting method?

Pretty much yeah. Social & demographic affiliation drives policy preferences much more strongly than the other way around. This is known sometimes as the "funnel of causality." It has been studied from a couple different angles by various political scientists but maybe the most notorious work is Democracy for Realists by Achen and Bartels.

Remember an election is about much more than the tabulation algorithm. There are months and months of campaigning with different social groups pooling resources (donations) and sorting into different camps. For pretty much any voting method, more money for campaigning and more popular support ----> higher chance of winner. Duverger's law is more about this process of partitioning voter space into two camps like this before the election even happens, regardless of the selection algorithm used.

If more than two parties are desired, there's pretty strong evidence to believe that can only be obtained by proportional representation. Condorcet, approval, IRV, and whatever else would be a great improvement to our democracy, but I do not think any would give us more parties represented in government.

While it is true that Nader was not a major party candidate, but was likely the Condorcet winner, I do not think this is a counterexample to Duverger's law, since we do not know how the positions & endorsements of the parties would have changed under the counterfactual where a Condorcet method was used---that is to say, one of the parties would likely move to wherever Nader was! My expectation is that better single-winner voting methods make a two-party system perform better and make parties more responsive to voters, but I do not expect to see more parties.