Ordinal Score Voting, Weighted Variation

cfrank

@rob Yes of course, it is a boundary case, but very often in mathematics the boundary cases give you the most information about a system. This is not a contrived scenario, rather it is an idealized thought experiment that I believe with good reason to be helpful for considering ways to tune the balance between majoritarianism and consensualism. Blurring the ballots will definitely help to more firmly establish a compromise, but isn't it a nice theoretical prospect that you might not even need them to be blurred? The system will work well to establish a compromise even in the case where there is hardly a middle ground. With the introduction of a middle ground, the compromise will be that much more reinforced. Isn't that what we mean by a "better system"?

Let me give a less borderline example. Say you have four candidates [A,B,C,D], and the voters score them as follows:

40% [0,1,2,5]
30% [5,4,1,0]
20% [0,1,5,2]
10% [1,5,0,0]

So the profiles of the candidates are:
(A): [1, 0.4, 0.4, 0.4, 0.4, 0.4]
(B): [1, 1.0, 0.4, 0.4, 0.4, 0.1]
(C): [1, 0.9, 0.6, 0.2, 0.2, 0.2]
(D): [1, 0.6, 0.4, 0.4, 0.4, 0.4]

Just as a somewhat realistic toy model, we can use the order statistics of the uniform distribution as the prior for the P-value of each score. The Kth order statistic of N independent uniform random variables over [0,1] has a Beta-distribution:

pK(X)=N!/((K-1)!(N-1)!)X^(K-1)*(1-X)^(N-1)

So for each candidate, each score S(K) contributes the integral of the corresponding distribution pK(X) from 0 to the P-value given by the candidate profile, and then weighted by 1/2^(K-1). Here are the scores for the candidates:

(A): 0.989742804656
(B): 1.06749731
(C): 1.08081248
(D): 1.0572628

So the winner would be (C) under just the weighted variant. That makes quite a bit of sense! With a STAR modification, the runoff is between (B) and (C), and (C) still wins with 0.6 to 0.4. The Score winner is totally not (C), it's actually the "majoritarian" (pluralistic) candidate (D). With the STAR modification, the runoff is between (C) and (D), and (C) wins with 0.5 to 0.4. So this geometrically weighted SP Voting variant converged upon the Standard STAR winner even without the STAR modification, and the STAR modification only reinforced the compromise already established by the system.

My purpose in developing this system was to eliminate the arbitrary nature of cardinal score values and to simultaneously balance consensus with majoritarianism. I am aware that it is more computationally complex than STAR, for example, but if this system tends to produce the same results as STAR then that is even more evidence in favor of STAR as a computationally simplified compression. I still do think that even standard STAR with exponentially diminishing cardinal scores is a great system, but I also like the distributions because they make it virtually impossible for a voter to predict the cardinal value their ballot will effectively give to a candidate without knowing the proportions of relevant ballots and also knowing the distributions. That makes the scoring process more subjective, which is good.

rob

@cfrank said in Ordinal Score Voting, Weighted Variation:

but isn't it a nice theoretical prospect that you might not even need them to be blurred?

My view is it is misleading when presented as it was, partly because you are expecting people to work out all this math in their heads, and when they get puzzled, they (prematurely) conclude something is wrong. I'm saying what is wrong is that you have described the unlikeliest of scenarios, but one in which it is a near tie, so it's not really a big deal if the answer seems ambiguous.

The bigger picture, though, is that I've participated in forums and email lists on the subject going on twenty years now, and we are no closer to fixing the problems with our democracy, while we are seeing violence (from people unhappy with election results) as a result of our broken system twisting people into hatred of the opposing faction. [1] So forgive me if I am prickly on this issue. I've seen contrived scenarios, that seem to be intentionally opaque, used to shoot down perfectly good systems over and over. It is like whack-a-mole. I feel it is an impediment to progress.

As to it being opaque, I am curious, who are you trying to persuade? I'm not a math idiot, but your posts make my eyes glaze over. I get through two paragraphs and say "holy crap this is too much work". I'm not convinced it is necessary to go so "turbo encabulator" on everything [2].

@cfrank said in Ordinal Score Voting, Weighted Variation:

40% [0,1,2,5]
30% [5,4,1,0]
20% [0,1,5,2]
10% [1,5,0,0]

Do you mean this?
40: A[0] B[1] C[2] D[5]
30: A[5] B[4] C[1] D[0]
20: A[0] B[1] C[5] D[2]
10: A[1] B[5] C[0] D[0]

I mean, you don't have to use my format, but if you use your own, point me to something that can parse it so I can actually do something with it. Again, that's why I did the codepens. I can plug the above into mine and analyze it in a dozen different systems, apply the aforementioned blurring, etc.

FYI C (let's call her Carol) is the Condorcet winner with that ballot set, while D (Dave) wins under STAR and Score. Personally, I think C is unambiguously the best winner, and further predict that as soon as any decent system is in place for a few years -- STAR, Score, and Approval included -- the chances of having the Score/STAR winner differ from the Condorcet winner would diminish even further, since it would discourage the most polarizing candidates from running in the first place.

Note that C is a pretty good compromise candidate, she has 60% of the people give her 2 or more, and only 10% give her a 0. No other candidate accomplishes that, but all of them have a lot more 5s than she has, which of course has less weight with any Condorcet method. (i.e. Condorcet methods are "median seeking," so outliers don't have the same pull as they do in an "average-seeking" method [3] )

If you add a bunch of middle ground voters with blurring, D is going to do better. D having a lot of 5's does make a difference with blurring, since that's going to pull up the new middle ground voters' scores for him. So, again, the more you try to make it realistically have middle ground voters, the more you will have Score and pairwise results agree.

****** Pairwise wins ******
C: 3
D: 2
B: 1
A: 0
****** Score ******
D: 240 (2.4000)
B: 230 (2.3000)
C: 210 (2.1000)
A: 160 (1.6000)
****** STAR ******
D: 60
B: 40

@cfrank said in Ordinal Score Voting, Weighted Variation:

So the profiles of the candidates are:

I don't know what you mean by "profiles", so I stopped there.

I predict the storming of the Capitol is nothing compared to what we are going to see surrounding future elections. Yes, I'm freaking scared.
Youtube Video
https://pianop.ly/voting/median.html

cfrank

@rob If you don’t know what I mean by the “profile” of the candidate, then you don’t have any business making claims about this system, because you didn’t take the time to learn about it. Your statements about the middle ground are not true for this system. The distribution of the middle scores do get pulled up, but since the candidate (D)’s score profile remains the same, the metric assigned to the candidate will diminish. I indicated the ordering when I wrote the candidates in bracket format as [A,B,C,D], they are also alphabetical and may as well be labeled [1,2,3,4], but yes, I did mean the same ballots you also indicated.

Clearly, I am making a case to persuade anybody who will take the time and effort to read and understand this system and my claims about it, I.e. anybody who is interested in opening themselves up to the possibility of persuasion. If you don’t want to take the effort to admit to my arguments more support than a straw man, there is no good reason to be having this discussion.

I am not trying to discredit Score, STAR, or Condorcet methods, I’m trying to develop a system that does a good job based on my own theoretical considerations, because I am also disturbed by the state of our voting system and the degree of polarization that exists in our society right now. If it so happens that this system mostly agrees with STAR or Condorcet in real world applications, for example, then as I said, it means that those methods are acceptable substitutes for me, and in my opinion is an argument in their favor because they would be simplified and efficient compressions of the admittedly more complicated considerations that I am making that personally make sense to me.

FYI, it very much irritates me when people make comments about my vocabulary being “technobabble.” I am an intelligent person with a technical background and this is how I speak. I’m sorry that I want to express myself with precision, but sometimes you have to do that when you are trying to create a technical mathematical model. I have a degree in pure mathematics, I’m not a salesman.

rob

@cfrank said in Ordinal Score Voting, Weighted Variation:

then you don’t have any business making claims about this system,

I wasn't making a claim about your system because I didn't read further, which was my point.

@cfrank said in Ordinal Score Voting, Weighted Variation:

I am an intelligent person and this is how I speak.

And that's fine, congratulations on being intelligent. We get it, really.

I'm good with code, but I don't put up a ton of code and expect everyone to read and understand it. (but for those who are inclined to, it is up there where they can run it, paste in data, edit the code, etc, with next to zero hassle)

But mostly I use the code to make things that help make concepts understandable to people.

I could have just not bothered replying, but my hope is that you'd be interested in why you lost at least one person.

cfrank

@rob I think I misinterpreted your last message, it’s easy to do that with text. Sorry. I wasn’t claiming that I am intelligent as a weird flex or to insinuate that you aren’t also intelligent, I corrected myself to indicate that I have a specific technical background. I’m definitely not looking for praise, I’m trying to state a fact. Definitely don’t take this the wrong way, but have you heard this quote of Dostoyevsky?:

“Tolerance will reach such a level that intelligent people will be banned from thinking so as not to offend the imbeciles.”

I am absolutely not calling you an imbecile, because to the contrary you are very intelligent, but the way you seemed to me to be taking offense to the expression of my ideas made me feel discouraged. Maybe I’m reading into this and inaccurately representing your response, and text is a bad medium. Anyway I am confident that the ideas are good, and so I presented them. I want them to be accessible too, though, not impenetrable. I have a hard time I suppose.

There is a 20 minute video explanation that can be compressed into 10 minutes if you double the speed, it more than amply and thoroughly explains the system with graphical displays, and I also shared and briefly explained the code. Please excuse my accusative statements, we should be cooperating and I was feeling irritable for other reasons.

rob

@cfrank said in Ordinal Score Voting, Weighted Variation:

have you heard this quote of Dostoyevsky?: “Tolerance will reach such a level that intelligent people will be banned from thinking so as not to offend the imbeciles.”

Well you're not winning points here by implying people who don't follow you must be imbeciles, whether or not you say "no offense I don't mean to call you an imbecile". If the quote is relevant.... um, ya kinda are.

(and I'm not claiming to be a super genius, but I do tend to be smart enough that, when I attribute a quote to someone, I do a quick search first to see if they actually said that. Just sayin' )

Regarding your video: I don't see a link to it.

But you know.... a 20 minute video is long. I don't generally watch 20 minute technical videos unless you first give me a really good indication that it will be worth my time, and that you've put the effort into making it efficient of my time. Like at the very least, communicate what problem you are trying to solve before expecting people commit to investing their time. I don't think I'm unusual in this respect....has anyone else watched it?

Regardless, I'll watch it if you link it.

But for starters, how about just a simple link to something that explains what you mean with what you call "profiles"? I searched around and found nothing close to that. I searched on electowiki, rangevoting.org, and just on google in general. Is this a concept of your own from a previous thread? If so, could you link to it? (Is it in the one with a few hundred lines of python code pasted in? Sorry that one lost me too.)

Really, all I'm saying is you seem to have high expectations for what people's attention spans are and the effort they are willing to go to to take in your message.

BTW, here's about 20 minutes of video I've made on voting stuff, while we're sharing.

https://youtu.be/NiS2A0QLeJU
https://youtu.be/eFbuBMdFSnA
https://youtu.be/jbGwR8fbVYM
Youtube Video

cfrank

@rob Good Lord you’re absolutely right. The internet is a bad place to get information, I just saw it attributed to him and didn’t look into it further. I haven’t read much Dostoyevsky, I just thought the quote was potentially relevant in terms of being met with hostility for expressing new ideas that are potentially complicated. But you’re right that I need to find a way to simplify the presentation.

I am not saying people who don’t follow me are imbeciles, I’m not trying to win points. I just don’t understand why there seems to be a general culture here of competition and shooting down ideas instead of cooperation and trying to build them up. It seems like we would all be making a lot more progress working as a team.

Anyway, I’ve already indicated the problems I was trying to solve. Most are the same problems we’ve all been trying to solve:

(1) How to balance the conflict between majoritarianism and consensualism;
(2) How to address the problem of strategic voting;
(3) How to create a system that is simple and meaningful and encourages compromise.

The last one is a pet peeve of mine, which is that (4) the cardinal values assigned to scores are arbitrary. This is in fact what led me to create this system, and why it is called an “ordinal score” system.

I believe I have successfully addressed all of the above things, but the “simple” part I suppose is a point of contention. The concept is simple, but the mathematics makes it look more complicated than it actually is, because it deals with probability distributions. The latest version of the system is just as good as if not superior to STAR, and with minor modifications it can be made even better. I’m not sure what else to say about it to make it appear worth looking into.

If you do watch the video, please excuse my brain fart at the beginning, I made it after a long day. @Jack-Waugh at least did watch it. I’ll definitely take a look into your material.

Here is the link: https://app.vmaker.com/record/SGSydGYcwOW9Vf6d

This is a first draft, what I am currently proposing is an additional weighting modification that directly addresses point (1), and it simultaneously improves (2) and (3) as well. I had actually considered weighted (S,P)-voting systems before, but I didn’t see the utility until @Marylander pointed out that the system is still more majoritarian than desired. The new system is explicitly indifferent between majoritarian and consensual influences in exactly the manner I explained in the first post of this topic, I.e. if half of the electorate scores one candidate the top score, but the whole electorate scores a different candidate the lowest nonzero score, then the system is typically indifferent between the candidates (to the first order of magnitude). Full consensual compromise with everybody mostly unhappy is comparable to half of the electorate happy and everybody else potentially fully disappointed.

rob

@cfrank said in Ordinal Score Voting, Weighted Variation:

I just don’t understand why there seems to be a general culture here of competition and shooting down ideas instead of cooperation and trying to build them up. It seems like we would all be making a lot more progress working as a team.

I agree and wasn't shooting down your idea, just pointing out my frustration with the presentation.

( I remain of the opinion that having regular votes on favorite methods would help a lot with this. The votes wouldn't actually "mean" anything, but I think we need to do our best to come to a consensus on what we want to get behind. After all, if we, of all people, can't figure out how to find a consensus, something is very wrong. )

I'll admit that my beef with what I referred to as a contrived scenario is because such scenarios tend to be used to dismiss other methods, whether those are fresh "ideas" or whether they are ones that have been around for a long time.

@cfrank said in Ordinal Score Voting, Weighted Variation:

How to balance the conflict between majoritarianism and consensualism;

I'd love to see these two concepts fully explained, including why you consider them in conflict. I tend to have issues with any use of the word "majority" when there are more than 2 candidates. I get that there are binary choices inside some methods (i.e. pairwise comparisons) but I just don't think the concept of "majority" is as meaningful there is implied by the term "majoritarian."

In some ways, I think it almost analogous to what I call average seeking and median seeking methods, which is detailed here in a simpler scenario (note that I linked this page above, and linked the embedded video in my list of 4 videos above.....). Anyway consider it an essential baseline concept: https://pianop.ly/voting/median.html )

cfrank

@rob that’s totally fair about the presentation, sorry. I was just trying to be explicit about the math I did to compute the metrics, although as you pointed out I may have made a calculation mistake with the STAR winner (I just used a Desmos calculator on my phone, not a program).

I definitely agree with you about scenarios or formal properties being used to dismiss systems. I don’t know if you read the discussion @Jack-Waugh and I had about “Frohnmeyer Balance,” or if you recall the discussion from the last forum. But basically that property is almost a meaningless triviality touted as a necessary milestone. I think we’re on the same page.

In terms of consensualism versus majoritarianism, I’m not sure if you watched the video and saw the definition of an “(S,P)-consensual” candidate, but I’m basically using that construct to try to measure consensuality and trying to choose the “most consensual” candidate.

What is interesting is that, unlike Frohnmeyer Balance, the concepts of (S,P)-consensuality and (S,P)-efficiency (see video) apply to any rank-order or score system, because more generally they are constructs that are well-defined in any ordinal score system. I would never use them to evaluate or dismiss a system for which the constructs don’t even make sense or are irrelevant, every (reasonable enough) voting system needs to be looked at in its own right. Anyway, (S,P)-efficiency is an even more basic construct than Frohnmeyer Balance anyway—almost any reasonable score system for example is automatically (S,P)-efficient.

But as I think you might agree as an informal definition, majoritarian system allows a narrow majority to overwhelm the rest of the electorate with its common preference or strategy. A consensual system successfully prevents this from happening when a sufficiently broadly-appealing compromise is available.

I can’t give a formal definition or a precise cutoff point between the two concepts. It’s a spectrum and I think a good system should find a “middle path” philosophically speaking. Practically speaking it just doesn’t seem likely that a good solution is on either extreme end of the spectrum—kind of getting meta here lol. The Federalist Papers starting from number 10 have a lot of very interesting things to say about majoritarianism and faction formation, and obviously why the U.S. should be set up as a federal democratic republic as opposed to a central authority, pure democracy or a confederacy. The line of reasoning there is similar to what I’m trying to suggest about an “ideal” voting system.

rob

@cfrank said in Ordinal Score Voting, Weighted Variation:

I can’t give a formal definition or a precise cutoff point between the two concepts. It’s a spectrum and I think a good system should find a “middle path” philosophically speaking.

I don't need a precise cutoff, I'm quite comfortable with concepts that lie on a spectrum.

What I think is under-explained is what is actually meant by it. How exactly is one system more majoritarian than another? Maybe a clear example?

Going back to the median vs average example (say, voting for the temperature to set the office thermometer at, and choosing the median choice or average choice), is median more majoritarian than another? What does "majority" even mean when there are a large number of choices and no single choice gets a majority of votes?

I'm also unclear on your concept of a method being rigid vs flexible. Flexible, to me, means it gracefully adapts to change. What kind of change are we talking about? Or is there some different way you would describe it?

It would be great if there were a web page you could point me to that explains these concepts in the way you view them, and gives examples. There must be one, if they are as universally understood concepts as they way you seem to treat them.

cfrank

@rob I suppose a generalization of majoritarianism would be pluralism, but for example if there is no available compromise, then the pluralistic candidate would actually be the best option. The systems I’m talking about being majoritarian/pluralistic select a large-faction-supported candidate not as a last resort, but as the M.O.

Arend Lijphart’s “Patterns of Democracy” goes into detailed description of one conception of consensualism versus pluralism/majoritarianism. He produces a two-dimensional framework for democracies with one dimension being “executives/parties” and the other being “unitary/federal,” with five distinctions made to indicate the position along the spectrum in either dimension. I’m paraphrasing here:

EXECUTIVES/PARTIES:
(1) Single-party majority executives/Multiparty coalitions
(2) Executive dominance/Executive & legislative balance
(3) Two-party system/Multiparty system
(4) Disproportional electoral system/Proportional representation
(5) Competitive interest groups/Coordinated interest groups

UNITARY/FEDERAL:
(1) Unitary/Federal (of course)
(2) Unicameral legislature/Bicameral
(3) Flexible constitution/Rigid constitution
(4) No judicial review/Judicial review
(5) Executive-dependent central bank/Independent central bank

Anyway, it’s a political science treatise but it does necessarily touch on the concepts I’m describing. He gives many specific examples and the book is a statistical analysis of the correlations between these properties and other socio-economic properties. Chapter 8 is called “Electoral Systems: Majority and Plurality Methods Versus Proportional Representation.” I think that may be an oversimplification because proportional representation is just one sort of indicator of a consensual system, since consensuality as I conceptualize it has more to do with the theoretical mechanism of a voting system than its specific outcomes in terms of proportionality.

The median temperature example: I would say that the median method is more consensual and less majoritarian than the mean method. The median method in my opinion is one that sits in the elastic territory—it is neither fully rigid nor loose, so it has a sort of blend of majoritarian characteristics and consensual characteristics. That’s the kind of system I think is good, I do think it leans toward rigidity and majoritarianism/pluralism. One could probably produce an even more consensual method by, for example, dividing the temperature scale into discrete bins, and voting on which bin to select according to a more ideally elastic method, which may be SP voting or perhaps even STAR. For example, what if there were some weird guy who preferred either very hot or very cold temperatures, but disliked the moderate? I think that’s a bit of a rabbit hole, but I also think it illustrates how one topology like a number line might be useful to compress certain or even most situations, but can’t be universal. No system can be universal, because there is not universal compression algorithm (https://en.m.wikipedia.org/wiki/Lossless_compression)

I don’t know whether or not the abstract rigidity/elasticity/looseness is a universally understood concept. They are concepts that I understand and that I am trying to explain, they probably fall in line with other more developed concepts but I don’t know what they would be called. Anyway, it’s all theoretical stuff. The fact is I have a voting system here that is well-defined and, based on my own exploration of it, to my knowledge it produces very good results.

rob

@cfrank Ok, I'll have to get back to this, I haven't had a chance to watch your video but I will.

I'm having trouble seeing a connection to the political science terms, and anything quantifiable (in a logical or mathematical or whatever way) in a voting method. I'm not saying there isn't, I was just hoping for something more concrete. For instance, voting methods typically don't have any concept of what a party is, so discussing parties isn't super helpful.

I've also seen "utilitarian" used as the opposite of majoritarian. Maybe some explanation there would be useful. Links are always great if there is somewhere this is explained.

As for the median temperature, I consider an implicit assumption that voters prefer temperatures closer to their single ideal temperature. That is, it's intended as a 1-dimensional variable. I think in the real world of human political candidates, things are much more multidimensional, but we shouldn't need to have wormholes in the fabric of "ideological space" to allow for people like your dude who likes it either hot and cold but not in the middle.

I think it is a useful thought experiment to think of how you'd set up the temperature example using any of our single candidate methods, and how we'd expect it to behave compared to the "vote for your favorite, we'll pick the median" method.

The temperature example, to me, is very similar to Condorcet methods, in that voters are treated as if each of them has a specific "direction" that they are pulling the results, but all are pulling with the same amount of force. They both intentionally ignore part of the data, so they don't incentivize exaggerating. It's just much easier to directly see this effect with the median example than it is with a Condorcet method.

What would you call that property?

cfrank

@rob unfortunately I think producing a concrete formalism would be a huge challenge. I could be wrong but as far as I can tell probably the best one could reasonably hope would be a collection of strongly correlated indicators. For example, some rigid/loose indicators of a voting system might include:

(1) Strict ballots/flexible ballots (ex: rank-order/score)
(2) Game theoretic stability/strategic “hall of mirrors”
(3) Large party formation/multiparty or corporatist group formation
(4) Majoritarianism/utilitarianism
(5) Simplistic algorithms/complex algorithms
(6) Winner-take-all/Proportional

I do think that those things tend to be correlated in voting systems, and that tendency I am referring to as the “rigid/loose” spectrum of voting systems. Do you agree? Does that make sense?

So that property of the median and Condorcet methods I would classify as rigid, since they have both strict ballots and game theoretic stability. My qualms with that is that it seems to me that the other two correlated constructs are what we want to avoid. Unfortunately the “hall of mirrors” effect (as you have described it, correct me if I’m wrong) of looseness also makes many more flexible and utilitarian systems undesirable as well (at least to me), so I am proposing that there is some sort of smeared out boundary between rigidity and looseness that I am calling the elastic region.

I think that Score falls in the loose region, whereas STAR falls in the elastic region, as an example. It has ballots that are flexible but are also game theoretically constrained to a degree due to the runoff, which also directly balances the majoritarian and utilitarian aspects against each other.

I also believe the system I am proposing (exponentially weighted SP Voting) is elastic. It has flexible ballots, and is not founded on utilitarianism or majoritarianism but rather on consensuality, which is quite different. I think it should be more or less game theoretically stable while simultaneously preventing excessive strategic behavior. My belief is that this would tend to curb large party formation and encourage sensible compromises to be made.

rob

@cfrank said in Ordinal Score Voting, Weighted Variation:

I do think that those things tend to be correlated in voting systems, and that tendency I am referring to as the “rigid/loose” spectrum of voting systems. Do you agree? Does that make sense?

I'm not seeing where rigid and loose come in. What makes a system rigid or loose? Are you saying that Cardinal vs Ordinal ballots is the difference? Otherwise I'm not sure how those terms apply.

Or are you speaking of things that aren't game theoretically stable (i.e. hall of mirrors) as being loose?

You also use the terms elastic and flexible, which I don't know what you are referring to. And you continue to use the term consensuality and distinguish it from utilitarian, but again, it is not clearly defined.