Talk:Arrow's impossibility theorem/Archive 2

Archive 1

Archive 2

Arrow claiming that rated voting doesn't have normalization-based IIA failures?

User:Closed Limelike Curves, I am responding because you have reverted my changes twice now. Last time you added the following quote to the CES podcast reference:

So I think Approval Voting is a little too coarse. I think if you had three or four candidates the incentives for this would be much less if you had three or four classes. There would be a tendency to approve candidates you don’t think very well of just to avoid somebody you think is a real catastrophe.

This is in reply to Aaron Hamlin asking if Approval would encourage the growth of third parties. The quote seems to be more related to strategy than to normalization problems. If it were related to normalization problems, it could just as easily be read the other way: "There would be a tendency to approve candidates you don't think very well of just to avoid somebody you think is a real catastrophe", implying that if the real catastrophe hadn't run, you wouldn't be approving those candidates you don't think very well of, hence a change in ballots would occur due to irrelevant candidates dropping out. So I would like to ask where you consider the quote, or the podcast reference in general, to imply that Arrow doesn't think Sen-type IIA failures will occur, i.e. that he "reversed his opinion later in life, coming to agree that scoring methods provided more useful information that make it possible for such systems to evade his theorem".

That sufficiently fine-grained cardinal ballots with sufficiently many candidates provide more ways to vote (more information) is not in contention, nor is that some cardinal methods like Range pass IIA when that extra data is held fixed. Indeed, the former is the reason Sen uses the wider concept of an SWFL and not just a plain SWF. Nor does the original Social Choice and Individual Welfare reference assume that data is being held fixed: Arrow directly refers to von Neumann-Morgenstern utilities and their invariance to positive linear transformations. Wotwotwoot (talk) 23:05, 19 April 2024 (UTC)

That’s correct. Yes. Now there’s another possible way of thinking about it, which is not included in my theorem. But we have some idea how strongly people feel. In other words, you might do something like saying each voter does not just give a ranking. But says, this is good. And this is not good. Or this is very good. And this is bad. So I have three or four classes. You have two classes is what you call Approval Voting. Just say some measures are satisfactory, and some aren’t. This gives more structure. And, in effect, say I approve and you approve, we sort of should count equally. So this gives more information than simply what I have asked for [... if] we don’t just rank the candidates. We say something like they’re good or bad or something. [...]

CES:But the system that you’re just referring to, Approval Voting, falls within a class called cardinal systems. So not within ranking systems. Dr. Arrow: And as I said, that in effect implies more information.

–Maximum Limelihood Estimator 02:57, 20 April 2024 (UTC)

Yes, Arrow does say that there is more information. That's why I said

That sufficiently fine-grained cardinal ballots with sufficiently many candidates provide more ways to vote (more information) is not in contention.

That much is clear. Sincere rated ballots based on vNM utilities give strength of preference, and ranks don't: that's the whole point. But he doesn't say nor does he imply that this invalidates Sen-type IIA.

We later have

But we have some idea how strongly people feel. In other words, you might do something like saying each voter does not just give a ranking. But says, this is good. And this is not good. Or this is very good.

Which implies again that there's a strength of preference ("not just give a ranking"), and that he sees a need for more ratings than just approve and disapprove, so that the voters may provide information about how strongly they feel about each candidate on some scale (good, very good, etc). But that is an orthogonal issue.

How fine-grained the rating resolution is doesn't itself validate or invalidate Sen-type IIA failure - unless you're thinking that Arrow, by using grade-like terms like "good" and "very good", is referring to Balinski and Laraki's MJ reasoning which is intended to reduce Sen-IIA type behavior. But that seems like somewhat of a reach. And were that the case, it's likely that he would have referred to MJ by name, given that he names Balinski elsewhere in the podcast.

In short, while the source shows that Arrow considers cardinal voting an improvement since such ballots can represent more information, Arrow does not specifically counter the point made in Social Choice and Individual Welfare. Wotwotwoot (talk) 17:31, 20 April 2024 (UTC)

I'm citing this to support the claim that Arrow, later in life, agreed cardinal voting is an improvement because it provides additional, meaningful information relative to ordinal ballots; and that this meaningful information allows such methods to evade his results. (Setting aside practical limitations like human psychology, which Balinski & Laraki take pains to minimize; I'll try and add your reference to their work back in.)

My issues with the previously-suggested edits:

1. The sentence raises a technical point that can only be properly discussed in the body (not the lede). The result is also only indirectly related to Arrow's theorem (enough to warrant mention in the article, but not the lede). Discussing these results in the lede serves to reinforce the common misconception that Arrow's theorem (either directly or in an only slightly-modified form) applies to cardinal methods.

2. The sentence makes it sound like the authors are proving another rigorous impossibility result, rather than raising a philosophical objection.

3. The sentence seems redundant, given the already-existing sections covering limitations caused by human psychology and philosophical disagreements about interpersonal utility comparisons.

4. The sentence makes it sound like rated methods are exempt under a technicality, or there's a similar result applying to rated methods under a different name (like how Satterthwaite technically doesn't apply to cardinal methods, but Gibbard's theorem proving only a slightly weaker result still does). Arrow, Vickrey, and Harsanyi would all disagree with the claim, and argue these kinds of numeric scores are meaningful in a way that allows score voting to avoid independence failures (up to practical failures). (Sen also might agree, since I've seen him argue elsewhere that interpersonal utility comparisons are possible.) Balinski and Laraki also showed cardinal utilities or grades proportional to aren't actually needed for IIA, so long as voters are allowed to rate candidates independently (median ratings only require ordinal information). –Maximum Limelihood Estimator 02:01, 21 April 2024 (UTC)

If you're using this citation to show that Arrow has changed his mind to that cardinal methods evade (i.e. aren't affected by) his own results, then it does not seem relevant to use this citation when discussing Sen's generalization. "Sen showed X, Arrow had an informal argument along these lines too, but later in life he changed his mind", makes it sound like the "later in life he changed his mind" part is relevant to the informal argument, which it isn't.

That's distinct from using the citation to reference that Arrow says that his own impossibility theorem does not apply to cardinal voting.

I now see you have removed the "informal argument" clause entirely, but I would like to restore it.

As for the rest, I've already given a summary above (under "Spoiler effects and IIA"), but I would like to add a few more points.

I am not proposing that the lead discuss the generalizations, only that it makes the reader aware that they exist. Actual discussion would take place in a separate section, not in the lead. If you'd like, we could add a contingency and say something like "generalizations exist that do apply to rated elections given additional assumptions".

My point is basically the converse of yours.

There are many places on Wikipedia where a footnote or caveat about rated voting says "the IIA result only holds if voters don't change their scales", or something to that effect. These exist because the consensus seems to be that some people do change their scales. The generalizations formalize the argument that if they do, then the broader election does fail IIA, giving a theoretical backing relevant to the theorem for what's being informally expressed in the caveats (as well as elsewhere, in Approval papers discussing how to vote, mean utility, etc.; or even right here on this page with CRGreathouse saying "I grant that there are normalization issues with cardinal voting systems").

So your concern is that discussing generalizations in the lead would risk people thinking that the standard Arrow's theorem applies to rated voting. Mine is that not doing so would risk them thinking that rated voting elections pass, just because the systems do when ratings are held fixed. The references to vNM utilities are intended to give a reasonably common theoretical model to explain such changes of scale. Responses could be dealt with in the section.

The reasons I gave under "Spoiler effects and IIA" give the theoretical relevance of the generalizations to Arrow's theorem. And the behavior seeming natural enough that there are caveats to this effect elsewhere indicate that it's practically relevant as well. Thus for two different reasons it deserves a more broad discussion than just a passing remark in the interpersonal comparison section. Wotwotwoot (talk) 11:17, 21 April 2024 (UTC)

These exist because the consensus seems to be that some people do change their scales. The generalizations formalize the argument that if they do, then the broader election does fail IIA, giving a theoretical backing relevant to the theorem for what's being informally expressed in the caveats (as well as elsewhere, in Approval papers discussing how to vote, mean utility, etc.; or even right here on this page with CRGreathouse saying "I grant that there are normalization issues with cardinal voting systems").

I'd grant there's an issue here in that some voters normalize their ballots, much like how behavioral economics has shown voters use a wide variety of heuristics to make even ordinal judgments (see decoy effect). In that case, no Smith-independent method actually satisfies ISDA: it's possible to introduce a new, strongly-dominated candidate who nevertheless changes the way voters rank other outcomes.

If we want to say "well, the system satisfies this axiom, but some voters act in a way that violates it", we'd have to apply that to every voting system and voting property, and every article's lede will quickly get very, very messy. (Also, we could no longer say that Condorcet methods uniquely minimize the rate of IIA failures.) –Maximum Limelihood Estimator 18:42, 22 April 2024 (UTC)

GA Review

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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This review is transcluded from Talk:Arrow's impossibility theorem/GA1. The edit link for this section can be used to add comments to the review.

Nominator: Closed Limelike Curves (talk · contribs) 22:22, 3 May 2024 (UTC)

Reviewer: Phlsph7 (talk · contribs) 08:32, 5 May 2024 (UTC)

Hello Closed Limelike Curves and thanks for all your improvements to this article. However, despite the improvements, the article fails criterion 2b since there are too many unreferenced paragraphs and a whole section lacks references. Examples are the section "Common misconceptions" and the paragraphs starting with "Arrow's theorem falls under the branch of welfare economics", "Arrow defines IIA slightly differently, by stating", and "Arrow's requirement that the social preference". According to criterion 2b, these passages require inline citations "no later than the end of the paragraph". The unreferenced section has the maintenance tag "Unreferenced section" and there are overall 6 "citation needed" maintenance tags in the article. I suggest that you add all the relevant references before a renomination.

A few other observations

• WP:EARWIG detects no copyright violations
• Arrow's requirement that the social preference $A\succ B$ only depend on replace "depend" with "depends"
• expressing social welfare, leading him focus his theorem on preference rankings add "to" before "focus"
• https://electionscience.github.io/vse-sim/ is probably an unreliable source

Phlsph7 (talk) 08:32, 5 May 2024 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.