https://drive.google.com/file/d/12FIZh6h65YB0u-Y3fxf69kGhnHpaa6FF/view?usp=sharing

]]>I found this presentation very interesting:

One point brought up by an audience member during the Q&A was that QV seems to illuminate the relative preferences of the electorate, which show up in the presenter's data as approximate Gaussian distributions and the grouping together of different strata of right-wing and left-wing groups, which does not occur without the quadratic cost.

]]>Quadratic voting is a somewhat poor quality method (imo) unfortunately.

]]>Bullet voting is maybe a good strategy for an econ voter who is not at all risk averse and are all-or-nothing for their top choice, but real people are risk averse with stratified preferences and will try to establish at least a Plan B in case Plan A doesnâ€™t work out.

Increasing the probability that oneâ€™s favorite wins as much as possible locally does not necessarily increase the probability of a more global â€śacceptable outcomeâ€ť as much as possible, which is what many real people try to accomplish, depending on their definition of what constitutes an acceptable outcome.

This does somewhat seem to lead to approval voting, which I donâ€™t think is a bad system actually. Iâ€™d have to learn more about it. Obviously it has its own problems, but at least burial is quite minor..

]]>Strategic agents will only submit values in {0,1} since by monotonicity any other value makes the chance of electing their favorite strictly lower.

Therefore the method must coincide with majority on {0,1}^n ballots, which is the entire domain of ballots from strategic agents

]]>In any case it could very well be that being a k-Condorcet winner when k=n is equivalent to being a unique candidate that is not SP dominated. Iâ€™m not sure! Still working through the paper.

I tried to give an explanation of the unweighted PFPP system a while back through a video. It may help, if you were interested, but I understand if itâ€™s not your cup of tea! This is the video:

https://app.vmaker.com/record/SGSydGYcwOW9Vf6d

Itâ€™s like 20 minutesâ€¦ 10 if you do x2, potentially less if you skip around.

On a related (maybe controversial?) note I take some issue with the Condorcet criterion. I also have noticed that ElectoWiki doesnâ€™t seem to be very objective about it. While a Condorcet winner has the majority support of the electorate over any other candidate in a pairwise face-off, the majority groups that support the winner from different face-offs can differ from each other dramatically.

In other words, I would say that there is no guaranteed stable locus of electoral consent for a Condorcet winnerâ€”it is rather like a stitching together of victories in various unrelated and somewhat gamified competitions, and to me this makes the Condorcet paradox less of a paradox. In line with the concluding remarks of the paper I think itâ€™s not at all obvious or necessarily correct that the Condorcet winner is the ideal choice even when one exists.

]]>PFPP may be equivalent to a positional scoring rule at each election, but the prescription of the particular scoring rule and how it is allowed to change from one election to the next according to informative distributions is what makes PFPP different

Ok, fair enough, but I can't comment on whether or not this is a good thing. It certainly would be a radical reform to current elections.

I sent the paper to the gmail attached to the google drive you shared, so let me know if you don't receive it. Unfortunately they don't do a ton of analysis besides introduce the concept of positional scoring dominance and then prove what types of dominance are actually constructible (what they call "Condorcet words"), but it is relevant I think nonetheless. In the language of this paper I think SP dominance corresponds to the k-Condorcet winner in the case where k = n.

]]>PFPP may be equivalent to a positional scoring rule at each election, but the prescription of the particular scoring rule and how it is allowed to change from one election to the next according to informative distributions is what makes PFPP different. For example, a thought I had earlier today was actually that if the distributions are allowed to update, then as fewer people over-use the higher score values, they become more potent when they are used. This can give voters an even stronger incentive against strategic bullet voting, since it will weaken their vote in the future when they may actually feel strongly about a candidate.

I'll read the PDF paper you linked and see if they coincide, if they do then I'll be happy because that means probably more analysis has been done on this system! Otherwise I'll try to illustrate points where I find that they differ. I think already the fact that the winner is called "generalized Condorcet" points to something different, since the methods I am proposing (at least on the surface, I could be wrong) have nothing to do with the Condorcet criterion.

]]>I agree that there can be interesting connections to probability theory. The Jury Theorem is a great example.

In this case, I am not convinced it is necessary or instructive to introduce any additional tools or definitions beyond what is already commonplace in voting algorithms.

In the words of Dijkstra

The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise

And I believe we already have the tools to be absolutely precise with regular old ranked ballots over finitely many candidates. Sticking with conventional nomenclature and concepts will help people understand your proposal much more easily, and will also help contextualize it and compare to other methods.

In particular, I believe SP dominance is in fact equivalent to Borda dominance, and I believe your weighted PFPP scheme is in fact equivalent (edit: not equivalent since we have to allow skipped rankings, but very closely related) to a positional scoring rule, but these connections are very hard to see underneath all the new definitions and unnecessary framework.

It's possible I am misinterpreting something and that the equivalence I suggest above is invalid, but if this is the case I would find it very helpful to my understanding if you could provide an example where they differ !

]]>I agree that it is not less rigorous, it's equally rigorous. It's just the way I think and express myself, probably because my background is in pure math. If you see a more apt way to describe the concepts I am proposing I would definitely like to hear that. I want to find a higher level of abstraction that can maybe unify some of the things we're looking at in voting theory, if I could find a good theoretical foothold I would be using category theory, but I don't want to go too far off into abstract nonsense that nobody wants to look into.

I don't actually agree that it is very much more complex than it needs to be. As I mentioned in the introduction of the pamphlet, connecting voting theory with probability theory is nothing new. Condorcet was one of the pioneers of voting theory and his Jury Theorem is a direct application of probability theory to voting theory. As another example, Nash's equilibrium theorem is a direct application of probability theory and topology to game theory.

The use of generalization is just that it is more general, and might be more amenable to application in other areas. I mentioned machine learning as one such area. And I want to point out that ordinal scores are different from weak orderings.

]]>You seem very intent on reformulating all the language, definitions, and algorithms used for voting in terms of probability measures and random variables.

Hopefully this won't come off as a pile-on, but this was something I was confused about as well (regarding a previous paper), it seems way more complex than it needs to be. I compared it to saying that a "randomly selected point in a glass had a 75% chance of being occupied by liquid," as opposed to simply saying that "the glass is 75% full." It strikes me as a very roundabout way of expressing a simple concept.

]]>It is not less rigorous to just use the conventional definitions used in social choice theory where ballots are weak orders and so on. Similarly, I do not see the use in considering generalizations of the voter / candidate sets to be of arbitrary (infinite) cardinality.

It could possibly be of interest to study limiting behavior of voting rules as the number of candidates or voters grow---for example, studying the probability of a tie or of a Condorcet cycle in the limit of these quantities---but I don't think that's what's happening here.

]]>If each candidate is also a probability distribution over those interests, choosing a candidate can be seen as more or less projecting/compressing the electoral distribution into the set of distributions determined by the candidate pool. With this conception a voting system functions exactly as a compression algorithm.

Real life is more complicated than that but I hope that illustrates my thinking better---a candidate's platform can be seen as a (high quality or lousy) compression of electoral interests.

]]>In terms of the probability measure, the electorate as I have defined it is a finite set of objects called voters, and any finite set can be equipped with a probability measure to turn it into a probability space. In terms of how it is used, that depends on the decision algorithm. It isn't easy to formalize the concept because decision procedures can get really wild, and I would have to restrict the scope to a specific kind of decision algorithm to say anything much more meaningful. I tried to connect it with Lewis Carroll's desiderata but it isn't formal. It might just be unnecessary.

For the SP consent ceilings you are correct that my meaning has an anomaly, you are also absolutely correct about the intended meaning.

Thank you for your input, I have these concepts floating around in my head so trying to put them down on paper and running them by other people who are knowledgeable and have a fresh perspective is very helpful.

]]>@cfrank said in Mathematical Paradigm of Electoral Consent:

The pamphlet is not complete and I intend to provide citations where appropriate, but Iâ€™m not sure what citations would be needed.

Your historical discussion on pages 2-4, for one thing. The definition of STAR voting also probably deserves a citation. Your discussion of the relevance of these ideas to politics also might merit some citations.

Pages 6-7:

The following stipulation is adopted: That if one intends to utilize probability

measures to establish a decision algorithm for an OSS in a democracy, any

utilized probability measure imposed on the electorate should be uniform.

In what way would you impose a probability measure on the electorate? What would you do with it?

In your definition of SP-consent ceiling, did you mean R >= S instead of R > S?

I assume that R needs to be an element of **S**. Using R > S can lead to some consequences that I am not sure if you intended. For example, in an Approval election in which a candidate gets 65% approval, (0, 0.65) is part of the SP-consent ceiling, as is (0, r) for r in [0.65, 1].

Pages 7-8: This seems to be a lot of loose threads. I think you need to find a point and stick to what relates (although not necessarily supports, discussing contrary perspectives is fine). Things like the role of decision algorithms in machine learning probably should go in its own discussion at the end that could discuss alternate applications of these ideas.

]]>Your proposal, and in particular "SP dominance" reminds me a bit of this idea.

]]>The broad overview is intended for people who are not necessarily familiar with voting theory, and the purpose is to establish the context of the document. I agree that some of it is tangential and I intend to make changes.

Also the formalism is not incorrect, it would just require an appropriate decision algorithm to select a winner from a continuum or may not allow certain criteria to be satisfied in certain cases as you indicated. But for example, if the candidate set is a collection of points in a plane, and the voters assign each candidate a score from a continuum according to distances from certain ideal points, then the decision algorithm might select a candidate that minimizes some chosen objective function of the scores.

But thatâ€™s not super relevant anyway, since only finite sets are considered.

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