Talk:Arrow's impossibility theorem

first sentence

I don't get that first sentence at all: "demonstrates the non-existence a set of rules for social decision making that would meet all of a certain set of criteria." - eh? (Sorry if this was discussed already, don't have time to read it all now, nor rack my brains on trying to decipher that sentence.) --Kiwibird 3 July 2005 01:20 (UTC)

That was missing an "of", but maybe wasn't so clear even with that corrected. Is the new version clearer? Josh Cherry 3 July 2005 02:48 (UTC)

From Dr. I.D.A.MacIntyre

I make three comments. Firstly the statement of the theorem is careless. The set voters rank is NOT the set of outcomes. It is in fact the set of alternatives. Consider opposed preferences xPaPy for half the electorate and yPaPx for the other half. ('P' = 'is Preferred to'). The outcome is {x,a,y} under majority voting (MV) and Borda Count (BC). (BC allocates place scores, here 2, 1 and 0, to alternatives in each voter's list.) The voters precisely have not been asked their opinion of the OUTCOME {x,a,y} compared to, say, {x,y} and {a} - alternative outcomes for different voter preference patterns. All voters may prefer {a} to {x,a,y} because the result of the vote will be determined by a fair lottery on x, a and y. If all voters are risk averse they may find the certainty of a preferable to any prospect of their worst possibility being chosen. This difference is crucial for understanding why the theorem in its assumptions fails to represent properly the logic of voting. As I show in my Synthese article voters must vote strategically on the set of alternatives to secure the right indeed democratic outcome. Here aPxIy for all voters would do. ('I' = 'the voter is Indifferent between'). Indeed as I show in The MacIntyre Paradox (presently with Synthese) a singleton outcome evaluated from considering preferences can be beaten by another singleton when preferences on subsets (here the sets {x}, {x,y}, {a} etc) or preferences on orderings (here xPaPy, xIyPa, etc) are considered. Strategic voting is necessary because this difference between alternatives and outcomes returns for every given sort of alternative. (Subsets, subsets of the subsets etc). Another carelessness is in the symbolism. It is L(A) N times that F considers, not, as it is written, that L considers A N times. Brackests required. In a sense,and secondly, we could say then that the solution to the Arrow paradox is to allow strategic voting. It is the burden of Gibbard's theorem (for singleton outcomes - see Pattanaik for more complex cases) (reference below) that the Arrow assumptions are needed to PREVENT strategic voting. The solution to the Arrow problem is in effect shown in the paragraph above. For the given opposed preferences with {x,a,y} as outcome voters may instead all be risk loving prefering now {x,y} to {x,a,} and indeed {a}. This outcome is achieved by all voters voting xIyPa. But in terms of frameworks this is to say that for initial prferences xPaPy and yPaPx for half the electorate each, the outcome ought to be {a} or {x,y} depending on information the voting procedure doesn't have - voters' attitudes to risk. Thus Arrow's formalisation is a mistake in itself. The procedure here says aPxIy is the outcome sometime, sometimmes it is {a} and sometimes were voters all risk neutral it is{x,a,y}. These outcomes under given fixed procedures (BC and MV) voters achieve by strategic voting. We could say then that Gibbard and Satterthwaite show us the consequences of trying to prevent something we should allow whilst Arrow grieviously misrepresents the process he claims to analyse

Thirdly if you trace back the history of the uses that have been made of the Arrow - type ('Impossibility') theorems you will wonder at the effect their export to democracies the CIA disappoved of and dictatorships it approved of actually had. Meanwhile less technical paens of praise for democracy would have been directed to democracies the US approved of and dictatorships it didn't. All this not just in the US. I saw postgraduates from Iran in the year of the fall of Shah being taught the Arrow theorem without any resolution of it being offered. It must have been making a transition to majority voting in Iran just that bit more difficult. That the proper resolution of the paradox is not well known (and those offered above all on full analysis fail to resolve these Impossibilty Theorems and in fact take us away from the solution) allows unscrupulous governments to remain Janus faced on democracy. There certainly are countries that have been attacked for not implementing political systems that US academics and advisors have let them know are worthless.

--86.128.143.185

Moved from the article. --Gwern (contribs) 19:43 11 April 2007 (GMT)

From Dr. I. D. A. MacIntyre.

I am at a loss to understand why other editors are erasing my comments. Anyone who wishes to do so can make a PROFESSIONAL approach to Professor Pattanaik at UCR. He will forward to me any comments you have and, if you give him your email address I will explain further to you. Alternatively I am in the Leicester, England, phone book.

I repeat: the statement of the theorem is careless. For a given set of alternatives, {x,a,y} the possible outcomes must allow ties. Thus the possible outcomes are the SET of RANKINGS of {x,a,y}. The other editors cannot hide behind the single valued case of which two things can be said. Firstly Arrow allowed orders like xIyPa (x ties with y and both beat a). Secondly if only strict orders (P throughout) can be outcomes how can the theorem conceivably claim to represent exactly divided, even in size, societies where for half each xPaPy and yPaPx.

Thus compared with {x,a,y} we see that the possible outcomes include xIyPa, xIyIa and xIaIy. In fact for the voter profile suggested in the previous paragraph under majority voting and Borda count (a positional voting system where,here 2, 1 and 0 can be allocated to each alternative for each voter) the outcome will be xIaIy. The problem that the Arrow theorem cannot cope with is that we would not expect the outcome to be the same all the time for the same voter profile. For for the given profile, and anyway, voters may be risk loving, risk averse or risk neutral. If all exhibit the same attitude to risk then respectively they will find xIyPa, aPxIy and xIaIy the best outcome. (Some of this is explained fully in my Pareto Rule paper in Theory and Decision). But the Arrow Theorem insists that voters orderings uniquely determine the outcome. Thus the Arrow Theorem fails adequately to represent adequate voting procedures in its very framework.

To repeat the set of orderings in order (ie not xIyPa compared with xIaIy, aPxIy etc). Thus all voters may find zPaPw > aPxIy > xIaIy > xIyPa if they are risk averse. ({z,w} = {x,y} for each voter in the divided profile above). The plausible outcome xIaIy is thus Pareto inferior here to aPxIy. In fact any outcome can be PAreto inoptimal for this profile. (For the outcomes aPxIy, xIyPa and xIaIy the result will be {a}, and a fair lottery on {x,y} and {x,a,y} repsectiively. The loving voter for whom zPaPw prefers the fair lottery {x,y} compared with {a} and hence xIyPa to aPxIy.

The solution is to allow strategic voting so that in effect voters can express their preferences on rankings of alternatives. Under majority voting such strategic voting need never disadvantage a majority in terms of outcomes, and as we see here, can benefit all voters. (Several of my Theory and Decison papers discuss this).

We are very close to seeing the reasonableness of cycles. For 5 voters each voting aPbPc, bPcPa and cPaPb the outcome {aPbPc, bPcPa, cPaPb, xIaIy} seems reasonable. This is not an Arrow outcome but one acknowleding 4 possible final results. But then the truth is, taking alternatives in pairs that with probability 2/3 aPb as well as bPc and cPa. What else can this mean except that we should choose x from {x,y} in every case where xPy with 2/3 probability.

(In the divided society case above if all voters are risk loving the outcome {aPxIy} is preferred by all voters to some putative {xPaPy, yPaPx}. The possible outcomes for voting cycles are to be found in my Synthese article.)

I go no further. Except to make five further comments. Firstly those who like Arrow's theorem can continue so to do, as a piece of abstract mathematics, but not as a piece of social science, as which it is appallingly bad. Arrow focuses on cyclcical preferences and later commentators like Saari have fallen into the trap of thinking opposed preferences not a problem for the Arrow frame. In fact both sorts of preferences are a problem for the erroneous Arrow frame. That is the way round things are. The Arrow frame presents the problems. The preferences are NOT problematic.

Secondly I reiterate strategic voting, which is a necessary part of democracy, need never allow any majority to suffer (see my Synthese article for the cyclical voting case). Indeed majorities and even all voters can benefit. Majority voting with strategic voting could, then, be called consequentialist majoritarian.

Thirdly, and if this is what is getting up the other editos noses then leave just this out because it is most important that everyone stops being fooled about MAJORITY VOTING by Arrow's theorem and his Nobel driven prestige, anyone who thinks Arrow has a point has been led astray. If US academics and advisors believe he has then why do we bomb countries for not being demcracies? And if no one does then why was the theorem taught unanswered to Iranian students here in the UK during the year of the fall of the Shah? If Iran is not a demcracy to your liking, I am speaking to the other editors, a good part of the reason is the theorems you are protecting, I can assure you. No one can be Janus faced about this. paricualrly not by suppressing solutions to the Theorem in a dictatorial way.

Fourthly to restrict the theorem to linear orderings which Arrow does not do is pointlessly deceptive. For it hides the route to the solution (keeping 'experts' in pointless but lucrative employment?). For even in that case the set of strict orders on the set of alternatives is NOT what voters are invited to rank.

Lastly the hieroglyths above are wrong too. The function F acts on L(A) N times. L does not operate on A N times as the text above claims. Brackets required!

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From Dr. I. MacIntyre : Of course any account of the Arrow Theorem and its ramifications is going to please some and displease others so I add this comment without criticism.

It seems to me that strategic behaviour in voting (and more generally) is such an important part of human behaviour that how various voting procedures cope with it will turn out to be the most useful way of distinguishing between them.

Indeed one could go so far as to say that strategic behaviour, properly understood and interpreted, also provides the key to resolving the Arrow 'Paradox'.

To that end, and anyway because of its importance I think it would be useful in this Wikipedia article to indicate, at least, the tight connection between the constraints Arrow imposes on voters in order to derive his theorem and what must be imposed on them to avoid the logical possibility of 'misrepresentation' or strategic behaviour. That is, the role of Arrow's assumptions in Gibbard's Theorem should, I think, be spelt out at least informally.

Many writers have suggested resolutions to the Theorem without paying any real attention to strategic voting. As a result they have missed what is certainly majority decision making's best (and I think decisive) defence. For under majority decision making strategic voting can benefit majorities, even all voters (sic!) (see my Pareto Rule paper in Theory and Decision) and no majority ever need suffer. No other rule (eg the Borda Count rule) defends its own constitutive principle in this way.

As a result of these omissions (of any acknowledgement of the ubiquity of strategic behaviour and of the Arrow - Gibbard connection) the technical literature in recent years has lost realism in its accounts of democratic behaviour and leaves its readers with the impression that democracy is best saved by abandoning majority voting. (As Borda Count does). Such an odd view of best voting practice is likely to encourage dictators and discourage even the strongest of democrats. Perhaps that is the intended effect. For one could argue that the way majority mandates have been de - legitimised is the worst legacy of the Arrow Theorem so that just redistribution has been thwarted in South Africa, Northern Ireland and elsewhere in localities better known by you readers than I.

I. MacIntyre n_mcntyr@yahoo.co.uk 27th April 2007

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.

Strategic spoilers

@Wotwotwoot Do you happen to have citations on strategic spoiler effects? (It seems obviously correct, but I don't have a source.) –Sincerely, A Lime 23:18, 10 May 2024 (UTC)

Previous lede…

[1] … wasn’t great, but it was better than what we have now.

More generally, the theorem isn’t limited to just voting theory and the lede completely misses that. Volunteer Marek 07:26, 13 May 2024 (UTC)

Lede included, generally speaking, this article would be shorter if it was longer. I have a CS background, and I've come back to this article like several times to reference it. A lot of conversations about RCV die with this article, because someone says "hey, RCV has these draw backs" and links here, and then it's really not clear to me (or probably anyone in the discussion) what the drawbacks actually are. I'd really have to print this out and follow most of the wikilinks and probably read the source material and other sources if I really wanted to understand the implications here. A longer article with additions from someone who has domain expertise would really go a long way... and/or if I read it and really dig in to it, I'll maybe try to add some clarifying language and citation material where I struggled. - Scarpy (talk) 20:17, 15 May 2024 (UTC)

This is the longer article with clarifying language and citations written by a domain expert :) I overhauled it a month ago.

The new version clarifies the main drawback of IRV is it tends to have a lot of spoiler effects—more than Condorcet methods, at least. Graded voting systems don’t have any spoiler effect at all, but some people have philosophical objections to them. –Sincerely, A Lime 03:56, 21 May 2024 (UTC)

Is your main complaint that the latest version doesn't cover ML and voting classifiers? I'd be happy to see more material on those added, but I don't think the earlier versions of the article covered this topic either. –Sincerely, A Lime 16:57, 21 May 2024 (UTC)

This may or may not be relevant and non-obscure, but it certainly does not belong in the lede. Volunteer Marek 23:43, 19 May 2024 (UTC)

I've modified the sentence to try and clarify the relevance, but further edits are welcome.

However, if your interest is in ML classifiers, I think the last sentence of the lede is probably the most relevant in the article. Arrow is offering an interpretation of his theorem as a proof that model averaging is better than using a voting classifier, because every voting classifier will violate Luce's choice axiom. –Sincerely, A Lime 00:15, 22 May 2024 (UTC)

No, that’s not the issue. The issue is that the lede is supposed to summarize the article and this doesn’t summarize anything in the article. Furthermore, the lede should be general and accessible rather than focusing on esoteric applications. Volunteer Marek 06:41, 22 May 2024 (UTC)

I don't know what you're referring to, then. The article contains an extensive discussion on research and results on the meaningfulness of cardinal information.

I'm not sure how voting is more esoteric than the other applications of Arrow's theorem. The vast majority of the hits for Arrow's theorem on Google Scholar are referring to it in the context of social choice. I know it's occasionally discussed in other fields, but most people who come to this article come from other voting theory articles. –Sincerely, A Lime 17:01, 22 May 2024 (UTC)

Ok, first problem - and it is a *really big* problem is that this entire article is written as if Arrow’s theorem is just about voting. It’s not. It’s about social choice in general. At the moment it would require way too much effort to rewrite the article to put it in a suitable state.

Second, I didn’t say voting “is esoteric”. What I said is that putting applications to *rated* voting are a bit esoteric.

And to get to basic issue here is the version of the article when I removed that stuff from the lede. The section on “rated voting” is much smaller and does not contain all the original research. I see you NOW added it [2], [3] AFTER I pointed out that the stuff in the lede wasn’t summarizing the article. So your statement of “I don’t know what you’re referring to” seems a bit off. At any rate, this seems backwards - add stuff to the lede, then when someone points out that it’s not summarizing anything in the main body, add it to the main body. Volunteer Marek 00:30, 23 May 2024 (UTC)

I think you missed this sub-subsection, in the previous version you linked, which still contains an extensive discussion on research and results on the meaningfulness of cardinal information. The links you provided show me adding a single paragraph referencing a relevant paper to this sub-subsection. –Sincerely, A Lime 03:01, 23 May 2024 (UTC)

Can you also explain how the two sources given support the stated text that “”Arrow himself initially dismissed such systems on philosophical grounds, but later considered this a mistake, describing score voting as "probably the best" way to avoid his paradox.” The given quotes do not support this text. The second source is by someone else, about something else. The first source is Arrow but where exactly is he saying that “rated methods” are “probably the best way to avoid his paradox”?

(this seems to be rather the standard issue of ordinal vs cardinal preferences) Volunteer Marek 06:46, 22 May 2024 (UTC)

I've added the exact quote in a separate reference –Sincerely, A Lime 20:28, 22 May 2024 (UTC)

Sigh, ok.

”Arrow initially asserted the information provided by these systems was meaningless” - where is this in the quote/source provided? All that Arrow does here is describe how consumer choice theory approaches its subject. If I’m not mistaken the given source does not even mention the impossibility theorem. Arrow was a very versatile economist who wrote on a very wide range of topics and just because he said something about one thing - consumer theory - does not mean he said the same thing about another thing - social choice theory. And he didn’t even say it! There’s nothing here where he says “information provided by these systems is meaningless”. This is original research at best.

”therefore could not prevent his paradox” - again nowhere in the quote (which isn’t even about the paradox) and not in the source. IF there is text somewhere else in the Binachi source which supports this, please provide it.

”he would later recognize this as a mistake” - where is that in the quote and or source provided? All that he says is that there’s different ways of thinking about social choice. And calling it a “mistake” is kind of a give away that this is just OR. It’s a theorem. It has applications given by its assumptions. A “mistake” would be if there was a mathematical error in it. There isn’t.

” describing score voting as "probably the best" way to avoid his theorem” - ok, you finally provided a source which is relevant (so the other two sources should be removed as they say nothing of the kind).

Except here is the actual quote:

” Dr. Arrow: Well, I'm a little inclined to think that score systems where you categorize in maybe three or four classes probably (in spite of what I said about manipulation) is probably the best.”

He says he’s “a little inclined”. And he says it’s “probably the best” but NOT that it’s a way of “avoiding his theorem”. What he is saying here is basically that given that his theorem tells you can’t have a perfect system, then having to choose from among all the imperfect systems, scored voting might possibly kind of be the best. That’s a different thing entirely. Volunteer Marek 00:30, 23 May 2024 (UTC)

The main problem with this article

… is that it presents the theorem as being exclusively about voting systems. But that’s neither how Arrow or standard texts on the subject characterize it. Arrow himself, in his famous paper, right of the bat mentions market exchange as an example of aggregation of individual preferences into social outcome. Look at how Stanford Encyclopedia of Philosophy approaches it [4]. The word “voting” doesn’t appear until the fourth paragraph in the very specific context of Condorcet’s Paradox. The whole point - as SEoP makes abundantly clear - of the impossibility theorem is that it’s NOT JUST voting (specifically majority voting) that is subject to anomalies like that of Condorcet, but *social choice* in general.

Presenting this subject as just about voting is both misleading to the reader and does quite an injustice to a very important, even fundememtal, result. Volunteer Marek 00:56, 23 May 2024 (UTC)

If you'd like to add more discussion of the social choice perspective, be my guest! There's a close relationship between voting and social choice—Arrow often referred to his theorem as being about either "social choice" or "voting" interchangeably—but I focused on voting because it's more concrete and easier to understand. –Sincerely, A Lime 18:22, 28 May 2024 (UTC)

Voting is just one way that society can make choices. Market exchange is another. The point of the theorem is to treat social choice at a highly general level.

I appreciate that different folks come to this subject from different backgrounds. At the same time we need to be aware of that and not let these backgrounds skew the presentation of the subject. The current problem is that the present version is SOOOOO skewed towards a particular version that it would truly be a great task to rewrite it appropriately. Volunteer Marek 04:57, 9 June 2024 (UTC)

I think I've improved on this.

Although, thinking about it more, it seems to me like Arrow's theorem—unlike other theorems of social choice—is in practice limited to voting. Markets etc. rarely (if ever) rely on pure ranking data; there might be a few situations where monetary transfers are prohibited like organ-matching, but generally social choice involves comparisons of utility. –Sincerely, A Lime 16:39, 10 June 2024 (UTC)

I think this article is much more clear for talking about voting in the lead instead of immediately plunging into the phrase "aggregation of individual preferences into social outcome". I am sympathetic to OP's view, but we must remember that Wikipedia has a pretty different readership than SEP. Mathwriter2718 (talk) 14:17, 18 July 2024 (UTC)

Non-nullity defined incorrectly?

As described here, a null voting system would be one that has an a priori ordering of all candidates, and always returns that ordering regardless of the votes. But there are other voting systems that do not meet this definition but still obey IIA. Here's one:

• Use an a priori weak order of the candidates, in which (among the entire field of potential candidates) each candidate has at most one other candidate with whom they are tied.
• Return a linear extension of this weak order, resolving ties between pairs of tied candidates by majority vote.

For a natural example of this, consider a voting system that always chooses the majority winner between the candidates from two major parties, and then lists the third parties in alphabetical order. There can be no spoilers, because they cannot affect the majority-vote tie-breaking system and nothing can affect the other comparisons. On the other hand, there are plenty of pairs of candidates for whom the voters are ignored. I think maybe the correct formulation of non-nullity is: for every two candidates, both outcomes are possible. —David Eppstein (talk) 08:58, 9 June 2024 (UTC)

The redefinition you proposed seems to be Wilson's weakened form of the citizen sovereignty (onto) condition, which he drops in the last section, but I think your counterexample is correct (which means I'm missing a condition somewhere). Closed Limelike Curves (talk) 18:42, 12 June 2024 (UTC)

Score voting

I am bothered by this article's blatant advocacy of score voting both in the lead and in the "Eliminating IIA failures: Rated voting" section, for three reasons:

1. it is off-topic.
2. I am not convinced it is neutrally presented.
3. In my personal experience (having seen this system in action in certain polarized committee votes) it is a very bad system, not because it can be gamed (all systems can be gamed) but because it is so blatantly obvious that it can be gamed as to put any honest participants at a severe disadvantage. Participants willing to game the system devolve to approval voting, honest participants spread their scores among different candidates, and the approval voters win. If you're going to enforce that voters spread their scores more uniformly you might as well just use Borda, and if you're going to allow approval voting then just use approval voting and put all voters on a more equal footing.
4. Our coverage of this gaming issue dishonestly mixes the two by talking about it as a voting system but then using sources such as Harsanyi that talk about aggregating utility (without opportunity for voters to misprepresent their preferences) rather than scored voting.

—David Eppstein (talk) 19:44, 9 June 2024 (UTC)

I've improved the lead, and will work on improving the rated voting section later. –Sincerely, A Lime 01:29, 10 June 2024 (UTC)

ArXiv Pre-prints are not Reliable Sources

Currently FN10: Holliday, Wesley H.; Pacuit, Eric (2023-02-11), Stable Voting, arXiv:2108.00542, retrieved 2024-03-11 is a link to this arXiv page which does not show a publication. This cannot be considered a reliable source as anyone can post there. Czarking0 (talk) 00:12, 19 June 2024 (UTC)

Per WP:ARXIV, Arxiv reprints are allowed/considered reliable if published by subject matter experts. That said you can also find a publication here:

https://link.springer.com/article/10.1007/s10602-022-09383-9 Closed Limelike Curves (talk) 14:18, 19 June 2024 (UTC)

Statement of the theorem, NPOV, and sourcing

I am concerned about how the theorem is stated.

1. The lead says that:

No rank-based procedure for collective decision-making can behave rationally or coherently. Specifically, any such rule violates independence of irrelevant alternatives.

This is highly problematic because a) one might not think that IAA is required for rationality or coherentness, b) there are other assumptions in the theorem statement besides IAA. It would be much more accurate to say that ranked-choice collective decision-making procedures cannot simultaneously satisfy several axioms that we intuitively think fair systems should satisfy. We ought to be careful to take an WP:NPOV and avoid making a definitive judgement about whether IAA is required for "rationality" or "coherentness".

2. The theorem statement in this article appears to say:

Total ordering + non-dictatorship + IAA implies contradiction.

It cites Wilson to support this. However, Wilson's paper does not support this!!! The assumptions are different.

3. The section "Intuitive argument (voting)" uses one source, Iain McLean's paper, to support several claims. However, these claims are more hyperbolic than they have a right to be. For example, it says "many authors" take a certain stance, and cites only that McLean takes this stance. It also says:

Given these assumptions, the existence of the voting paradox is enough to show the impossibility of rational behavior for ranked-choice voting.

which I again believe is an NPOV problem.

I think these problems must be resolved before this article can be considered a Good Article. Mathwriter2718 (talk) 14:10, 18 July 2024 (UTC)

So, I'll first mention on the topic of rationality/coherence that in decision and social choice theory, these have a specific meaning, given by the von Neumann–Morgenstern axioms (including IIA); I've tried making that more clear by linking to them. IIA is considered a requirement for rationality because violating it implies your behavior will be self-contradictory (see spoiler effect) and opens you up to Dutch books.

On Wilson, could you explain to me how I've been misunderstanding him? I thought I was missing something but the paper says he drops the assumption that the function is onto. Closed Limelike Curves (talk) 17:58, 18 July 2024 (UTC)

1. You probably know more about this than I do, but my impression is that the von Neumann-Morgenstern independence axiom should not be thought of as "equivalent to" the IIA axiom. If nothing else, the von Neumann–Morgenstern axioms are about individuals, and IIA is about societal aggregation. Even if the idea is the same, the mathematical content and context are quite different, no? There could be other analogues of von Neumann-Morgenstern independence that also seem reasonable to require. For example, the relevant SEP page (linked below) has more than one non-equivalent formulation of IIA. Now, when I read IIA, it seems like a really strong assumption compared to von Neumann-Morgenstern independence. My impression is that some authors resolve the Arrow dilemma by rejecting that IIA is required for rationality, but I've never heard of someone rejecting von Neumann-Morgenstern independence. (To be clear, I'm not endorsing or rejecting this view, I'm just saying what I believe to be the case in the field.) For some evidence besides just my impression, the relevant SEP page ([[5]]) discusses IIA as if it needs justification. For example:

Gerry Mackie (2003) argues that there has been equivocation on the notion of irrelevance. It is true that we often take nonfeasible alternatives to be irrelevant. That presumably is why, in elections, we do not ordinarily put the names of dead people on ballots, along with those of the live candidates. But [IIA] also excludes from consideration information on preferences for alternatives that, in an ordinary sense, are relevant. An example illustrates Mackie’s point. George W. Bush, Al Gore, and Ralph Nader ran in the United States presidential election of 2000. Say we want to know whether there was a social preference for Gore above Bush. [IIA] requires that this question be answerable independently of whether the people preferred either of them to, say, Abraham Lincoln, or preferred George Washington to Lincoln. This seems right. Neither Lincoln nor Washington ran for President that year. They were, intuitively, irrelevant alternatives. But [IIA] also requires that the ranking of Gore with respect to Bush should be independent of voters’ preferences for Nader, and this does not seem right because he was on the ballot and, in the ordinary sense, he was a relevant alternative to them. Certainly Arrow’s observability criterion does not rule out using information on preferences for Nader. They were as observable as any in that election.

2. Maybe I'm the one misunderstanding, so I'll explain my reading of Wilson and we can discuss. I assume the theorem you are referencing is Theorem 3: Every social welfare function is either null or dictatorial. Take any set $S$ and call the set of preferences on $S$ (complete and transitive binary relations) $\Pi$. Then to Wilson, a "social welfare function" is any map $f:\Pi ^{n}\to \Pi$ satisfying two axioms:

I (IIA). If $R,R'\in \Pi ^{n}$ agree on a subset, then $f(R),f(R')$ also agree on that subset.

II. If $x,y\in S$, there exists $R\in \Pi ^{n}$ such that $xf(R)y$.

To summarize, my reading is this: the section Arrow's_impossibility_theorem#Formal statement says

IIA + non-dictatorship $\implies$ contradiction

but Wilson says

IIA + II + non-null + non-dictatorship $\implies$ contradiction.

Mathwriter2718 (talk) 18:56, 18 July 2024 (UTC)

To be clear, I'm not inherently against using the word "rational" or "coherent" to refer to principles such as von Neumann Morgenstern independence that it is widely accepted a rational agent must obey. Instead, I am questioning whether it is really widely-accepted that any social aggregation function violating IIA is incoherent or irrational. Mathwriter2718 (talk) 19:07, 18 July 2024 (UTC)

There's some ambiguity here in what we mean by "rejecting IIA". First, for every widely-accepted axiom there's some fringe philosopher willing to argue against it (same for VNM's IIA).

Second, if it's impossible to behave completely rationally (because you don't have cardinal information), violating IIA becomes second-best and therefore "rational" in a sense. (Assuming you care about >1 person's welfare). If you decide you want to reconstruct the utility function from the orderings, you have to give up IIA. e.g. if you have two ballots, with the first ranking A > 24 candidates > Z, and the second ranking A > Z > 24 candidates, we can't logically prove the 1st prefers A > Z more strongly than the 2nd, but we could reasonably infer it by looking at all of the "irrelevant" alternatives sandwiched between A and Z in the first one. But it would still be better to have the actual utilities for each candidate, so we don't have to use heuristics like that. David Pearce has a wonderful discussion here. Closed Limelike Curves (talk) 20:50, 18 July 2024 (UTC)

I found Pearce's discussion of Gorgias's "On Nonexistence" very amusing. Anyway, by this point maybe we could just find a reputable citation about whether or not IIA is viewed as a necessary condition for coherence/fairness, or just as a possible condition for coherence/fairness one might reject. (To be honest, I find the use of "rational" to refer to a social aggregation function and not an agent a bit strange.)

I am interested to know if you agree or disagree with my reading of this article and of Wilson. Mathwriter2718 (talk) 22:19, 18 July 2024 (UTC)

I believe Wilson says in Section (not theorem) 3 that he drops the assumption of citizen sovereignty (that the SCF is onto), but I'm actually a bit confused, because I'm not sure what he replaces it with. Closed Limelike Curves (talk) 22:51, 18 July 2024 (UTC)

I looked pretty hard at the article again today. I found new discrepancies. A) Wilson talks of complete and transitive binary relations (which he calls preferences and Wikipedia calls total preorders), but the article talks of total orders, which are antisymmetric total preorders. B) Wilson's requirement of non-dictatorship also requires that there is no "inverse dictator" whose preferences are always the exact opposite of those of the function. C) Wilson is extremely fussy about exactly what assumptions imply exactly what conclusions. The article theorem says that stuff implies IIA is violated, but neither Wilson nor the arguments on this page take that logical path. Wilson himself takes the path of IIA and surjectivity implies either null or dictator.

Wilson says in the abstract quite clearly that he drops surjectivity and still proves Arrow's theorem. However, his only relevant theorem (Theorem 5) is simply not the promised result. Perhaps if you do WP:OR, you can see how Theorem 5 gets you the desired result. But I think the prudent thing to do is to not say in this article that you can drop surjectivity. Mathwriter2718 (talk) 13:30, 19 July 2024 (UTC)

Speaking of WP:OR, the proofs of Arrow's result on this page are apparently "simplified versions" of proofs in the literature. I'm not sure if this "simplification" is OR or not. Mathwriter2718 (talk) 13:33, 19 July 2024 (UTC)

@Closed Limelike Curves I agree with many of the changes in your recent edit. But there are some I disagree with, including some reverts you made of my edits.

1. The Arrow quote in the lead: I removed the link to Condorcet paradox because there is already a link to it only a few sentences ago, and it's not clear to me that Arrow was even talking about the Condorcet paradox. Seems more likely he was talking about IIA violations.
2. Removing Voting paradox from See also: this is just a redirect to Condorcet paradox, which is already in the See also.
3. Neutrality does not imply Non-imposition: the null voting method that is indifferent between all alternatives is neutral but not surjective.
4. I am not so sure about calling neutrality a "free and fair election". To me, "free and fair election" means more about how the election is administered, whether or not some candidates are arrested, whether or not everyone in society is allowed to vote, etc. The lead for free and fair elections supports this view:

A free and fair election is defined by political scientist Robert Dahl as an election in which "coercion is comparatively uncommon". A free and fair election involves political freedoms and fair processes leading up to the vote, a fair count of eligible voters who cast a ballot, a lack of electoral fraud or voter suppression, and acceptance of election results by all parties. An election may partially meet international standards for free and fair elections, or may meet some standards but not others.

A social choice function on the other hand doesn't even need to be an election. I feel less sure about calling anonymity "one vote, one value". The slogan "one vote, one value" seems to me to imply that anonymity is somehow counting up votes, when it really just requires the function to treat each voter the same, and the function a priori might not have a natural interpretation in terms of voting. But the page for one man, one vote says it is about "equal representation", which feels right on point with what anonymity is.

It looks from that edit like you agreed with me that it is prudent to not drop surjectivity. In that case, I think I should add the surjectivity requirement to the formal statement (it is the only requirement in the Non-degenerate systems section that is not in the formal statement). Mathwriter2718 (talk) 13:32, 20 July 2024 (UTC)

On the name of the surjectivity requirement: is there a source in the literature that calls this "Non-imposition"? As I'm sure you already know, Wilson just calls this a "weaker version of Arrow's condition of Citizen's Sovereignty", which is not super helpful. I would really like to not come up with our own name for this, but it seems like we have to. I feel like "weak Citizen's Sovereignty" or "surjectivity" are both names that are minimally new, so those are the ones I support at this moment. Mathwriter2718 (talk) 13:39, 20 July 2024 (UTC)

One last concern similar to the one for "non-imposition": is there a source in the literature that defines the term "non-degenerate ranked choice voting systems" as ones satisfying every Arrow hypothesis except for IIA? I couldn't find this term in Wilson or Arrow. I worry it may be an invention of Wikipedia. Mathwriter2718 (talk) 14:02, 20 July 2024 (UTC)

Sorry for the large volume of posts, but one last thing I just spotted: non-imposition/weak Citizen's Sovereignty/surjectivity is currently defined as "it is possible for any candidate to win", but this is a weaker statement than surjectivity. Mathwriter2718 (talk) 14:07, 20 July 2024 (UTC)

I very much agree with these NPOV concerns. The suggested edit in #1, or something along those lines, seems good to me. Gumshoe2 (talk) 18:47, 18 July 2024 (UTC)