Quantile-Normalized Score
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@jack-waugh I don’t think it’s a simple mathematical problem. Math can help, but only after a model is chosen, and I can’t claim to know what the correct model should be. I see it as a philosophical problem, and my approach to it is fully consequentialist, where formal theory is only a concern to me as a proxy.
What I mean is that to me, the core of the problem is in the domain of continental philosophy, with analytical philosophy as a secondary supplement.
After a brief chat with ChatGPT, here is how it categorized the philosophical disposition I call “social consequentialism”:
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Classification
Ethical Foundation: Rooted in consequentialism, with an emphasis on the outcomes of actions but uniquely focused on social consequences.
Philosophical Tradition: A fusion between analytic and continental traditions, using analytic clarity and rigor to address the broad, context-sensitive concerns typical of continental philosophy.
Social and Political Philosophy: Places a strong emphasis on the social implications of actions, potentially aligning with theories of social justice, communitarianism, and democratic ethics.
Questions for Deeper Understanding
Scope of Social Consequences: How do you define or delimit "social consequences"? Are there specific aspects of social life (e.g., justice, equality, freedom) that are prioritized in your assessment of outcomes?
Methodology for Assessing Outcomes: How do you propose to assess or measure the desirability of social consequences? What methodologies or criteria are used to evaluate the complex outcomes of actions on society?
Handling Conflicts and Trade-offs: How does social consequentialism navigate conflicts between individual and social outcomes, or between different social goods (e.g., equity vs. efficiency)?
Theoretical Influences and Precedents: Are there specific philosophers, theories, or movements within continental or analytic traditions that particularly influence your thinking? How do you position your theory in relation to utilitarianism, social contract theory, or critical theory?
Practical Implications and Applications: What are the practical implications of social consequentialism for policymaking, social activism, or individual ethical decision-making? How does it inform actions in concrete scenarios?
Temporal and Cultural Context: How does social consequentialism account for the variability of ethical judgments across different cultural contexts and historical periods?
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I thought the classification was pretty decent and also found the questions pretty central.
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@cfrank said in Quantile-Normalized Score:
failure to satisfy independence of clones
What are the grounds to understand this as a problem?
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@jack-waugh the grounds by which to say that failure of independence of clones is a problem is the collection of negative and avoidable social consequences it leads to. We have to agree on a state of affairs for us to recognize the same problems. For example, I think we both agree that political divisiveness, externally imposed conflicts of interest on the public, lack of accountability of public officials to the public, and all associated corruptions and abuses of power are problems, don’t we?
There is a delineated complex of issues that contribute to this vicious cycle, one of which is vote splitting, which is a simple consequence of the failure for clones to be independent. In contrast, I know of no clearly delineated complex of negative consequences that involves Frohnmayer’s formalization of balance. If you can identify a complex for me that stands up to scrutiny, I would be more convinced. But the state of it to me is that Frohnmayer balance appears to be a purely theoretical, arbitrary formalism with no clear tangible consequences, and therefore that using it as a criterion by which to judge the merit of voting systems is a fallacy.
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@cfrank, under the conditions that I am peppering you with questions, I obviously have an obligation, as someone who would be seen as honest and fair in discourse, to at least try to answer the questions that you put to me in the discussion.
Do I agree that divisiveness is a problem? Let me take an example. There is division of opinion among US citizens in that some support genocide and others oppose it. If the condition were such that all supported genocide, the division would be gone, but the situation would be worse relative to my values. As I see it, the ideal situation would be that all the citizens oppose genocide, and the worst situation would be that all support it, and the divisive situation where some oppose it and others support it, is at an intermediate level of quality between the two extremes. This example illustrates why I do not agree that divisiveness is a problem. Division happens because people disagree about what is right and wrong and about what is desirable.
Do I agree that externally-imposed conflicts of interest on the public constitute problems? Yes. When AIPAC provides money support to candidates who support genocide, that is a conflict of interest with the needs of the citizens of the US such as food, clothing, and shelter, so that is an externally-imposed conflict of interest, and I do agree that that is a problem.
I agree that lack of accountability and the associated corruptions and abuses of power are problems.
What is the mechanism whereby failure of independence of clones leads to social consequences?
Please evaluate the following systems as to their acceptability or desirability relative to one another:
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system 0: Approval Voting;
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system 1: There is no "system 1" mentioned in this discussion.
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system 2: The tally is as in Approval, but you are constrained to approve strictly fewer than half the count of candidates, unless there are only two. So for example, if six candidates are running, you can approve one or two, but not three or more, because three is not strictly less than half of six.
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system 3: The tally is as in Approval, but you are required to approve strictly more-count than half the count of candidates, unless there are only two.
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system 4: Choose-one Plurality.
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system 5: You are required to approve all but one of the candidates.
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@jack-waugh you’re right that divisiveness is not a problem in itself, but I suppose what I mean is that discourse is focused on issues that are divisive rather than on issues that can be addressed with a reasonable measure of consensus, and the controversial issues hold the reigns of policy rather than the agreements that exist.
And also, if everyone agrees that genocide is desirable, including the race of people who would be exterminated, we enter a pretty absurd and peculiar ethical territory. The agreement you’re talking about presupposes a disenfranchisement of a group of people, which itself goes against the preservation of liberty and choice and natural rights. It’s complicated and there may be strong arguments that are concise, but I won’t try to make any because more likely it would have glaring objections. On the whole and sophistry aside (by that I refer to my own sophistry), I think we agree. I’ll think about your points.
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@cfrank Can you explicitly explain this method, rather than making us infer it from a related Wikipedia article?
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@toby-pereira yes sorry! There was an explanatory video I shared with @Jack-Waugh, I just put it at the top next to the Wikipedia link, also here: Youtube Video
What’s sort of interesting about this is that there is no need for even a set scale, in principle people could put whatever scores they like, even negative numbers, and the normalization would still put everybody on the same scale and footing.
This kind of normalization is really common in computational biology to correct for batch effects and outliers, for example when trying to infer highly expressed genes in one sample of cells relative to another based on count data from RNA sequencing.
I’m running a small toy election about GitHub icons with my lab-mates, and comparing score to quantile normalized score, I like it. I’ve just been asking ChatGPT to quantile normalize and it does so successfully, but there are definitely more guaranteed, readily accessible methods that do this normalization efficiently even for huge data sets like an actual state or national election.
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@cfrank Cool, thanks for that. My previous message was probably a bit blunter than I intended it to come across.
As a voting method I'm not sure it seems ideal to me, as it comes across a bit Borda-county. If there are e.g. 4 candidates, then regardless of whether one person scores them 10, 9, 1, 0 and another scores them 7, 6, 5, 4 they will be normalised to e.g. 8, 6, 4, 2 (or whatever the normalised values end up being). So it seems it will have all the problems Borda has, including failure of independence of clones.
Also, a voter might think the actual scores they give matter little because the distribution will be set by the whole population of which they are just a tiny proportion. That might make the scores even more like the Borda count as they just separate each candidate by 1 to make it easier.
How does it handle equal scores? Give them the mean of the scores they would occupy?
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@toby-pereira yes it can handle equal scores. There’s a slight peculiarity in terms of discreteness of distributions, but equal scores will stay equal after the normalization, since they have the same quantile.
I agree that it’s Borda-like, but I think there’s a difference between set scores at the outset and scores that adjust depending on the ballots. I understand your concerns, and what the resolution to the questions you have might be isn’t wholly obvious to me at the moment. I think the only way to comprehend the differences is to do an actual results comparison over different ballot sets, for example, between this method and Borda and score. My guess is that they’ll be fairly different, but I could be wrong, and there might be weird effects of the normalization that I don’t see immediately.
In terms of scores not mattering, this is a makeshift idea, but if a score range is fixed, and if something like pseudo candidates (who cannot win) are entered with all minimum and all maximum scores for every voter, it is still advantageous for a voter to utilize the whole range to maximize the expressiveness of their ballot, which eliminates some of the arbitrary nature of selecting scores.
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Do you agree that it is possible for two voters to have directly opposed positions on the candidates, such that each of the two voters hates the candidates the other one loves, and loves the candidates the other one hates?
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@jack-waugh yes of course I do, but there isn’t any particular mathematical formalism naturally associated with that situation.
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@cfrank Then how can you justify giving one of these voters more power to sway the outcome than you give to the other voter?
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@jack-waugh in any voting system, we can look at the algorithmic mechanism and produce attributions of final or intermediary objects constructed by the algorithm to the inputs of the algorithm. For example, in score voting, we could look at the final aggregated scores of candidates, and take a correlation between the scores given by individual voters and the final outcome. Some voters are liable to have more positive correlations than others, but in this case, correlation is not necessarily causation.
Can we find a suitable attribution that succeeds in measuring “causation”? This is a central question in the theory of attributions, for example, in machine learning, and there isn’t a universal answer. Different attribution methods will yield different results. It’s a voting system all over again, in this case, it’s “who had the most power”?
In an election with two voters, if the question really is as simple as “love” and “hate,” then the voters should maximize their scores of loved candidates and minimize their scores of hated candidates. In this case, both score and quantile-normalized score (with an adjustment for discreteness) should produce a stalemate.
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@cfrank said in Quantile-Normalized Score:
I agree that it’s Borda-like, but I think there’s a difference between set scores at the outset and scores that adjust depending on the ballots. I understand your concerns, and what the resolution to the questions you have might be isn’t wholly obvious to me at the moment.
Ultimately I think this might be a fun theoretical idea, but I can't see it having any real practical use.
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@toby-pereira why do you think it wouldn’t have the same practical use of score voting? It may even perform better in practice by reducing the effects of bullet voting.
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@cfrank Because people can't choose the score they end up giving to candidates, Borda count style. If someone has a big difference between e.g. their 3rd and 4th favourite candidates and someone else has a very small gap, why should they both be forced to give the average gap across all voters? I don't see any need for it. It seems like a solution in search of a problem.
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@toby-pereira the problem is bullet voting and a synthesis of rank and score systems, this is one way to address it. Also, many very difficult problems in the real world are solved by "solutions in search for a problem," including most of the most impactful drug treatments.
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@cfrank said in Quantile-Normalized Score:
take a correlation
In this discussion, there is no need to resort to statistical measures.
two voters
I'm not talking about an election with only two voters. I'm talking about an election with 300,000,002 voters. Let's say that two of the voters we will designate them as A and B. All 300,000,000 voters except A and B cast their ballots. A and B are on their way to the polls. We can validly for any voting system talk about what outcome it would produce with the first 300,000,000 voters and without the votes from A and B. Let's say that is a certain outcome. Now A arrives and casts his vote. This changes the outcome. A candidate has gone from losing to tied or from tied to sole winner, or from sole winner to tied, or from tied to a definite loser, because of the effect of A's vote. We don't have to resort to statistics or philosophy to attribute causation. All we have to do is observe that the election would have a different outcome without A's vote and has a different outcome with his vote. There's nothing fuzzy about that determination. It consists of cold, hard, irrefutable facts. Now let B finally arrive at the polling place, and just in the nick of time. B casts her vote, and is unable to reverse the effect of A's vote. Clearly, the voting system is granting A more power over the outcome than it is granting B. If they had equal power, their effects would balance, which we could observe by seeing the outcome shift back to what it would have been without the last two votes. How can you justify accepting a system that imposes such a bias, that B does not get as powerful a vote as A? This kind of inequality is the root of the spoiler effect.
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@Jack-Waugh I understand your argument and it is compelling logically to justify the conclusion. However, there is a nuance to this, as we have discussed before, which is that the situation where “B cannot reverse the effect of A” can trivially be made “possible,” even if the probability of it is virtually zero. In fact, the probability can be made arbitrarily small, and thus the mere “possibility” of cancellation surely cannot be what is actually important.
To clarify, the consideration is as follows: Any ballot format can be augmented with, for example, an extra, mandatory ordered pair of strings to be input by each voter, the first of which is a “password” string, and the second of which is an “attack” string. After ballots are closed, let the augmented voting system then randomly prefix a unique identifier string to each password string, so that prefixed passwords are unique.
Then, the augmented system searches through the attack strings and prefixed passwords for matches in the sense that if voter A’s attack string matched voter B’s prefixed password, and vice versa, then the ballots of A and B will be removed, in which case A and B can be called a “cancellation pair.” Note that it is impossible for any voter to belong to more than one cancellation pair, since prefixed passwords are unique.
This augmented system satisfies your requirement that voter B can always cast a ballot that reverses any effect of voter A, even though the probability of that event can be made virtually zero. This artificially augments the voting system into one that is 99.9999% guaranteed to operate exactly the same as the un-augmented system, but where the premise of your argument is actually impossible.
I’m not merely contorting an argument against you, I’m trying to comprehend what I see as a legitimate paradox about “cancellation”: it requires voters to act based on information they cannot access (the choices of others), which contrasts with the principle that voters should make decisions based on their preferences and public knowledge. And therefore the question seems to be about how much information any individual voter should have a right to about the ballots of others, I.e. the real object of interest seems to me to be a balancing point between privacy and transparency.
As a summary, if a voting system does satisfy cancellation, that in itself actually implies virtually nothing about power imbalances, which are intrinsically tied to information imbalances. So then, probably, what you want is a combination of cancellation and a “well-judged” compromise between privacy and transparency.
I was about to say that if cancellation can be satisfied trivially, then the compromise between privacy and transparency is important, but in a sense, it does seem difficult to trivially satisfy cancellation without infringing, albeit also in a very trivial way, on the balance between privacy and transparency.
What I mean to say is, each voter can in principle make their “password” as intricate as possible, and it can be understood that direct cancellation is correspondingly unlikely, which depending on judgment may well be an acceptable aspect of privacy in a well-judged compromise. I have another orthogonal point to make about your argument, but maybe I’ll stop there for now.
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@cfrank The conclusion of your argument is not that the balance constraint is not necessary, but merely that it is not sufficient. You show how to construct a system that obviously has no merit but meets the formal balance constraint. This shows that the constraint is not sufficient.
I say that whether a system meets the balance constraint is telling when it meets an additional constraint or two.
Choose-one Plurality Voting is a variant of Score Voting, in the sense that a total score is accumulated for each candidate by adding up the votes and whoever scores the highest wins. The difference between Choose-One Plurality and Approval can be found in restrictions on what votes are legal to cast. And the difference between Approval and other ranges of Score Voting is merely the granularity. All these systems are what I want to call "additive". A vote can be interpreted as a mathematical object of some kind that obeys addition laws, and the tally depends solely on the sum of the votes so interpreted. In the case of Score Voting with or without restrictions, this mathematical object that obeys addition laws is a vector, where the indices along the vector correspond to the candidates.
In some hybrid systems where preferences figure in and more than one round of tallying is required (e. g. STAR), the mathematical object that interprets a vote is a combination of the aforementioned vector and a preference matrix. The preference matrix for the electorate as a whole is the ordinary matrix sum over the preference matrices for the voters.
I assert that:
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Frohnmayer balance is necessary for a voting system that resists the spoiler effect;
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When it comes to value going beyond mere necessity, Frohnmayer balance along with additivity is pretty telling, if the decisions made at the ends of the rounds are made at the level of candidates, not voters, and only depend on candidate totals made by summing up parts of the mathematical interpretation as I described for the votes.
The construction you described does not meet that. It's a game where if a voter wants to cancel another voter's vote, she has to guess the other voter's password and some salt that you append to it. With additivity, cancellation happens not because someone is trying to cancel, but as a natural result of the voters casting opposite votes.
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