Quantile-Normalized Score

[cfrank]

@jack-waugh Sure! This is Python code that will perform the normalization:

import pandas as pd
import numpy as np

# Create the raw score dataframe
data = {
    "VOTER_1": [27, 21, 25, 26, 17, 49, 65, 18, 45, 75, 9, 31],
    "VOTER_2": [40, 40, 40, 50, 25, 60, 75, 2, 85, 95, 90, 75],
    "VOTER_3": [19, 21, 20, 50, 20, 65, 95, 3, 90, 85, 95, 10],
    "VOTER_4": [0, 0, 0, 0, 0, 100, 0, 0, 0, 90, 0, 0]
}

df = pd.DataFrame(data)

# Define the quantile normalization function
def quantile_normalize(df):
    rank_mean = df.stack().groupby(df.rank(method='first').stack().astype(int)).mean()
    return df.rank(method='min').stack().astype(int).map(rank_mean).unstack()

# Apply quantile normalization to the raw score dataframe
normalized_df = quantile_normalize(df)
normalized_df

That's ChatGPT4 code, pretty compactified. But basically, it (almost) sets every voter's q-th quantile score to the mean of the q-th quantile scores across all voters. If score distributions were continuous, the previous "almost" could be removed, but sometimes effects of discreteness can't be avoided non-arbitrarily. I think it could also make sense to increase smoothing with an interpolation method to further mitigate the discreteness effects. In one respect if we're normalizing scores we might as well go there, but I can also understand why that prospect might feel even less palatable.

[Jack Waugh]

The array associated with each voter gives the score by candidate index?

What do pandas and numpy do?

I am a JavaScripter, not a Pythonista.

[Jack Waugh]

@cfrank said in Quantile-Normalized Score:

every voter's q-th quantile score

What does this term mean?

[Jack Waugh]

@cfrank Voter 1 doesn't look realistic. She hasn't used both extrema of the range 0 and 100. Why would a voter voluntarily give up political power? Please use examples based on an assumption that voters are in conflict and want to push policy in the direction they favor.

[Jack Waugh]

Does it conform to Frohnmayer balance?

[cfrank]

@jack-waugh actually these are real score ballots from an actual election (albeit a low-stakes one). I wondered the same thing, but there you have it!

[Jack Waugh]

Anything that I don't understand, I am downvoting to the max.

[cfrank]

@jack-waugh that’s right. Pandas is an API for working with data-frames, basically it just formats data into queryable tables. Numpy is an API for various mathematical operations, especially for matrices. I don’t see exactly where ChatGPT used numpy, but it probably does somehow… maybe?

[cfrank]

@jack-waugh if q is a real number in [0,1], a q-quantile of a random variable X is a value x such that Prob(X<=x)=q. For example, a median is a 0.5-quantile (aka 50th percentile).

I don’t really care for StatQuest, since I find it to be fairly patronizing… but here’s this: https://youtu.be/ecjN6Xpv6SE?si=XGugEzrGvUX078VF

[cfrank]

This post is deleted!

[cfrank]

@jack-waugh I doubt it. But I'm still not sold that Frohnmayer balance is either meaningful or desirable. At the heart of the criterion is cancellation, which treats the election in some sense as a zero-sum game. I don't think it's necessary to do that.

[Jack Waugh]

@cfrank said in Quantile-Normalized Score:

not sold that Frohnmayer balance is either meaningful or desirable

Do you object to Choose-one Plurality for a single winner? On what grounds?

[cfrank]

@jack-waugh I surely do! The main reason I object is that voters intrinsically suffer from untenable conflicts of interest when casting their ballots. This is also inextricably tied to the consequences of the system and its failure to satisfy independence of clones in particular, leading to the formation of a political duopoly wherein those exact conflicts of interest presented to voters via the character of the “choose-one” ballot structure are, passively or otherwise, exploited to enable the propping up of arbitrary, parasitic, pseudo-democratic power structures that ultimately have, often not without intention, minimized within unavoidable constraints their accountability to the electorate.

While I do think that there needs to be a degree of balance in a reasonable electoral system, I don’t think that Frohnmayer’s formalization of balance actually means what it intends to, or even that it is always well-defined.

[Jack Waugh]

@cfrank So it looks as though you have two main lines of reasoning in opposing choose-one plurality, and they start with, respectively:

• conflicts of interest experienced by the voters;

• dependence on clone candidates.

How can we observe or reason that voters face conflicts of interest? Is this about each voter individually, or about some aggregates from among them? What is the conflict, and how does it cause a problem? Do alternative voting systems take away the conflict of interest? How?

[cfrank]

@jack-waugh essentially, there can never be a total removal of conflicting interests, which is more or less a consequence of Gibbard’s theorem. People strategize only to navigate conflicts of interest. In choose-one voting, the conflict of interest is manifest in spoiler candidates and choosing the “lesser of evils” candidate, or a popular candidate, rather than the candidate a voter would actually hope to win the election. In my mind, the goal is, roughly, to find a way to minimize these conflicts while preserving choice, even including conflicts regarding issues of participation and ballot complexity.

The way I think of to reason about conflicting interests is to consider oneself as a generic voter and to conceptualize the task of making a decision on how to vote in relevant circumstances in order to achieve various goals. When there are clear, significant goals that require incompatible solutions, that introduces a conflict of interest. Obviously a choice of A versus B must be made somewhere, but the question is one of the needless frequency and severity of certain conflicts that can otherwise be mitigated by adopting a system that is superior in that regard.

I think the fact that “choose-one” is rated so low in all of our little tags here is an indication that we can easily see when conflicting interests are too frequent and severe to be tenable when reasonable alternatives exist. This is a pretty philosophical topic and I’m not confident there will ever be a clear answer. But one thing is for sure: failing independence of clones opens up a significant dimension to be populated with conflicting interests.

[Jack Waugh]

@cfrank said in Quantile-Normalized Score:

minimize these conflicts

If it's possible to minimize the conflicts, that implies some sort of measure for how severe the conflict is. What do you want to minimize, mathematically?

[cfrank]

@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”:

‘’’

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?

‘’’

I thought the classification was pretty decent and also found the questions pretty central.

[Jack Waugh]

@cfrank said in Quantile-Normalized Score:

failure to satisfy independence of clones

What are the grounds to understand this as a problem?

[cfrank]

@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.

[Jack Waugh]

@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:

• system 0: Approval Voting;

• system 1: There is no "system 1" mentioned in this discussion.

• 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.

• 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.

• system 4: Choose-one Plurality.

• system 5: You are required to approve all but one of the candidates.