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![]() | Arrow's impossibility theorem is currently a Mathematics and mathematicians good article nominee. Nominated by –Sincerely, A Lime at 20:46, 27 May 2024 (UTC) Anyone who has not contributed significantly to (or nominated) this article may review it according to the good article criteria to decide whether or not to list it as a good article. To start the review process, click start review and save the page. (See here for the good article instructions.) Short description: Result that no ranked-choice system is spoilerproof |
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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)
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
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!
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
This article, and the spoiler effect article, should carefully distinguish between an election method as such not changing its output when a candidate who didn't win is removed from all input ballots; and the stronger effect that a candidate deciding to enter the race where a particular election method is being used, and then losing, does not change who wins. It's a subtle distinction, but normalization issues can cause cardinal voting methods to fail the latter even though they pass the former, even if the voters use von Neumann-Morgenstern utilities.
It's a bit off topic for this article, but the article's phrasing should still make clear that the former definition is what's being talked about, and that it doesn't imply the latter.
The rank-order voting section of this page might be relevant. Wotwotwoot (talk) 13:10, 29 March 2024 (UTC)Reply
But if we're using Arrow's axiomatic approach to back a statement that Condorcet only has minimal spoilers instead of no spoilers, we should also use Arrow's axiomatic approach to back a statement that elections with Range only has minimal spoilers instead of no spoilers. Cardinal elections shouldn't be able to get off on a technicality just because their method is asking for information that the voter can't supply with the absolute rigor that is necessary for the Arrow-style implications to hold.
Such a definition doesn't exist for rated ballots because even von Neumann-Morgenstern utilities fail to pin down what the one correct expression for the vote is.
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:
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)Reply
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.
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").
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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Nominator: Closed Limelike Curves (talk · contribs) 22:22, 3 May 2024 (UTC)Reply
Reviewer: Phlsph7 (talk · contribs) 08:32, 5 May 2024 (UTC)Reply
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
Arrow's requirement that the social preference only depend onreplace "depend" with "depends"
expressing social welfare, leading him focus his theorem on preference rankingsadd "to" before "focus"
Phlsph7 (talk) 08:32, 5 May 2024 (UTC)Reply
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.
@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)Reply
[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)Reply
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)Reply
… 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)Reply
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:
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)Reply
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:
—David Eppstein (talk) 19:44, 9 June 2024 (UTC)Reply
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)Reply