Quantile-Normalized Score
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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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@jack-waugh yes, you’re right, if ballots are directly elements in a vector space (or maybe more generally an Abelian group), and the first operation of the voting system is to compute the sum of the elements directly and have no future reference to the original ballot set, there are definitely appealing consequences. And score ballots after quantile normalization almost surely fail to have inverse elements, which is the issue under consideration.
I think I’m coming around to your point of view, I just want to understand it as transparently as I can. My concern is that it seems to impose very rigid restrictions on the kinds of transformations or normalizations we should be able to do to ballots, for example it seems to restrict to linear projections or group homomorphisms, and then aggregation, which would have the same effect as if aggregation were performed before the transformations. Maybe that’s just the way it is.
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@cfrank Well, maybe some of that is more restrictive than necessary.
The ballots can be referred to again for rounds of tallying (maybe impractical for public office because of integrity issues, but applicable in organizations that trust their IT departments) but again treated additively, just in the context of some candidates already having been eliminated from consideration by prior rounds. I included this in a recent proposal here.
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@jack-waugh yes, so that is in one sense multiple voting systems passing information in series to perform projections, such as eliminating certain candidates based on the result of a preceding system. That does add complexity, it’s essentially a hierarchical model of additive systems, and kind of like a one-pass neural network.
One of my concerns is that the end result seems heavily utilitarian, and that borders on majoritarianism to me, the issue is complicated. I’ll check out your proposal!
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@jack-waugh another point about “cancellation,” is whether it is actually necessary in practice. I think it’s arguable that effective or practical cancellation is just as good as perfect theoretical cancellation in almost all instances. For example, while a subset of N voters may not be able to perfectly cancel each other out, they can still on aggregate amount to little more than noise, which is a form of practical cancellation. I think there’s an argument to be made that sacrificing perfect cancellation for practical cancellation is justified to obtain other desirable properties. Playing devil’s advocate, one could even say that noise is a positive thing, since it’s a robust way to decide effective ties, which is more or less what would have occurred in your 3,000,002 voter illustration.
In practice, small margins victory lead to recounts anyway, which has its own practical noisiness.
And here is an adjustment to quantile normalized scoring inspired by your proposal: we could force the final quantile values to be symmetrical. I think that preserves the structure of an abelian group.
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@cfrank said in Quantile-Normalized Score:
@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.
I'm not aware score has a problem with bullet voting. And while drugs are often found by some sort of trial and error, they have to be tested. So I suppose I can be patient and wait to see what this method offers!
Edit - I see there's a new version now so I'll have a look.
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@jack-waugh said in Quantile-Normalized Score:
Which voters are going to have their votes diluted by such a procedure?
In reply to this, you might be interested to note that, based at least on my own observations of the symmetrized version, it is the bullet-voters and min-maxers who will have the advantages of their tactical behavior reduced in favor of a broader consensus, in a fashion very similar to what would be expected in a Condorcet method. However, the method is not Condorcet compliant, I found a counter-example.