Quantile-Normalized Score

cfrank

This seems like a hybrid approach between rank and score that automatically places ballots on a somewhat equal footing.

https://en.m.wikipedia.org/wiki/Quantile_normalization

In this case, just to be clear, normalization would take place over the set of ballots, not over candidates, which would be absurd.

Quantile normalization would automatically force the effective scores of voters to have equal spread. More specifically, it would force effective scores of each voter to have, roughly, the same distribution.

Jack Waugh

Sounds like you want to massage the votes to coerce them into some sort of statistical constraint over them all. Which voters are going to have their votes diluted by such a procedure?

cfrank

@jack-waugh hi Jack! I actually don't think it makes sense to speak of dilution, constraint or coercion in this context. It's simply a voting method. For example, dilution is only a sensible concept relative to an alternative that enables a faithful comparison. One can very well reverse any argument to the effect that method B "dilutes" the influence of certain voters relative to a comparable method A without any loss of validity to the argument; that is, whatever can be argued to be "diluted" by the transition from method A to method B can just as well be argued to be "inflated" by the transition from method B to method A---yes?

Based on my observations, this is a pervasive issue that plagues discourse in this field. So many arguments made to one effect can be totally reversed without any loss of validity, merely by changing out one arbitrary measuring stick for another. I think it's a pretty serious issue. As long as we're making these kinds of arguments without first agreeing on or at least recognizing what measuring stick to use and its sphere of valid applicability, I think that we won't be able to make any further headway on subsequent agreements. Unfortunately, the choice of measuring stick is precisely the issue under question. This is by no means a criticism of you in particular, but just a general phenomenon that I observe. I truly think that as a collective, we need to be more careful to check that our claims can't so easily be reversed, otherwise we won't actually be saying much at all that is ultimately substantial about the field.

My opinion is that our approach should be more practical and consequentialist, and generally less theoretical. Even if theory informs a method, the theory itself can't be used as a legitimate feather in that method's cap. At best it is a speculative feather, and only consequences of actually applying the method can ever be effective data about it.

Jack Waugh

My language is incendiary and it shows my bias in favor of plain Score Voting or at least toward regarding it as the norm based upon which variants should be justified.

Would you please lay out the tallying algo?

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?