Talk:Arrow's impossibility theorem/Archive 1

Archive 1

Archive 2

Misc

Added reference to the original paper: "A Difficulty in the Concept of Social Welfare" (according to [[1]] Sdalva 08:26, 8 September 2005 (UTC)

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I removed the link

• A lecture on Arrow's impossibility theorem

because it is dead.

• Thank you.

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"their" vs. "his or her" has nothing to do with political correctness. It's the way the language is moving, at least in parts of the US. -- Zoe

That was a speculation as to why it hapened. Regardless, it can cause confusion on occasion - and here, where it's rather important to distinguish between individuals and collectivities, it makes a large difference. It would be all too easy to read "they" as referring to the collectivity. (I came across this important distinction in game theoretic stuff like the Tragedy of the Commons, where it actually drives what is going on.)) I think I missed some of the ambiguous usages, so I'll go back and tidy up further. PML.

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Where there are only two choices, Arrow's theorem doesn't appear to hold. You can simply apply first past the post, which meets all the criteria mentioned when there are only two choices (meet universality by letting the first voter decide where there is a tie). Do we need to change "among several different options" to "among at least three options"? Martin

Read the theorem again (it's below the list of desired properties), and you'll see that it doesn't claim anything if there are fewer than three options. --Zundark 18:46 2 Jul 2003 (UTC)

Ahh, missed that bit, thanks. But "and the society has at least 2 members" - huh? If the society has only one member, Arrow's theorem holds by non-dictatorship and citizen sovereignty alone, surely? Martin 19:42 2 Jul 2003 (UTC)

You're right that the theorem is true without the at-least-2-members restriction. (But you do need monotonicity to prove this.) The no-dictatorship property can hardly be considered "desirable" in the 1-member case, though, which is probably why the restriction to two or more members was included. You can remove it if you think it's confusing. --Zundark 21:18 2 Jul 2003 (UTC)

Oh yes - of course you need monotonicity - otherwise the method could simply give the reverse of the society member. Neat. :)

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Politicians-and-Polytopes The pseudo-theorem is not there if it can't be translated into statements of quantifier logic over inequalities. The weights of the votes are unconstrained. The voting power of an individual may be much greater than the voting power of the surrounding society. It seems almost that the IIA above 'deletes' candidates and/or papers since saying "subset of options", and the word "compatible" might mean that the same winners result. Arrow is completely in error to say that that is desirable. A fairer far superior axiom than Arrow's dumb and useless 1951 IIA check for no/little change, is a rule that prohibits all checking of changes of the win-lose state of all candidates not named on the ballot papers involved in the change. In the IIA above, if there are delicate back scratching coalitions, and a single individual holding over 95% of the voting power was removed, then that IIA is clearly a rule that no one actually wants. The rule named "non-dictatorship" looks like it is supposed to stop individuals from being too powerful and it is so lacking in a good definition that no one would believe that the non-dictatorship rule was actually able to keep the power of a single individual small enough. The "95%" could be replaced with a sequence of percentages that get smaller and smaller. We start with a certainty that IIA is undesirable since any removing of a super-powerful voter can change the victories of all the other winning candidates. At some point, while the percentage is dropping, the non-dictatorship rules changes from saying something to saying nothing. A conclusion is that the pseudo-theorem is very unimportant since it is finding incompatibilities between statements, some of which we definitely want to reject. The non-dictatorship rule is totally inconsistent with the aim of proportionality which says things like "1+1+1+1 > 1".

Cool, Craig Carey has been here! If you think that the above is completely unintelligible, you're right! If you're a fan of this type of writing, never fear. Craig has an email list dedicated to him pretty much talking to himself! Rejoice! -- RobLa 03:50 8 Jul 2003 (UTC)

Ahh, fond memories :) Martin

I was sure you were nearby Rob since one of those seemingly corrupt decisions to delete far superior definition of monotonicity had been made. I wrote to debian-vote in the 1st third of the year. You said I was at your mailing list. Like everyone else I was being lied at you never hawving a word privately except to defend unlimited lying in preferential voting to the extent possible of a man who never actually appears to ever read received e-mail.

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Should something be added about the theorem's applicability to elections?

Strictly, the theorem says you can't always find a "preference order among several different options" given certain criteria. But this isn't the problem that an election poses. In an election the aim is to select one candidate out of a set of candidates, rather than rank the candidates in a preference order.

How about a real answer? An election could instead of asking for only your top choice could instead ask for your whole preference. It could then use this to try and come up with a social preference, which it would take the most perfered (the first place candidate) and elect them. That's how it could apply.

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"This statement is stronger, because assuming both monotonicity and independence of irrelevant alternatives implies Pareto efficiency."

I found this confusing. Does this refer to the theory as a whole with that replacement criterion or ...? Where does "assuming" come into play?

I agree. It sounds to me that this adjustment is implied by the other criteria already. --Starlord 4:37, 29 April 2005 (UTC)

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After seeing people misuse Arrow's theorem, would anyone have a good way of wording the limitations of Arrow's theorem? Some misuses use arrow to say all voting schemes are equal or there is always a dictator. These are simply false, the first is obvious because there are voting schemes that use more of the information in a preference than others. The second is because of the way the proofs are carried out. All the proofs construct a set of individual votes that have a dictator. In quantifiers: for all social preference functions there exists a set of individual votes where there exists a dictator. A simpler way of getting at this is that in Condorcet voting the Smith set is the criteria for a dictator to exist: only when the Smith set contains more than one candidate must a dictator exist (which is where the different versions of Condorcet diverge).

No randomness allowed

I clarified the condition of "universal domain" to say that no randomness is allowed in the result. You could say, I suppose, that the wording of all the conditions assumes that the result is deterministic, but I think it needs to be specified. Otherwise, there is a method that meets all of the conditions. Call it "Randomly-completed Condorcet" or "Smith//Random":

Choose a voter at random, before the vote, and don't disclose who this voter is. Then hold the vote. Elect the Condorcet winner if there is one; otherwise, find the Smith Set, and ask the random voter to choose one member of the Smith Set as the winner.

It's not dictatorial, even when it's up to this random voter, because everyone's votes are taken into consideration to determine who is in the Smith Set.

RSpeer 23:09, Oct 14, 2004 (UTC)

Monotonic is not the same as strategy-free

I removed the text

Equivalently: an individual should never be able to get a preferred result by misrepresenting his or her true preferences.

from the monotonicity criterion, because this is not equivalent. That text describes a method being strategy-free, which basically no method accomplishes unless you relax the criteria in some way. Lots of methods, however, are monotonic.

RSpeer 19:30, Jan 19, 2005 (UTC)

Condorcet?

I really don't understand the phrase:

With a narrower definition of “irrelevant alternatives” which excludes those candidates in the Smith set, some Condorcet methods meet all the criteria.

First this seems to say that if there is a set of preference oderings which conflict with the irrelevant alternatives requirement then they should not be treated as irrelevant - that seems like cheating. Second, Condorcet methods are designed to choose a single winner, not a societal preference order; generalising the method to the whole order means changing the definition of a Smith set and other elements. So I think it doesn't belongs in the article. --195.92.40.49 16:27, 8 Feb 2005 (UTC)

Range voting and universality

It was probably justified to remove the remark about range voting. Indeed, range voting does not satisfy the criteria; the one it misses is universality, which is not one of the "big three" conditions people usually quote. But the reason it fails universality is pretty well hidden, based on the definitions given. Range voting is a cardinal ranking system, so that the voter specifies more than the preference order, and this fails universality because the same preference order does not always give the same result.

I'm not entirely sure how to clarify this in the article. A sentence or two could be added saying that Arrow's theorem only applies to voting methods based on preference order. But then a plausible response by the reader would be "oh, okay, let's just stick with plurality then". Arrow's theorem hits plurality as well, as long as you assume that each voter has a preference order that plurality is not letting them express. Which, I suppose, has some philosophical implications to it. So this is difficult to clarify. RSpeer 22:50, Feb 15, 2005 (UTC)

Range voting doesn't violate universality, because universality doesn't apply. Universality (as it is defined here) only applies to ordinal rankings. In fact, it's precisely because universality is discarding the cardinal rankings information that range voting appears to violate universality in the first place, so the restriction to ordinal rankings is clearly relevant. --Wclark 20:33, 3 September 2006 (UTC)

Math, not politics

An anon added the opinion that Arrow's theorem is "flawed" to the page, couched in weasel words ("criticized by many voting theorists" to mean "I don't like it").

This is an opinion that I have encountered before. It does not belong in Wikipedia, because not only is it expressing a point of view that favors certain voting methods over others, the statement is not even about Arrow's theorem.

Arrow's theorem is a theorem, a mathematical statement that has been proven. Since the proof has undergone peer review, it is almost certainly not a flawed proof. So the theorem, which states that no voting method satisfies a certain five or so criteria, is not flawed.

There is a certain popular, non-mathematical rephrasing of Arrow's theorem, which you will not find on Wikipedia: "Every voting method is flawed." This is what the anon was attacking, apparently, but it is not the subject of this article.

RSpeer 00:38, May 13, 2005 (UTC)

I'm the anon, sorry for the lack of account. My objection is that Arrow's theorem is routinely used to justify the thought that it is "impossible to design a set of rules for social decision making that would obey every ‘reasonable’ criterion required by society." (which is in the article). Elsewhere, you said that this could be minorly edited to "seemingly reasonable", but this isn't sufficient. All Arrow's theorem does is arbitrarily take a set of criteria and demonstrate it is impossible to meet all those criteria at once. It does not prove that each of those criteria are required for a reasonable voting system, however. Isn't a discussion of the flawed usage of a theorem fair game for a page on the theorem? (It's true that I personally favor Condorcet methods but I'm not being a zealot about it or anything.) I don't think anyone contests that Arrow proves the set of criteria are contradictory, but contesting its "conclusion" that it is impossible to design a voting system that obeys societal requirements is fair game for this page. Perhaps it could be added instead to a section that discusses how the theorem is used?

That is reasonable. There could be a section for "Interpretations of Arrow's theorem" pointing out that these interpretations are not mathematically rigorous, with criticisms of them. RSpeer 04:03, May 13, 2005 (UTC)

ok. Please note, I am not "a poor man blames" - I didn't do the revert. But I (or someone else) will refashion the reversion into a discussion section soon. In addition, the opening summary should be softened. (later) all right, hopefully that is an improvement.

Request for clarification

Can an explanation be given for those of us a non-mathematical bent? I am no wiser as to what is involved, apart from it being "something to do with voting".

This weekend I have some time and will try to add a better introductory section to explain the concept with less formality. I may also add a proof, which of course would not help with this particular problem, but may be nice anyway. --Zarvok | Talk 18:26, 5 August 2005 (UTC)

The structure of the article should be changed so that it is more accessible to laypersons. For example, one could : (1) add one or two sentences from section "Interpretations ..." in the opening paragraph, (2) make the section "statement of the theorem" shorter and less formal (there is a formal section later anyway), illustrating the requirements by real-life example like "favorite of most people does not satisfy monotonicity". In general, an article for the general educated public does not need that much mathematical language. PhS 10:48, 12 March 2006 (UTC)

Does not "favorite of most people" satisify monotonicity? How it can hurt a candidate if you rank her higher? I would think that it is IIA that is not satisfied. If a new candidate conmes along that I rank first and everybody else last this may cause old winner to lose since he might have needed my vote. —The preceding unsigned comment was added by 130.235.35.193 (talk • contribs) .

The Theorem's skewed language and point of view

It might help to better clarify the situation by changing the language behind the terms used. In place of “Dictator”, the correct term should be “Belwether”. Then what the theorem states is not the impossibility of a certain kind of social consensus system, but the inevitability of visionaries, seers, wise men (and wise women) who seem to be at the center of all consensus — the Belwether.

The US Supreme Court had a near-realisation of Arrow's Theorem not too long ago, by the way. Nearly every 5-4 decision over the earlier part of this decade was determined by a single justice. One, therefore, gets an image of a giant statue of a woman (the one who was de facto in charge of the Supreme Court), blindfolded, holding in each hand a scale, with 4 tiny justices standing on the left scale and 4 tiny justices on the right. —Preceding unsigned comment added by 205.232.226.178 (talk) 21:49, 4 December 2008 (UTC)

A deciding vote is not the same as the "dictator" in Arrow's theorem. It's also not the same thing as "Belwether" suggests, since there is no well-defined "consensus" -- that's what voting systems are for measuring in the first place! Scott Ritchie (talk) 22:13, 5 December 2008 (UTC)

Dead link, anyone have a new one?

I removed the link to "A Pedagogical Proof of Arrow’s Impossibility Theorem" because it was dead. I would, however, be interested in seeing this proof - does anyone have a new link? I wasn't able to immediately find anything by google, but didn't have time for an in-depth search. --Zarvok | Talk 18:22, 5 August 2005 (UTC)

I've found in http://ideas.repec.org a link to this article (ftp://weber.ucsd.edu/pub/econlib/dpapers/ucsd9925.pdf) Sdalva 10:03, 30 August 2005 (UTC)

Non-dictatorship

Could someone clarify what non-dictatorship is?

If it just means that no voter's preference order cannot equal the social choice function's order, then I don't see why this would make things "unfair." -Grick(^{talk to me!}) 02:09, September 1, 2005 (UTC)

Indeed, that would be fairly silly. Non-dictatorship can be expressed like this, I believe: a group of all but one voter should always have a way to change the result away from that voter's preference. RSpeer 04:38, September 1, 2005 (UTC)

Yes. That's right. But just to be formal: the voting system is a dictatorship by individual n if for every pair a and b, society strictly prefers a to b whenever n strictly prefers a to b. Perhaps the article could be improved if the Theorem is rewritten in mathematical form (along with an intuitive explanation about what every criteria means, of course). Or better(?): two sections, one is math, the other is natural language. Sdalva 20:42, 1 September 2005 (UTC)

I'd put it like this: there is no individual whose choice will always (we might almost say, by definition) be the same as the group's choice. This does not mean that the group won't occasionally produce a result which one or more individuals also chose, but simply that the group's "lining-up" with those individuals is an accident of everyone's choices rather than a decision that the system will follow, say, Joe Bloggs.

Does this mean that the mathematical definition of non-dictatorship is missing something? When I read it first time, I thought it might be saying what Grick evidently thought it was, but on reflection, I concluded it was missing a clause to the effect that there is no i etc. *for all possible $R_{i}$*. I'm not quite sure that that's how to put it, though. (Wooster, 14-09-2005, not signed-in)

But how do I know it is an accident or a real dictatorship? Can anybody provide an example that satisfies all of Arrow's conditions, but that is a dictatorship?--Ezadarque 22:06, 20 September 2006 (UTC)

That's a strange question. Dictatorships are always Pareto efficient (if everyone agrees, then so does the dictator), independant of irrelevant alternatives (because only the top of the dictator's list matters), "citizen sovereign" (everyone can win, all it takes is for the dictator to top-rank them). Dictatrorships are also monotonic, if your version requires that. CRGreathouse (t | c) 22:24, 20 September 2006 (UTC)

Perhaps if I give an example I will make it clearer. Suppose that we have three individuals, 1, 2 and 3, which have to rank A,B and C. Suppose that 1 and 2 rank A>B>C, while 3 ranks B>A>C. Then the social function would be A>B>C. Are 1 and 2 dictators in this case?--Ezadarque 02:31, 21 September 2006 (UTC)

You misunderstand. A dictatorship is a voting system. Given your three individuals and your three choices, there are 3!³ (216) possible ways for the individuals to rank the choices. A voting system, then, is a function that maps from each of these possibilities to a unique social preference order; there are 6²¹⁶ (2.8 trillion) possible voting systems. A dictatorship is a voting system wherein there exists some individual (1, 2, or 3) such that the social preference order is always equal to that individual's ranking of the choices. (There are three possible dictatorships.) So in your example, you've ruled out all but 6²¹⁵ (470 billion) of the possible voting systems, but you've still not given us nearly enough information to know whether the voting system is a dictatorship. Does that make sense? Ruakh 02:51, 21 September 2006 (UTC)

P.S. The answer is no; it's possible that 1 or 2 is a dictator, but it can't be that both are dictators, as there can only be one dictator. Ruakh 02:51, 21 September 2006 (UTC)

I think I got it. Thanks.--201.51.241.12 03:15, 21 September 2006 (UTC)

Proved?

I'm questioning the verbage on this line:

The theorem is named after economist Kenneth Arrow, who proved the theorem in his Ph.D. thesis and popularized it in his 1951 book Social Choice and Individual Values.

Is there no debate over if it is proven or not? There has got to be some amount of debate as demonstrated here.

That article indicates only that there is debate about whether independence of irrelevant alternatives should be considered a criterion for fairness. This has no bearing on whether it can be satisfied at the same time as other criteria. People who believe that independence of irrelevant alternatives is not necessar for fairness do not believe that Arrow's theorem is false, just that it is not fatal to fairness. Josh Cherry 12:02, 1 October 2005 (UTC)

Not proved. I have a contradiction in hand if the special constraint is used that each voter has a complete, ordered preference for each possible choice (no ties allowed!). If any choice is irrelevant, it can be dropped from the list. In many of the nasty cases there is no obvious winner, but there is always an obvoius looser. That looser can be found by SUM(position on list for each individual, where 1 is best). The loser is irrelevant to the eventual winner. Recurse until one choice is left. -- (unsigned)

Congratulations...just write that up in a paper, get it published in a journal, and the Nobel prize is practically yours. I'll hold my breath while I wait for it! I'm so excited! -- RobLa 07:33, 19 May 2006 (UTC)

Sorry. No Nobel prize in mathematics. This mechanism is to computationally expensive to use, so it won't solve any problems elsewhere. -- (unsigned (same user))

Arrow's Nobel prize was in Economics, not math. You'd be up for a Nobel prize and possibly eventually an Abel prize if you dispoved Arrow's Theorem. CRGreathouse 06:26, 12 July 2006 (UTC)

Your system is just Baldwin's method, which doesn't meet the IIAC. Consider 5 votes ABC, 4 votes BCA, and 3 votes CAB. If only A and C run then C wins, but if B runs the winner changes to A. (C has 26 'badness points' under your method vs. 23 and 21 for A and B, so is 'irrelevant' for the purpose of IIAC.) CRGreathouse (talk • contribs) 04:37, 5 August 2006 (UTC)

Reference for the Sen/Pareto remark?

Does anyone have a reference for the assertion in the very last paragraph that Sen showed that the Pareto principle is incompatible with private-domain liberty? There probably ought to be one, plus I'm curious :-) --Paultopia 15:42, 2 October 2005 (UTC)

Check http://ideas.repec.org/a/ucp/jpolec/v78y1970i1p152-57.html It's in the Journal of Political Economy 78 (1970), vol 1, pp. 152-157. I thought of adding an entry about it myself, soon. Sen proved that Pareto is incompatible with any ideological restriction of the range of the social choice rule. You can't just impose some rule on the way preferences are aggregated, and then expect people to be happy with it whatever their preferences are. He gave a nice example: Suppose individual liberty, which means that at least two people are allowed to veto some results. You want to read a bad book. I don't want you to read it, I would rather read it myself (and suffer). You would really enjoy seeing me suffer from reading it. If we're both given the right to prevent interference with our private affairs, then it will be impossible to force that I read the book and suffer. We would get the "rational" result that you read the book, though both of us would rather I suffer from reading it. The problem isn't with liberalism. Liberalism is exactly intended on preventing things like that from happening: a liberal would be against killing a person even if that person is suicidal and everybody else hates him. It's an incompatibility that you have to deal with, like with Arrow's theorem. You can't expect Pareto efficiency (the way it is defined above) to yield "reasonable" results without limiting the domain of your function to "reasonable" preference profiles. mousomer 19:16, 2 October 2005 (UTC)

I read "Rationality and Freedom" by Amartya Sen and he actually says that reasonableness or rationality limit the choices. Pareto efficiency together with other reasonableness criteria limit the choices to so little that none exist. But if you start relaxing these constraints then "Social Choice" is actually possible. I think this should be included in the main post. 141.212.110.114 (talk) 17:29, 6 January 2008 (UTC)Sushant.

non-imposition was missing

Starlord already remarked above that the sentence "This statement is stronger, because assuming both monotonicity and independence of irrelevant alternatives implies Pareto efficiency" was confusing. It was also wrong, since this is true only assuming non-imposition. Any constant social welfare function (i.e. one that is independent of the individual preferences) is monotonic and independent of irrelevant alternatives, but neither non-imposing nor Pareto-efficient.

I find the statement confusing. It first speaks of replacing non-imposition and monotonicity with Pareto-effeciency. Then it claims that the second version of the theorem is weaker since Pareto and non-imposition togther does not imply monotonicity. This is confusing since according to what came before we have dropped the assumption of non-impostition and it does is irrelevent what it implies. —The preceding unsigned comment was added by 130.235.35.193 (talk • contribs) .

That monotonicity, independence of irrelevant alternatives and non-imposition together imply Pareto efficiency can be seen as follows: Assume a social welfare function is monotonic, independent of irrelevant alternatives and non-imposing, but not Pareto-efficient. Then there is a preference profile in which alternative a is preferred to alternative b by all individuals but not socially. Due to monotonicity, swapping a and b in any subset of the individual preferences cannot cause a to be socially preferred to b, and due to independence of alternatives, moving a and b without swapping them cannot do this, either. Thus, no matter how we move a and b around, a will never be socially preferred to b, contradicting non-imposition. Joriki 14:43, 5 October 2005 (UTC)

Excelent, but you don't need the IIA. Non-imposition+monotonicity --> Pareto.

I have a reference to Malawski and Zhou (1994) that says IIA + non-imposition --> weak Pareto or inverse weak Pareto. CRGreathouse 17:09, 20 July 2006 (UTC)

social choice function/social welfare function

I replaced "social choice function" by "social welfare function" because:

• The "Formal statement of the theorem" section used that, so one or the other had to be changed.
• Texts that use both expressions (e.g. [2]) use "social welfare function" to mean what we mean and "social choice function" for a function that assigns a single chosen alternative, not a preference order.

The article on social welfare functions treats them as providing not just an ordinal structure but a numerical measure of welfare. The Stanford Encyclopedia of Philosophy says that this is what philosophers tend to do, whereas economists tend to define it as a preference ranking. Since this article is categorized under economics theorems, that would seem to fit. I added a paragraph to the article on social welfare functions to explain this difference. Joriki 15:38, 5 October 2005 (UTC)

Restricted domain (request for references)

Does anyone have a reference for the following assertion in the section "Some possibilities"? IMHO such refernce should be presented in the article.

Indeed, many different social choice functions can meet Arrow's conditions under such restricting of the domain. It has been proved, however, that any such restriction that makes any social choice function adhere with Arrow's criteria, will make the majority rule adhere with these criteria

--Y2y 08:38, 21 February 2006 (UTC)

I'll second the request for a reference there. Especially since it's the lead-in to the sentence, "So the majority rule is in some respects the fairest and most natural of all voting mechanisms.", which startled me with its sweeping judgement. (At the very least, I'd suggest changing to "Under these conditions, the majority rule is a fair voting mechanism", if not excising entirely - especially if no reference can be found for the lead-in.) --anon reader 17:33, 11 August 2006 (UTC)

I've changed the wording (no "fairest", {{fact}}, added qualifier, grammar):

Indeed, many different social welfare functions can meet Arrow's conditions under such restrictions of the domain. It has been proved, however, that any such restriction that makes any social welfare function adhere with Arrow's criteria will make the majority rule adhere with these criteria. Under peaked preferences, then, the majority rule is in some respects the most natural voting mechanism.

This should make the paragraph better for now. This isn't meant to be the final fix -- we really needs a reference or it should go -- but this takes the 'edge' off the paragraph. CRGreathouse (talk • contribs) 19:39, 11 August 2006 (UTC)

I may have found the reference:

"May (1952) provided the first axiomatic characterization of majority rule as a social welfare function. May’s characterization is based on Independence of Irrelevant Alternatives, Neutrality, Anonymity, and a strong positive responsiveness axiom. Maskin (1995) substituted the Pareto criterion for the last of these. He proved that any social welfare function satisfying the four axioms will fail to be transitive-valued at any individual preference profile at which majority rule violates transitivity, and, unless it is majority rule itself, will fail to be transitive-valued at some individual preference profile at which majority rule is transitive-valued. This was also established by Campbell and Kelly (2000) with a less demanding set of axioms."

• Donald E. Campbell and Jerry S. Kelly, "A strategy-proofness characterization of majority rule", Economic Theory, vol. 22, No. 3 (2003), pp. 557–568.

The papers referenced are "Maskin, E.S.: Majority rule, social welfare functions, and game forms. In: Basu, K., Pattanaik, P.K., Suzumura, K. (eds.) Choice, welfare, and development. Oxford: The Clarendon Press 1995" and "Campbell, D.E., Kelly, J.S.: A simple characterization of majority rule. Economic Theory 15, 689–700 (2000)". CRGreathouse (t | c) 01:36, 16 September 2006 (UTC)

Excellent! Now, can someone translate that to English?  :-) Mdotley 19:08, 18 September 2006 (UTC)

Added criticism

Note that the page still states "Arrow's theorem states that there is no general way to aggregate preferences without running into some kind of irrationality or unfairness." This is nonsense, but I left it for the time being, to allow others to digest the criticism. Colignatus 01:58, 3 March 2006 (UTC)

• I agree that that statement is nonsense, so I removed it. Your additional section was written as an opinion, however, and I reverted it. The widely-held criticisms of Arrow's theorem already appear under "Interpretations of the theorem". It's better Wikipedia style to ensure that the whole article is neutral - separate "advocacy" and "criticisms" sections are a last resort for particularly contentious topics. rspeer / ɹəədsɹ 03:38, 8 March 2006 (UTC)

• No, the text that I entered is not an opinion but a reasoned statement. It is not an interpretation but a reasoned statement. It may be that you are right that there should not be a separate criticism section, but, given the first part of the article, this reasoned statement can be put here. My suggestion is that more people think through what the article would look like but including the reasoned statement in the main body of the article. It is useful to have the text available in the article before the whole is re-editted, to prevent innocent readers to get a wrong impression. Colignatus 22:32, 10 March 2006 (UTC)

• I noticed that Rob now added a notification on original research or verification. This is better than simply removing the text, but there remains a confusion. You all must distinguish me (1) as a scientist who for you is a third party and who provides sources, e.g. see me as the writer of VTFD in 2001, (b) as a scientist who helps you, just now, to get this article straight. You should note that the verifiability condition concerns facts that need be checked by various sources. But the text that I provided (in the disputed criticism section) contains a reasoned argument that you can check by the logical faculties of your minds. Thus the condition does not apply in that sense. Also the reference to original research doesn't apply, since I didn't enter original research. The research originated in 1990 and was published in 2001. The only thing I do now is correct the misleading element in the article, and provide the reader (and you as editors) with a condensed reasoned statement, that you again can check by using the logical faculties of your minds. Also, the verifiability page mentions that the sources should be reliable. Well, again this is a non-issue for this paragraph. For, the point is that a reasoned argument is provided, so that it doesn't matter where it is from, and you have to use the faculties of your minds. If the washing lady says 1+1=2 then this has the same value as when the bishop says so, even when the bishop doesn't know what he talks about. Which is exactly why I added the suggestion, above, that when the reasoned argument is understood, the criticism might perhaps best be included in the main text, revising the main article. I didn't do that myself, since all of you would be completely shocked, not having digested that reasoned statement. Some other comments: PM 1: I doubt whether you will find many other sources on criticism other then I already included in the list of literature in the links that I provided. If someone feels like unpacking those links and transferring those references into the main body, feel free to do so. PM 2: Now that Rob has eliminated that one sentence, that he agreed was nonsense, perhaps the header Criticism is too strong, and it may be perhaps Evaluation or something like that. PM 3: If the consensus is that this group of editors cannot take responsibility for using their logical faculties of minds on the reasoned argument, then you may also attribute it to me, see Colignatus at wikinfo, and perhaps research a bit how reliable I am. Simply a Google scolar is not enough, though. PM 4: If all this fails, I would like a comment from someone who follows the logic of the reasoned argument and have suggestions how it can remain available for readers, then in another format or place. Colignatus 20:22, 11 March 2006 (UTC)
• Colignatus, did you bother to read the rest of the article? If I understand you correctly, you claim that the "paradox" is just demonstrating the incompatibility of Arrow's criteria. This claim is not new, and is explicit in the text we wrote. Is it not explicit enough? Look at:
  • "So, what Arrow's theorem really shows is that voting is a non-trivial game, and that game theory should be used to predict the outcome of most voting mechanisms. This could be seen as a discouraging result, because a game need not have efficient equilibria, e.g., a ballot could result in an alternative nobody really wanted in the first place, yet everybody voted for." mousomer 08:38, 12 March 2006 (UTC)

Slight error, I think

Am I right in that there is a slight error in one of the formulae? The one that says that no individual voter's preference should always prevail. The formula begins: (I don't write latex) there exists no i belonging to N such that...

Should it not rather be: there exists no i belonging to [1,N] such that... since N is a number?

Is this just informal notation or have I misunderstood? But it seems like an error to me.

• • The theorem works for any set of voters - be it infinite or finite (as long as there are at least 2 of them). So, when N is a 'set of voters' - rather than a number, we need that no voter 'in' N will be a dictator. mousomer 08:02, 9 April 2006 (UTC)
    • I think there's actually an inconsistency in the article with respect to this. Some parts work with N a natural number or a set, e.g. $\mathrm {L(A)} ^{N}$. But some work only with N a natural number, e.g. "The n-tuple $(R_{1},...R_{N})$" (which should actually say "N-tuple"). That's why at some point I changed "N a set of voters" into "N a number of voters". But the part that 65.94.43.126 objected to only works for N a set. Even if we consider a natural number to be equivalent to a set, this abstraction would be unnecessary unless we also wanted N to stand for other, possibly infinite sets, and then the tuple notation wouldn't make any sense. So I agree that something needs to be changed. Joriki 21:00, 30 April 2006 (UTC)

Did Arrow actually prove the infinite case? If he didn't, it would probably be best (for simplicity as well as accuracy) to consider only that case. The infinite case can be mentioned with reference in another section, or it can go in another page. CRGreathouse (talk • contribs) 04:56, 15 August 2006 (UTC)

Talk:Social Choice and Individual Values

Announcement: The above is the discussion tab for a new article Social Choice and Individual Values. Input is welcome through the article, the Talk page, or to me. The plan is to gather comment, corrections, or suggestions for probably at least a couple of weeks, make final changes, then go from there. Links to related articles (indluding the present one) would come after revision. Thanks for your help.

Thomasmeeks 22:54, 27 May 2006 (UTC)

Extending Arrow's Theorem

I have made a mockup of a section on extentions to Arrow's theorem. I would like feedback on this:

• Is the section usable as written?
• Should it be more or less technical?
• What approach is best for putting this content on Wikipedia: as a section in Arrow's impossibility theorem, as its own article, or as a series of articles?
• Are there any major results I'm missing? In particular, is there anything on social choice functions not generated by SWFs?
• For the last result (on voting rules and social choice functions), do you know of an earlier reference?

Also, I'd like thoughts on what else I should do before putting this into the main namespace. Thanks! CRGreathouse (talk | contribs) 06:19, 24 August 2006 (UTC)

I would love more detail but I understand that it is hard work and others might think it would become to long. It think the most important result you mention is the one about single-winner function (social-choice functions). This is the most commonly encounterd situation in real life and it is good to point out that there is a version of Arrows theorem for that situation as well. I find the section readable. —The preceding unsigned comment was added by 130.235.35.193 (talk • contribs) .

Non-manipulatabiliy and IIA

This paragraph doesn't seem sensible. Gibbard-Satterthwaite + Duggan-Schwartz shows that of universal ordinal systems, only dictatorships are non-manipulatable. Universal ordinal systems that are non-manipulatable thus all have IIA, but only trivially (because they're all the same system, a dictatorship).

Relaxing the IIA criterion, though popular, has a distinct disadvantage: it can result in strategic voting, making the voting mechanism 'manipulable'. That is, any voting mechanism which is not IIA can yield a setup where some of the voters get a better result by mis-reporting their preferences (e.g. I prefer a to b to c, but I claim I prefer b to c to a). Clearly, any non-monotonic social welfare function is manipulable as well. If one uses a manipulable voting scheme in real life, one should expect some "dishonest" voting. What this means is that the real-life implementation of most voting mechanisms results in a complicated game of skill. The Gibbard-Satterthwaite theorem, an attempt at weakening the conditions of Arrow's paradox, replaces the IIA criterion with a criterion of non-manipulability, only to reveal the same impossibility.

I've removed this for two reasons. First, it seems clearly untrue or trivial as discussed above. Second, were it nontrivial and true (or at least widely believed) it would belong in the IIA article (or perhaps strategic nomination etc.). CRGreathouse (t | c) 00:04, 4 September 2006 (UTC)

Pareto ambiguity

The Pareto principle is not the same as Pareto efficiency. Mdotley 22:11, 14 September 2006 (UTC)

(Simple) English please

Hi. I consider myself a pretty bright guy and I am sure that I could understand this if I took the time to study it but it would be nice to have a simplified abstract of the problem in the intro. And if you think that is what we already have, well all due respect but I beg to differ. I came here from paradox and just wanted to understand the nature of this and perhaps why it might be counter-intuitive (which would probably be obvious if I had any idea what it was about). Thanks --Justanother 14:39, 27 October 2006 (UTC)

According to the first sentence of the article, the theorem "demonstrates that no voting system based on ranked preferences can possibly meet a certain set of reasonable criteria when there are three or more options to choose from." Which part of that do you not understand? (It's hard to simplify when we don't know what needs to be simplified.) Ruakh 16:23, 27 October 2006 (UTC)

Yeah, I was afraid of that (laff). I read most of it and have a better understanding. I will propose some language later unless someone else does it first. --Justanother 16:46, 27 October 2006 (UTC)

Which statement is IIA

The article references statement IIA several times, but no such statement is defined in the article. In fact the statements aren't numbered. —The preceding unsigned comment was added by Arnob1 (talk • contribs) .

That stands for independence of irrelevant alternatives:

independence of irrelevant alternatives: if we restrict attention to a subset of options and apply the social welfare function only to those, then the result should be compatible with the outcome for the whole set of options. Changes in individuals' rankings of irrelevant alternatives (ones outside the subset) should have no impact on the societal ranking of the relevant subset. This is a restriction on the sensitivity of the social welfare function.

Does that help? CRGreathouse (t | c) 03:05, 21 November 2006 (UTC)

Thanks, I will put (IIA) in parenthesis next to that statement to make this clearer. Arnob 04:47, 21 November 2006 (UTC)

Relaxing the IIA criterion

I removed this addition to the article:

Relaxing the IIA criterion, though popular, has a distinct disadvantage: it can result in allowing strategic voting, making the voting mechanism 'manipulable'. (See also: Gibbard-Satterthwaite theorem).

Since the G-S theorem shows that non-dictatorial, non-imposed voting systems are all manipulatable, I don't see the connection to IIA.

CRGreathouse (t | c) 22:09, 4 March 2007 (UTC)

1. I didn't mention direct connection between G-S theorem and IIA, but only via manipulability. But you are right, my text can be misunderstood.

But what about the same text whithout link to G-S theorem?

2. I think G-S theorem (as closely related to the Arrow theorem) should be mentioned in this artcile not only in "See also" section. Possibly you can find the right way to mention it?

--Y2y 22:58, 4 March 2007 (UTC)

My point is that you can't get non-manipulatable results even by keeping IIA, unless you're imposed or dictatorial.

As for G-S, it is very closely related to Arrow's theorem. I would fully support content along these lines; perhaps starting with a simplified combined proof would be in order, or at least a description of how to modify the proof of Arrow's theorem to get G-S. I can provide a reference if we don't have one already (and you don't have one).

CRGreathouse (t | c) 02:09, 5 March 2007 (UTC)

> My point is that you can't get non-manipulatable results even by keeping IIA, unless you're imposed or dictatorial.

Agree, if you add "or non-deterministic or with restricted domain" (Explanation: 1) these conditions implies non-pareto-efficiency; 2) non-pareto-efficiency and non-imposition implies non-strategy-proofness).

And the first fact (IIA and deterministic and unrestricted domain and non-dictatorial => non-pareto) is evidently formally equivalent to the statement of Arrow's theorem itself (it's "second version" in the article).

But the proposed text is about other (and informal) question: why IIA is desirable? And I think it is very important to emphasize the connection between IIA and strategy-proofness which may be (and logically should be) informally clear before Arrow's theorem. (Without this it's quite difficult to understand WHY Arrow's theorem has these "strange" condition).

BTW a similar text was in the article before your deletion on 4 Sep 2006. (I have restored it only partially, because, agree, in the previous form "it would belong in the IIA article...").

Well, four tildas to you. (Or squiggles) My keyboard boasts neither.

Dr. MacIntyre again. (Throughout in xPy P means 'is preferred to', whilst xIy means 'the voter is indifferent between x and y'

Lets start with Gibbard's theorem which is for single valued outcomes which makes it interesting only from the point of view of contention rather than as a description of voting systems which allow ties. Further, voters can only express strict preference orderings. This may be standard but Gibbard assumes that they represent adequately voter preferences. (If you find two candidates equally satisfactory in Gibbards REPRESENTATION you have to say xPy or yPx).

This is vital for Gibbard's result. Now if IIA is violated the outcome over some pair must change from some x to y whilst preferences over x and y remain unchanged. We consider the agenda just of x and y. Now change the voters profiles in that overall change one at a time. One voter must change the outcome from x to y. But the voter was, during the change in preference, still of the opinion that xPy or that yPx. If xPy throughout changing back from the step bringing about the change x to y improves the outcome for him/her. If yPx, the reverse is true. Thus the procedure is manipulable here (the word used to describe strategic voting by (a coalition of) one voter. This way of proving this sort of result was developed to my best knowledge by Professor Pattanik. The reult is proved for whne xIy is not the voter's true opinion.

Now even if Gibbard were to allow ties so that the outcome on x and y could be a tie, his result still holds. For now things are as above here or one of the outcomes is xIy. It is worth checking (and quite fun so to do) that whether the voter finds throughout that xPy or yPx that one direction of the change constitues one person strategic voting.

But when indifference by voters is allowed the violation of IIA can be taken to be solely due to a voter for whom xIy so that any change of outcome no matter what (the three possible outcomes are xPy, yPx and xIy on the pair x and y now of course) leaves the voter for whom xIy indifferent (!) in the sense that the results x and y are equivalent for the voter and a fair (in fact every) lottery on x and y has the value of x (and of course of y).

Lastly three observations. Here at least I think it worthwhile pointing something out which I know is not obvious to everybody but may be to many!. When we are talking about outcomes say on three alternatives we sometines talk of an order on the {x,y,z} set as the outcome. Sometimes we talk of a subset of {x,y,z} being hte outcome. (Say, {x} or a tie like {x,y}. There is an obvious set of correspondneces between these two formulations. Intuitively singleton outcomes from the set of THREE alternatives like {x} or {z} correspond to the orderings xPzIy, xPzPy, xPyPz ofr {x}, zPxIy, zPxPy, zPyPx for {z} etc. Meanwhile doubleton outcomes like {x,y} correspond uniquely with xIyPz, etc. and the unique triple {x,y,z} as an outcome is (uniquely ) equivalent to xIyIz.This means among other things that one (readers and reseachers) can expect that results in the one frame will turn out to be true in the other.

Secondly I would like to stress to you the importance of consequentialism in talk about strategic voting. If IIA is violated in a way which results in 1 person strategic voting as described here the system cannot be majority voting on x and y because the vote hasn't changed but the outcome has. As I say in my discussion on the material presented here on the Theorem itself, for the same underlying preferences on alternatives under majority voting one may want (regard as desirable) different outcomes in theory. However they are achieved by presenting to the voting system different EXPRESSED preferences on the alternatives to be decided upon. The changes here come about with the same preferences on the alternatives, just x and y, to be voted upon. Thus a majority may be disadvantaged, restitution impossible. This is singularly not the case with strategic voting under majority voting.

From a consequentialist point of view then strategic voting under majority decison making is thoroughly desirable. Indeed it makes democracy work. Majority decison making respects IIA. The cases of IIA being violated here if ever advantage are so for a system unnecessarily disadvantageous in the first place or are downright disadvantageous. IIA does good and its violation can be harmful to democracy.

Given some of the poor quality of discussion not least in prestigious journals of economics which should know better (JET comes to mind) and seeing the Wikipedia formal account of the Arrow Theorem it is easy to see that US - UK advice on democracy to the rest of the world should stop. Their best academics seem confused by the subject(the social choice theorists I have in mind are in America and Europe). Everything these academics say indicates they cannot mandate the US and UK to criticise other polities. Correspondingly the US and UK only have the right to keep their bombs and their advice at home.

>As for G-S, it is very closely related to Arrow's theorem. I would fully support content along these lines; perhaps starting with a simplified combined proof would be in order, ...

Agree, but I consider even more important to show the informal connection between conditions of these two theorems (via connection between IIA and strategic-proofness).

--Y2y 09:52, 5 March 2007 (UTC)

You write: Agree, if you add "or non-deterministic or with restricted domain". Actually, the only nondeterministic and non-imposed method that is not manipulatable is a random weighted dictatorship (Pattanaik and Peleg 1986). (But see my example in my discussion of the Theorem itself. I. MacIntyre) Murakami (1961) shows that a version of Arrow's theorem holds under domain restrictions (using monotonicity and weaker dictators).

I still don't agree that there's a strong connection between IIA and manipulability, given these strong impossibility results. Further, if a connection could be shown and sourced properly, it would belong in the IIA article (or possibly in the G-S article), not here.

CRGreathouse (t | c) 17:25, 5 March 2007 (UTC)

You wrote: "I still don't agree that there's a strong connection between IIA and manipulability".

1. Connection between IIA violation and strategic nomination is already shown by example in the section "Interpretations of the theorem".

2. Connection between IIA violation and strategic voting can be shown in the following way:

For simplicity let us assume that only the winner does matter. Let consider a voter with such preferences: A > B > C (A is preffered candidate for this voter). A situation may arise when their sincere voting will result in victory of B, but misrepresenting his preferences as A > C > B will result in victory of A. (For example see push-over). IIA obviously forbids such situations (alternatives A and B have the same order in these two preference profiles).

3. Please note that without the proposed text the last paragraph of "Interpretations of the theorem" ("So, what Arrow's theorem really shows...") is inappropriate. Because "So" in that paragraph related to the assertion of connection between IIA-violation and manipulability.

--Y2y 21:19, 9 March 2007 (UTC)

But IIA systems are also manipulatable, unless they're dictatorial or fixed (Campbell and Kelly 1993 discusses this "trade-off"), so IIA doesn't get you anything. CRGreathouse (t | c) 17:22, 11 March 2007 (UTC)

> But IIA systems are also manipulatable, unless they're dictatorial or fixed

1. This does not mean that IIA and non-manipulability are not connected. (I could say that IIA denies only some kinds of manipulability).

> Campbell and Kelly 1993 discusses this "trade-off

2. No. Their statement is that any satisfying IIA social welfare function is (partially) dictatorial or (partially) fixed. (See citation and link below).

And this statement supports the point that IIA seriously decreases manipulability (if does not eliminate it): dictatorial or fixed (imposed) functions are hardly manipulable.

Citation (reworded a little) from the Introduction:

K.J. Arrow proved that an social welfare function must be dictatorial if it satisfies the Pareto criterion and IIA condition. We show that there is very little to be gained by relaxing the Pareto-efficency criterion: every social welfare function satisfying IIA either gives some individual too much dictatorial power or else there are to many pairs of alternatives that are socially ranked without consulting anyone's preferences.

3. So I think we may restore the paragraph about relaxing IIA and manipulability?

4. Thank for the reference. I think it should be mentioned in the article too.

--Y2y 22:20, 11 March 2007 (UTC)

(Resetting indentation) In response to #1: If you're saying that IIA prevents some forms of manipulability, state those and give a reference. That would actually be useful. For #2, do I read you right in that you're supporting IIA because dictatorial and fixed functions are non-manipulatable? In that case we should probably skip the confusing IIA and state that dictatorial and fixed functions aren't manipulatable. For #3, I don't see any reason to restore the paragraph: this is the wrong place, and the wording does not display NPOV. As for the references, feel free to add whatever you see fit to add. CRGreathouse (t | c) 23:56, 11 March 2007 (UTC)

#1. Well, I'll think how to reword.

#2. No. 1) I do not "support IIA". I think that we should clarify why such not very evident condition may be desirable. (Se above: 5 March). 2) "Dictatorial and fixed" does not belong to the conditions of Arrow's theorem, but IIA does belong. So we should speak about IIA.

#3. I do non see POV. I have seen no reference really denying connection between IIA and manipulability. You have mentioned: Campbell and Kelly 1993, Pattanaik and Peleg 1986, Murakami (1961). But as I see none is about manipulability. (And the first really in some degree indirectly supports my point of view, see above). As for Gibbard-Satterthwaite theorem. 1) See above: 5 March. 2) G-S postulates non-imposition. But IIA decreases manipulability even without non-imposition (see my previous message).

But I'll think how to reword for make the statement more clear. Thank for help.

--Y2y 09:13, 12 March 2007 (UTC)

Is the theorem still valid?

I'm a PhD student in the TLCs area and came upon this short bio: http://www.iiis.org/iiis/Nagib-Callaos.asp I don't have the necessary skills and background, but it appears that someone (Prof. Nagib Callaos) succesfully confuted Arrow's theorem. Maybe some hints or links could be provided in the article.

CB —Preceding unsigned comment added by 192.167.209.10 (talk) 07:56, 10 September 2007 (UTC)

That's the guy whose conference accepted the fake paper, right? [3] I wouldn't expect much. Considering the simplicity of the theorem, it's hard to imagine a mistake slipping past everyone all these years. Arrow's original proof may have taken a chapter, but if you Google for it you can probably find several one-page proofs of the theorem.

Without actually seeing the Calloas paper I can't say much more, but if you have a link it would be an interesting exercise to see whether it disproved something other than Arrow's theorem or whether it was itself flawed.

Edit: see [4] [5] [6] [7].

CRGreathouse (t | c) 13:57, 14 October 2007 (UTC)

Recent removal/revert

The bold portion of the quote below was removed, then readded, from and to the article:

The reason that the IIA property might not be realistically satisfied in human decision-making of any complexity is twofold: 1) the scalar preference ranking is derived from the weighting—not usually explicit—of a vector of attributes (one book dealing with the Arrow theorem invites the reader to consider the related problem of creating a scalar measure for the track and field decathlon event—e.g. how does one make scoring 600 points in the discus event "commensurable" with scoring 600 points in the 1500 m race) and 2) a new option can "focus the attention" on a different attribute or set of attributes, changing the tacit weighting and thus the resultant scalar ranking for the previous options. For example, suppose one were offered jobs in Montreal and Vancouver, B.C. and decided that (the jobs being considered equal) one preferred Montreal based on a more lively night life; but then one was also offered a job in Winnipeg and this reminded one — the winters in Winnipeg being harsh — that the winters in Montreal are far more severe than in Vancouver, causing one to choose Vancouver on the basis of a milder climate. (Edward MacNeal discusses the instability of a scalar ranking of "most livable city" with regard to different weighting of a vector of criteria in the chapter "Surveys" of his book Mathsemantics: making numbers talk sense, 1994.) It should be pointed out that there is still a problem in this argument though, namely that the formal IIA statement would say that if Winnipeg was again removed as an option then the preference would 'flip' back to the original ordering, making the system act in a seemingly irrational manner.

I found that the removed portion was actually the only sensible portion of the entire section. IIA does not allow for a change in focus or a second voting -- it only considers re-voting with the information already submitted. I actually can't think of a better way to improve the section than by removing it. Thoughts?

CRGreathouse (t | c) 13:53, 14 October 2007 (UTC)

Yeah, I think I agree. —Ruakh_TALK 21:28, 17 October 2007 (UTC)

The passage was recently re-added by an anon after I removed it. This time it mentioned MacNeal

(Edward MacNeal discusses the instability of a scalar ranking of "most livable city" with regard to different weighting of a vector of criteria in the chapter "Surveys" of his book Mathsemantics: making numbers talk sense, 1994.)

and Herbert Simon

Herbert Simon has noted that studies which appear to show that political campaigns are relatively ineffective in indoctrinating voters with new ideas may be missing the point — political campaigns can be quite effective in focusing voter's attention on a certain set of issues of which they already have some awareness, and hence convincing the voter that these are the issues on which the election should be decided.

Neither of these addresses the crucial issue of IIA. IIA is in no way about changing the information of the voters; as already mentioned above, it would require preferences to change back if the candidates changed back, which would not happen under either scenario added by the anon.

CRGreathouse (t | c) 22:32, 20 October 2007 (UTC)

Let me be as brief as possible. The section that I wrote on this was not intended to follow the Arrow Theorem exactly -- maybe I should have made this clearer -- but to give a plausible mathematical/cognitive model of why the IIA property is not necessarily "reasonable" in real life. (Models supported in the literature -- hence the reference to Herb Simon's work.) The IIA property seems reasonable only when we think of preferences as simple, atomic (i.e. non-decomposable), intrinsic things -- that's why the Sydney Morgenbesser anecdote/joke in the IIA article about violating IIA works, because most of us don't (and can't) analytically decompose our pie preferences. In many other important preference rankings -- cities we want to live in, jobs we want to work at, people we want as friends, politicians we want to vote for -- it is much more obvious that the preferences arise from some weighting of different attributes and are not atomic and intrinsic. So an apparently "irrelevant" alternative can remind you of different attributes of the "relevant" choices and flip your relative ranking of these choices, as per my example. 137.82.188.68 05:31, 10 November 2007 (UTC)

But that's not what IIA says. Your example is a change from a linear order M > V (with W unknown, in one of the 5 possible positions) to a linear order V > M > W. But this is nothing but a change in preferences; if W was later discovered to be infeasible, then the preferences would be V > M -- a change from the original. Under IIA there is no change, and given only M and V preferences would be M > V. The example has nothing to do with IIA.

Sure, people can change their minds and this is sometimes (rarely) analyzed in social choice theory. But it's a different thing entirely from IIA.

CRGreathouse (t | c) 14:55, 10 November 2007 (UTC)

My understanding of the key feature of the IIA condition -- derived from reading H.W. Lewis's Why flip a coin? and also illustrated by the Morgenbesser pie-choice anecdote -- is that given, say, some preference ranking for choices (A,B) then given another choice C, C could appear anywhere in the new preference ranking of (A,B,C) but would not reverse the relative ranking of A with regards to B. The point is not that people change their minds, it's that they are not "supposed" to change their minds (in a "single election", speaking loosely) about the relative ranking of the pre-existing choices due to the introduction of another choice. Are you claiming that the Morgenbesser pie-choice joke/anecdote about a violation of IIA is essentially misleading about the nature of IIA? 137.82.188.68 02:15, 11 November 2007 (UTC)

I'd support such a claim. Arrow's Theorem is about aggregating a collection of fixed individual preference orders into a single societal preference order. If you have an aggregating algorithm violates IIA, and you start with three alternatives and remove one, then the societal ordering of the other two alternatives might suddenly flip. A person following such an internal algorithm would be saying, "I preferred A to B as long as C was in the race, but then C dropped out, so I voted for B over A." Now, there does seem to be some connection between IIA and your example, but it's not completely clear to me what it is, and it seems like OR for us to come up with our own theory. —Ruakh_TALK 02:30, 11 November 2007 (UTC)

"would be saying .." Yes, but this is only a verbal reformulation, not any kind of explanation in terms of an underlying cognitive/mathematical model. What kind of explanation do you imagine an articulate and self-aware person would give to justify this flip-flop? And in terms of real-world applications/interpretations of Arrow's Theorem it should be fairly obvious that removing an existing choice is not symmetric to adding a new choice --whatever the formal mathematical model -- because human beings possess the capability called "memory." You might note that no management/labor union bargaining sessions (that I am aware of) start with management presenting its absolutely best contract offer and then systematically removing benefits as the bargaining proceeds! As for the dreaded WP:OR, all the pieces of the argument are covered (and not disjointly, either) in the literature. If I can give a reference for "A implies B" and another for "B implies C" then am I allowed to say that "A implies C" is supported in the literature? For some, I know, the answer is "No."

137.82.188.68 04:03, 11 November 2007 (UTC)

(de-indent) What you're trying to post may be considered OR, but I'm not concerned about that -- often a bit of research (even if original) will improve a page, and this is what WP:IAR is for. My problem is that your research is bad -- or rather than insofar as it's good it's inapplicable. What you write about is about information sets, which are obviously not symmetric with respect to addition and subtraction. This article is about a phenomenon which is symmetric. A candidate considering dropping out of the race (changing no one's information by so doing) in order to swing the result would be a relevant example; irreversibly changing one's mind would not be.

In your example being offered a job in Winnipeg is a canard -- the person may have simply seen Winnipeg's harsh winter on the news and switched preferences from M > V to V > M. This preference alteration is not permitted in Arrow's framework, which assumes static preferences. If the person was then offered a job in Winnipeg, the preferences could be enlarged IIA-wise from V > M to V > M > W. IIA isn't about changing one's mind, it's about individual transitivity mapping to societal transitivity.

CRGreathouse (t | c) 04:27, 11 November 2007 (UTC)

We seem to have reached an impasse. Here is a quote from The Mathematics of Behavior by Earl Hunt (Cambridge University Press, 2007), from a chapter where he describes (and proves) the Arrow Theorem in some detail (pg. 168, note that Hunt also uses a flavor preference example a la the Morgenbesser joke/anecdote; my interjection in double parentheses):

According to the independence of irrelevant alternatives axiom, once the relation Z < X (or Z > X) has been asserted, it cannot be changed by changing the value of some other choice. ((Or equivalently, introducing some new choice, especially if it is the least-favored choice.)) If you decide that you prefer chocolate to vanilla, vanilla to strawberry, and chocolate to strawberry, that is rational. Changing your mind to prefer strawberry to vanilla should not affect your assertion that chocolate is preferred to strawberry.

(and, speaking again of IIR, pg. 173, my underline emphasis) ... the requirement that the choice between two alternatives not be affected by the presence of a third alternative. (Earl Hunt)

I believe my example speaks to exactly this point. If you don't think so, I guess we have to agree to disagree. I won't attempt to re-insert it if you're dead set against it. I'd be interested in the opinions of several more people familiar with the Arrow Theorem, though; given the structure and dynamics of Wikipedia a vote of 2-1 against is not an especially compelling reason to revise one's own considered opinion. Regards, 137.82.188.68 01:01, 12 November 2007 (UTC)

One more point which I can't resist making -- it is assuredly not a canard for the purposes of my example that the person is offered a job in Winnipeg -- and hence "runs their mind" over the total possibility/pattern of living in Winnipeg -- rather than merely being reminded of harsh Winnipeg winters by a newscast, because it is the collision/comparison/contrast/context of multiple vectors of attributes that is the point at issue with regard to IIA. In my simplified example, Winnipeg refocuses the attention on a single attribute (mildness of climate) which reverses the decision between Montreal and Vancouver, but it could easily be on two or more attributes -- e.g. Winnipeg is both cold and flat so it reminds one Montreal has harsh winters and that there are no local mountains for skiing in Montreal as there are in Vancouver. I think there is some reasonable analogy here with neural-net models of pattern recognition/recall -- although it is beyond my current competence to give details. Regards, 137.82.188.68 02:38, 12 November 2007 (UTC)

So far I am still "dead-set against it", but at least we're talking. How about this: I'll look for a copy of that book, read it, and see if it changes my mind. In the meantime if others are swayed to your perspective they can change it as they see fit. I will also look at the Ray article to see what flavor of IIA Hunt is using, if that's relevant. I've now seen at least three or four different types in social choice theory alone (plus an econometric one), depending on whether the one in Schwartz's book The Logic of Collective Choice is distinct from the ones in the aforementioned paper.

As to your last paragraph, I honestly don't follow. Couldn't hearing a newscast on Winnipeg remind one of its flatness in addition to its winters? I can see an argument that additional candidates could inform a neural net, but removing them from consideration does not remove the knowledge -- so I can't see the parallel. Am I missing something?

CRGreathouse (t | c) 03:24, 12 November 2007 (UTC)

You can change the "vote of 2-1 against" to "3-1 against". A voter changing their mind is irrelevant to Arrow's Theorem. VoteFair 07:09, 12 November 2007 (UTC)

Regrettably, VoteFair, your vote must be weighted as zero in my own voting function because you have misunderstood the IIA point at issue. The Morgenbesser anecdote, again. And, CRGreathouse, as for the "newscast on Winnipeg" point, this is itself a canard. It's not how you gather the information on attributes that is at issue, it's how your attention is focused on a particular set of attributes -- and this is done most effectively in many cases by explicit contrast and comparison. (Cf. The anecdote from Richard Feynman's Surely You're Joking, Mr. Feynman! where his Brazilian physics students could regurgitate the textbook perfectly but couldn't apply this formal knowledge in the real world at all. It's entirely possible that one could know that Winnipeg has harsh winters but not make any use of this information in a comparison of Montreal and Vancouver, until Winnipeg is included as a choice under consideration.) Will there always be a magic newscast for choices based on a complicated tacit combination of attributes -- say, to decide whether to propose marriage to one of {Angela, Barbara, Cathy}, all of whom you are dating (and then you start dating Darlene)? Regards, 137.82.188.68 00:17, 13 November 2007 (UTC)

Perhaps we don't understand, because I'm still stuck with same the IIA issue. In miniature:

• Me: Isn't this the same as a newscast making the guy change his mind?
• You: No, adding candidates can change his mind too.
• Me: But IIA isn't about people changing their mind.

I followed your Morgenbesser story, your Winnipeg story, and have read the Feynman anecdote. But I still don't see how at the core of any of those there's any principle beyond voters changing preferences. IIA is about how society's preference changes in an 'irrational' way when the set of alternatives changes but preferences remain fixed. I simply don't see the connection; the situations seem almost as different as possible within social choice theory.

CRGreathouse (t | c) 03:37, 13 November 2007 (UTC)

(de-indent) O.k., I think I see the crux of our (apparent) disagreement -- we've been talking at cross-purposes. Your point (and Ruakh and VoteFair's as well), as I understand it, is that the Arrow Theorem uses the IIA condition on the Social Welfare Function, not individual preferences, and so (among other things) my "focus of attention" argument is not obviously applicable to a SWF. Quite true. My point was only (but non-trivially, in my opinion) that the reasonableness of the IIA condition on the SWF is justified by examples from individual preference rankings of "simple" things (food preferences are an effective example, if one is not an expert taste-taster), but one can see that this condition is already not so realistic for a single individual with a plausible model of preference ranking arising from a vector of attributes. I promise to say nothing more about this unless I've read Arrow's book and really mastered the proof presented by Earl Hunt. (And maybe not then, either.)

Best Wishes, 137.82.188.68 05:06, 15 November 2007 (UTC)

I think you're looking at it the wrong way; yes, humans are subject to constraints that resemble Arrow's Theorem (but are a bit different, firstly because our subpreferences can have scalar values and not just rankings, and secondly because we're not deterministic in the same way that Arrow's Theorem requires), and yes, this sometimes means we choose to violate IIA. But this doesn't mean that IIA is unreasonable; rather, it means that as humans we don't always have the option of behaving only according to reason, and IIA is one reasonable criterion that we're willing to sacrifice. (Further, the fact that we don't switch back after an option disappears means that we still regard IIA as reasonable; we'll violate it when we have to because a new option turns up and affects our thought process, but we won't willfully violate it by intentionally forgetting things we've come to consider relevant.) But I think the article already covers this sufficiently with the passage that begins "Various theorists have suggested weakening the IIA criterion as a way out of the paradox." —Ruakh_TALK 06:16, 15 November 2007 (UTC)

Arrow's Example of IIA

I read some of Arrow's book and this is his example of IIA (pg. 26, 1963 edition):

... For example, suppose that an election system has been devised whereby each individual lists all the candidates in order of his preference and then, by a preassigned procedure, the winning candidate is derived from these lists. (All election procedures are of this type, although in most the entire list is not required for the choice.) Suppose that an election is held, with a certain number of candidates in the field, each individual filing his list of preferences, and then one of the candidates dies. Surely the social choice should be made by taking each of the individual's preference lists, blotting out completely the dead candidate's name, and considering only the orderings of the remaining names in going through the procedure of determining the winner. That is, the choice to be made among the set S of surviving candidates would be independent of the preferences of individuals for candidates not in S. ... (Kenneth Arrow)

With this example in mind I see that one is naturally led to objections of the sort offered by critics of my "reminder" example.

Regards, 137.82.188.68 (talk) 06:16, 18 November 2007 (UTC)

I've read the book (only the 1963 version; the 1951 version had some serious mistakes) and recall the example. Actually it's a rather good one, I think; some of Arrow's examples are not good (as Ray points out). This doesn't seem to me like your other examples. If you wanted to add this to the article I would have no objections whatever. CRGreathouse (t | c) 06:33, 18 November 2007 (UTC)

Below I comment on two versions of IIA in circulation; I wrote that before noticing the above. Ironically, the version above may not be what Arrow actually used in the theorem. Range satisfies the description above and thus is a counterexample to Arrow's theorem (and this interpretation would simply mean that Arrow's theorem depends purely on ranked ballots and does not apply to cardinal ratings methods). What it does not satisfy is what Arrow might have thought equivalent: that the election function's generated social ordering for a set of candidates would not change if some set of ballots were changed without altering the preference order for those candidates on those ballots. In other words, the common discrepancy we find in definitions of IIA can be traced to Arrow himself. He clearly, in the passage above, is *only* contemplating pure preference order ballots; and he claims "all election procedures are of this type." --Abd (talk) 03:40, 12 January 2008 (UTC)

Introduction

I noticed that the introduction of this article presents a unique and non-standard description of the theorem. That is the inseretion of the concept that it only applies to voting systems based on ranked ballots. I looked back and found that this odd element was inserted by an editoer WClark (nothing on his/her talk page) in September 2006. This same editor was also working on the Range Voting article at about the same time. It is an important claim of advocates of Range and Approval Voting that they are exempt from Arrow's Theorem because they don't use ranked ballots. This is obviously debatable (since the voters may still HAVE preferences, whether the voting system allows them to express them or not). So I have edited the introduction slightly to make it more accurate. I did not go all the way back to the original (basically that there can be no perfect voting system), since I know the Approval and Range advocates won't let that long standing version survive. Instead I tried to make it accurate without using the generally accepted short-hand description...but eliminating the notion that some favored voting methods are exempt simply because they ignore the ranked preferences of individuals. Tbouricius 21:20, 17 October 2007 (UTC)

Please be bold. :-) —Ruakh_TALK 21:26, 17 October 2007 (UTC)

Editor Tbouricius is an advocate of Instant-runoff voting and proponents of IRV have wanted to cite Arrow's theorem as part of a political argument, it's a defense against critics of IRV who note, say, monotonicity failure for IRV; the response is essentially, "So what? Arrow's theorem proves no voting method is perfect."

There is a lot of confusion about Arrow's Theorem, as can be seen by the many differing descriptions of the theorem, its conditions, and what it supposedly proves. The theorem itself is math, but the application isn't. That is, given Arrow's stated assumptions, his conclusion, that no voting method can satisfy a set of supposedly reasonable criteria, isn't controversial. The problem is in the assumptions or in the reasonableness of the criteria.

The problem is most blatant with IIA, Independence of irrelevant alternatives. There are two ways that this criterion has been stated, and Range voting satisfies one of them and not the other. Arrow, I think -- I don't have the paper but Warren Smith notes this -- stated it such that Range doesn't satisfy it. Yet the other way of stating it is closer to what I'd assert would be most people's intuition of what "Independence of Irrelevant Alternatives" would mean, and, in fact, this is the criterion as stated in the current article (permanent link: [8]).

Arrow's way: Criterion of independence of irrelevant alternatives. If one set of preference ballots would lead to an an overall ranking of alternative X above alternative Y and if some preference ballots are changed without changing the relative rank of X and Y, then the method should still rank X above Y. Now, any method which considers preference strength is likely to fail this criterion. Consider a Range 100 election, and 51% of the voters vote A:100, B:1, C:0, and 49% vote A:100, B:0, C:99, the resulting preference order is A>C>B. If, however, The 51% change their vote to A:100, B:99, C:0, the original preference rankings have not changed, but the determined order is A>B>C.

The other statement of what IIA means is: If a candidate is added to the ballots, the output preference order among the other candidates does not change from what it was before the addition. Range does satisfy this because the score of each candidate is independent of the score of all other candidates. Note that some raise the issue of strategy here. It's irrelevant. Arrow's theorem is about what's on the ballots, not about shifts in what voters *would* put on the ballot if some new candidate comes in. Sometimes this criterion is stated with reference to candidate removal rather than candidate addition (which is clearer in a practical sense).

Basically, Arrow's theorem simply does not contemplate the use of preference strength; it is *as if* Arrow had stated that an ideal election method would neglect preference strength expression. So does Arrow's theorem apply to Range voting? Sure it does. Range fails IIA as defined by Arrow, but this does not mean that Range is therefore defective in that way. It actually means that IIA, Arrow's version, is defective, it was a criterion that seemed obviously desirable as long as one was only thinking of pure ranked ballots; hence the common opinion that Arrow's theorem does not apply to Range voting

Now, the analysis above was with respect to Range Voting. What about Approval voting? The "preference order" allowed on Approval ballots is maximally primitive, but it *is* a preference order. This discussion would take us to a topic Tbouricius and I have worked on extensively: what does "preference order" mean? Is it some opinion held by the voters, but not necessarily expressed on the ballots, or is it only that which is on the ballots? If it is not what is on the ballots, then we must have some means of translating the opinions to votes, or we cannot determine how an election method will behave.

Considering ballots only, is Approval a counter-example to Arrow's theorem? Here is a statement of the conditions and criteria from one web site[9]:

1. Universality. The voting method should provide a complete ranking of all alternatives from any set of individual preference ballots.

Warren Smith seems to claim that Arrow's ranking does permit equal ranking in both input and output. If so, then Approval satisfies this.

2. Monotonicity criterion. If one set of preference ballots would lead to an an overall ranking of alternative X above alternative Y and if some preference ballots are changed in such a way that the only alternative that has a higher ranking on any preference ballots is X, then the method should still rank X above Y.

Approval is monotonic.

3. Criterion of independence of irrelevant alternatives. If one set of preference ballots would lead to an an overall ranking of alternative X above alternative Y and if some preference ballots are changed without changing the relative rank of X and Y, then the method should still rank X above Y.

Approval satisfies this.

4. Citizen Sovereignty. Every possible ranking of alternatives can be achieved from some set of individual preference ballots.

Approval satisfies this.

5. Non-dictatorship. There should not be one specific voter whose preference ballot is always adopted.

Approval satisfies this.

What's wrong with this picture? Well, it's pretty likely that I copied a defective description of Arrow's criteria, and the most likely culprit is the first, possibly in its interpretation with regard to equal ranking. And if that is the case, then we must really say that Arrow's theorem did not contemplate the general case of voting methods, but only a particular class of methods, those using pure ranked ballots, it just happened to be that nobody at the time was thinking about other possibilities, or at least not Arrow. --Abd (talk) 03:31, 12 January 2008 (UTC)

General Irrelevance

Should discussion of Gordon Tullock's On the General Irrelevance of the General Impossibility Theorem (1967) be added to the criticism section? Mathematically, the theorem is correct but the magnitude of the paradox of voting becomes fairly trivial in many real-world circumstances, especially as the number of voters increases. —Preceding unsigned comment added by 128.239.180.11 (talk) 06:02, 22 October 2007 (UTC)

That seems appropriate. I'm somewhat surprised, though, since I've seen papers taking the exact opposite side -- that the problems in Arrow's paradox are almost unavoidable at large sizes. CRGreathouse (t | c) 14:19, 22 October 2007 (UTC)

We already mention the work done by Duncan Black, "Duncan Black has shown that if there is only one agenda by which the preferences are judged, then all of Arrow's axioms are met by the majority rule. " I believe the paper you refer to expands that concept. If you want a voting system that can help with more than two parties, however, Arrow's criteria are still rather interesting. Paladinwannabe2 22:52, 23 October 2007 (UTC)

I guess we should add some words about it. If I remember correctly, Riker in "liberalism against populism" makes the assersion that when the number of agendas and voters becomes lagre, the chances of having a problem approach 1/3 - which is very high. mousomer 10:32, 26 October 2007 (UTC)

edit re IIA and "voters" vs. "ballots."

Arrow's theorem is often cited as proof that "no voting system" is "perfect," or the like. Often this is used as a justification for a system failing some criterion like Favorite betrayal criterion or the Condorcet criterion. However, there are assumptions in Arrow's theorem that do not necessarily apply to all voting systems. The problem has become fairly clear to me; it has to do with how internal preferences of the voter are translated into votes.

We may assume that voters have some internal ordering of candidates. Arrow assumes a strict ordering, if I'm correct, where preference strength is irrelevant. However, when preference strength is below some level, it is actually irrelevant, the voter does not care if either candidate beats the other. Now, some voting methods don't allow full expression of preference, but require some equalities. For example, Plurality requires equating all candidates but one, placing all but the "favorite" in the bottom rank, on the ballot.

No voting system can make decisions based on voter preferences if the voters don't express the preference. Arrow, to my knowledge, does not confront the problem; but many others, working with Arrow's theorem and various election criteria, begin to do so. They require that the voter vote "sincerely." Now, in ranked systems, this means that the voter does not reverse preference, and many ranked systems don't allow the expression of equal preference. But what if a system does allow that? By voting equal preference, a voter conceals a possible preference that may be asserted as the basis for a criterion failure. Is that a "sincere" vote? Within ordinary meanings, yes, it can be. But, remember, "sincere" meant, here, that the voter expresses the necessary preference being considered by the Criterion, or does the best that the system allows. (So, in Plurality voting, the voter must vote for the favorite, and not for any other, if there is a favorite, and thus, say, the Majority criterion can be applied and Plurality passes. On the other hand, a voter may, with Approval, conceal a preference, without actually reversing any. This is an option of the voter; the voter may choose to express strict preference, as in Plurality, or may choose to express something else, perhaps for strategic reasons. Does Approval satisfy the Majority criterion? Depends on the exact definition, and the early definitions did not address the issue, so, by choosing the definition to use, one may make Approval pass or fail, making mincemeat of the idea that election criteria are "objective," and, so, I've seen an election theorist standing on his head, using what is a concealed double-negative definition relying on the fallacy of the excluded middle to show that Approval fails. Why? Well, everybody agrees that it fails, and it's his feeling that it fails, so we must define the criterion to make it so.

Now, to the present point. IIA can be understood in terms of voter preferences, as some internal abstract, or in terms of expressed preferences on a ballot. Failure of IIA based on expressed preferences is one thing, based on internal preference is another. Arrow was really talking about the translation of individual preferences into a social ordering, so, for him, ballots are beside the point; the input is individual preference orders and the output is a social order. When we are talking about an actual voting system, however, it could be argued, the input is the ballot. And thus we can look at IIA in a very simple way: IIA is satisfied if we take all the ballots and add a new candidate, that candidate does not change the social ordering found through the election process, among the other candidates. There are methods which allow such an insertion where there is no effect on the outcome. Range voting, for example, because Range generates a candidate score which depends only on the votes for each candidate, which are (as expressed), independent. Borda count, even though it resembles Range in some way, does not, because adding a new candidate to the ballot shifts the rank of some candidates, but not others, and thus can affect their relative ordering. If, however, we took Borda Count, and modified it so that there were a fixed number of ranks, and more than one candidate could be assigned to a rank, we would have, in fact, Range voting. The votes for each candidate become independent.

However, another consideration is then raised. Suppose the voter changes votes for other candidates because of the presence of this new candidate? Approval satisfies the expressed-ballot version of IIA. But suppose that a voter, seeing some really disliked candidate whom the voter fears might win, the voter might change the vote to add an additional approval for another candidate, thus, possibly, reversing an outcome (between two candidates other than the "irrelevant" one).

Arrow's theorem does not address this issue at all, because it isn't really about election methods, though it can be applied to them -- with caution. Arrow proceeds from an assumed preference order, and makes no assumptions about intermediate expressions. Obviously, though, the preference order must be "visible" to the method, even if it only selects some information from it. Otherwise, for example, we could assert that Plurality voting fails the Majority criterion if a majority of voters prefers one candidate, but, perhaps fearing some negative outcome and being unaware of its position.

An edit just inserted the word "ballot" into the article. I haven't reviewed the context yet, but the change was clearly designed to refer to the ballot expression interpretation of IIA. It raises the issue again for me, so I'll be reviewing the article and editing accordingly. To my knowledge, peer-reviewed consideration of Arrow's theorem has hewed to the strict preference order interpretation, and ballots are irrelevant. The problem here is that editors and others want to explain Arrow's theorem to the public and to draw conclusions from it that do not necessarily follow from the theorem itself. Ultimately, we will need to tighten up on sourcing and faithfulness to source.--Abd (talk) 18:09, 20 February 2008 (UTC)

Okay, I edited the article. However, what I put in is not sourced (except that an argument is attributed to Warren Smith and can be sourced as such. On the other hand, neither is the material in the article sourced which implies that Arrow's theorem applies to "all voting systems." --Abd (talk) 18:29, 20 February 2008 (UTC)

I also took out the example for IIA with Dave et al. If all voters rank a candidate last, nearly every voting system will satisfy IIA, and, in fact, I can't think of a practical system that wouldn't. Perhaps, if needed, a better example could be found where a new candidate does shift the results; but the problem with IIA is that it really isn't crisply defined in such a way as to make its application to real election methods clear. Is the voter's "rank ordering" something that is on the ballot, or is it some internal order? (Arrow simply assumes a complete rank ordering, and avoids the question entirely of how the voter translates it. So we may imagine translations that will cause any "voting system" to fail IIA. Thus ... we either stick with what's on the ballot as being the "preference order" -- which creates a certain set of problems -- or we must define how the votes are translated into the method; and if the translation varies with the method, we are simply designing pass or fail for IIA -- according to no standard. Arrow's theorem, then, can be seen as narrow, not sufficiently specified to use it for real election methods. As noted, with some definitions of the criteria, Range Voting satisfies all the criteria, and with others, it does not. It is not merely that we might "weaken" IIA, rather, the whole concept wasn't clearly defined in the first place, and we can define it strongly or not. Arrow assumed that rank orders were complete, and full (no equalities).

Another way to state the problem is that a "voting system," if it allows voters to do anything other than completely rank all candidates, must include some algorithm for translating the complete rankings presumed to exist for the voters into actual votes. As I wrote, attempts have been made to do this, by, for example, requiring that votes be "sincere," but this, too, hasn't been clearly defined, since there can be, with methods which allow equal ranking or rating, more than one kind of "sincere" vote. The solution I've seen is essentially that any votes are allowed which are "not insincere," i.e, do not reverse preference, but, as Warren Smith notes, votes are not neatly categorized into sincere and insincere, there is a middle, votes which are non insincere in the classic ranked sense (reversing preference), but which are not fully sincere either (as they conceal a material preference). --Abd (talk) 02:07, 21 February 2008 (UTC)

Okay, somebody please explain dictatorship and deterministic

Suppose there are three voters and three candidates. These are the first two preference orders:

A>B>C
C>B>A

Comes the third voter, and the voter can determine the outcome: The voter can cause A, or C, in most systems that I can think of. What about B? Well, if the voter can make B win (as with some systems), this voter is a "dictator," as defined. Yet this doesn't seem objectionable. If B wins as a result of the vote, then we can say that B is everyone's first or second choice. Not bad. The only reason the B voter has apparent dictatorial power is the balance of the electorate.

This shows, I think, the hazard of applying Arrow's theorem (which is about specifying a complete result order deterministically) to practical voting methods. With Range Voting, for example, the preference orders above could be expressed as

A:100 B:50 C:0
A:0   B:50 C:100

This is a three-way tie with Range, and the third voter can pick the winner, and, indeed, the entire social order will reflect that voter's preference order. Is this "unfair"? I fail to see it. The society, except for the voter, is in a balanced situation, so whatever the voter wants is the determining factor. Arrow's criteria are arbitrary, and were merely historically significant, for most thinking up until Arrow seems to have been that preferential voting could be arranged so that there was some universally just result.

As to deterministic, if there are N equal factions of voters, and they rank N candidates symmetrically (i.e., there is a cycle involving all candidates; in the case of two candidates, they are exactly tied), then most systems will declare a tie of some kind, either no result or a chance result. This cannot be resolved fairly, deterministically. Practical voting systems can't be completely deterministic based only on the votes. Please explain, someone who knows Arrow's theorem well. --Abd (talk) 22:04, 21 February 2008 (UTC)

A dictator in the Arrovian sense is a voter who can select the winner in all cases, not just in a particular case. The article's wording might need some cleanup so this is evident. CRGreathouse (t | c) 01:22, 22 February 2008 (UTC)

Can someone define "transitivity"?

The article mentions it several times, but it's not clear what it means Fry-kun (talk) 02:10, 19 March 2008 (UTC)

See http://en.wikipedia.org/wiki/Transitive_relation —Preceding unsigned comment added by 69.232.40.177 (talk) 00:07, 17 November 2008 (UTC)

Relationship between single-peaked and cyclic preferences?

One thing not made clear in the article is when cyclic preferences don't occur. If a group of people have single peaked preferences for a single issue axis (say, where to set the volume on the music), then it's impossible for Condorcet cycles to occur. This makes Black's majority argument and the IIA argument essentially the same. Scott Ritchie (talk) 23:00, 21 March 2008 (UTC)

What part of the article are you referring to? I'm not quite sure what the "IIA argument" is. Of course there are no cycles when there are single-peaked preferences, that's precisely Black's theorem. But what can/should we change in the article to better reflect this?

Where I see Black mentioned in the article it's as a 'way out' of Arrow's theorem, along with 'having just two candidates'. Certainly the latter is a special case of the former... is this what you mean?

CRGreathouse (t | c) 20:03, 22 March 2008 (UTC)

I got confused by the whole "interpretations of the theorem" and "other possibilities" being two entirely separate sections. It reads fairly poorly - stuff like "the preceding discussion" made me lose focus. The way the article is now, it looks like Black's argument is completely separate from the discussion of cycles. Scott Ritchie (talk) 05:28, 23 March 2008 (UTC)

Proof broken, please fix

Now if each person moves his preference for C above A, then society would prefer C to A by WP. By the fact that A is already preferred to B, C would now be preferred to B as well in the social preference ranking. But moving C above A shouldn't change anything about how B and C compare, by independence of irrelevant alternatives. That is, since B is either at the very top or bottom of each person's preferences, moving C or A around doesn't change how either compares with B. We have reached an absurd conclusion.

That's not what IIA says. By IIA, if you take a subset of choices from each voter's preferences, those choices will be ordered the same way in the outcome independently from how the rest of preferences are ordered. If you, for example, change A-B-C to C-A-B, IIA isn't violated in any way. The underlined sentence is simply nonsense. Also, in the given situation, the profile 1 is assumed (right?), so everyone is forced to C-A-B. Which neither breaks IIA.

The whole idea of the B would move directly to the top seems dubious. For instance, suppose everyone would have $R_{i}=[A,C,B]$. That would mean that after $n/2$ voters would have changed to $R_{i}=[B,A,C]$ we could take some kind of averaging scheme in the vote evaluation and come up with the change $[A,C,B]$ → $[A,B,C]$, disproving the conjectured premise.

--82.181.53.129 (talk) 17:08, 3 July 2008 (UTC)

Actually, this part is fine. You misunderstand the setup here. We assume in profile 1 that B is at the bottom for every voter. We assume in some unnamed profile (call it "profile 4") that B is at the top for every voter. If we go through one voter at a time in some order, moving B from the voter's lowest ranking immediately to their highest ranking, then there is a first voter for which moving B from the bottom to the top turns the societal order for B from the bottom. Suppose we call the profile when this happens "profile 5". In this part of the proof, we suppose that the SOCIETAL order for profile 5 is A>B>C. We now imagine everyone changing their preferences from profile 5 so that C is above A, so that each person either has B>C>A or C>A>B (call this "profile 6". Since A is irrelevant in society's ranking between B and C (by IIA), and the relative ranking of B and C by each voter is unchanged going from profile 5 to profile 6, we have that B>C in profile 5 precisely if it is so in profile 6. But in profile 6, since C>A for every voter, it follows from unanimity that C>A for society as well. Yet in Profile 5, C<A.

As to your "disproof": The concept of "averaging" is clever, but there is no guarantee that the average of a voting scheme is meaningful: how do we add voting schemes? And how do we divide by n? In such a way that IIA and unanimity, etc. continue to hold? Another response is that we have not mentioned ties. Geanakoplos's article, and Arrow's article, handle this nicely, but it's possible that an "averaging" scheme, if one existed, would make them all ties.

I have another problem with this proof, though, that should be fixed. I should note that this is not a problem in the original, cited at the end of the article (Geanakoplos's article); only in the way it is written in the Wikipedia article.

The problem is this: as stated, the voter n that determines the switch from B being on the bottom to B not being on the bottom MIGHT depend on everybody's relative rankings of A and C. Thus, when we change everyone's relative rankings of A and C, we might be changing which voter is voter n.

In Genanakoplos's article, this is solved by showing that in general, if no one puts B in the middle of their rankings, then society does not put B in the middle, either. This is done before discussing moving B one voter at a time from the everyone's bottom to everyone's top.

In this case, closer adherence to Geanakoplos's article would be better, and in fact, would remain clear. —Preceding unsigned comment added by 69.232.40.177 (talk) 00:05, 17 November 2008 (UTC)

Formal statement of the theorem

Question about the formal statement of the theorem - If n is the number of voters and k is the number of candidates in the election, does the theorem say that for every choice of n and for all k>=3, the only Arrow system is a dictatorship, or does it only say that for all k>=3 there exists some n (>=2) for which the only Arrow system is a dictatorship? It's not clear to me in the formal statement whether or not n is allowed to vary.

For example, does the theorem imply that: -there's no Arrow system that can combine 2 voter preferences in a 3-candidate election, and -there's no Arrow system that can combine 3 voter preferences in a 3-candidate election, and -there's no Arrow system that can combine 4 voter preferences in a 3-candidate election, and etc., or does it only imply that there's some n for which no Arrow system can combine n voter preferences in a 3-candidate election? Random Widget (talk) 03:36, 7 August 2008 (UTC)

For any given n and k > 2, the only Arrovian system is a dictatorship. CRGreathouse (t | c) 04:33, 7 August 2008 (UTC)

Though, to be fair, if there's only one voter it'd have to be a pretty weird counting system to not be a dictatorship. Scott Ritchie (talk) 07:47, 9 August 2008 (UTC)

For n = 1, k > 1, you could use an anti-dictatorship (choose what the sole voter likes least) or any of k systems with a fixed outcome. But of course for n = 1 a dictatorship is 'the best' possible system, just as for n = 2 a simple majority system is 'the best' by May's Theorem. CRGreathouse (t | c) 19:01, 9 August 2008 (UTC)

Proof broken?

There is one point, where I am unable to follow the proof: In the second part "Profile 2" is defined as the profile where every voter up to n ranks B at the bottom and the rest rank B at the top. A few lines later it is said that "this organization is just as in Profile 2, which we proved puts B below A". However, in part one this has been proven for a different kind of profile, namely one, where every voter up to n puts B at the top and the rest at the bottom. The same holds for "prifle 3". Since at this point of the proof the "arbitrary but specific order" in which we run through the members in society does still matter, "Profile 2" should rather be defined as the profile where every voter up to n ranks B at the top and the rest (including n) ranks B at the bottom. Likewise for "Profile 3".

Eckhart --132.180.58.38 (talk) 13:56, 16 March 2009 (UTC)

: Hi, part one proves this:

when {1,2,...n} have B at the bottom and {n+1, ...} have B at the top, then B is at the bottom of society's ranking

when {1,2,...n-1} have B at the bottom and {n,n+1,...} have B at the top, then B is at the top of society's ranking

Is this part clear? So for the A–B ordering, Profile 2 is just like the first of the two above. Shreevatsa (talk) 14:12, 16 March 2009 (UTC)

Oops, sorry, you were right. The meanings of "top" and "bottom" were switched in the first paragraph of Part Two; does the proof make sense now? Shreevatsa (talk) 14:30, 16 March 2009 (UTC)

Thanks for editing the proof. I did not feel confident enough to do it myself right away. To me it seems correct now. So I can go on recommending the wikipedia page on "Arrow's Theorem" to my students ;) 77.191.11.78 (talk) 17:27, 16 April 2009 (UTC) Eckhart

Linear orderings and permutations

There's been a little confusion over whether the linear orderings and permutations on a finite set are "equivalent". (This pertains to the formal statement of Arrow's Theorem.) Let me try to clarify the situation.

For any finite set A, we have the set L(A) of linear orderings on A, and the set P(A) of permutations of A. They have the same cardinality (namely, the factorial of the number of elements of A). So, there is a bijection between L(A) and P(A). But there is no canonical bijection between L(A) and P(A). In other words, in order to specify such a bijection, you have to make an arbitrary choice - roll a die, if you like. To see this, note that in order to specify a bijection, you must, in particular, specify which linear order on A corresponds to the identity permutation. In other words, you have to choose a particular linear order on A. And since A is given only as an abstract set, there's no canonical way to do that.

You might say "ah, but can't we say without loss of generality that A = {1, ..., n}?" The answer is "no", because A does not come equipped with a distinguished linear ordering, whereas {1, ..., n} does. And Arrow's Theorem is all about linear orderings, after all, so it's important to be careful on this point.

If you don't like words such as "canonical" and "arbitrary", here's a perfectly precise statement of the non-equivalence of linear orderings and permutations. It uses the language of combinatorial species. There is a species L assigning to each finite set A the set L(A) of linear orderings on A, and another species P assigning to each finite set A the set P(A) of permutations of A. It is a theorem that these two species are not isomorphic.

I'll now edit the "formal statement" section, deleting the assertion that linear orderings and permutations are equivalent. 86.165.44.73 (talk) 13:13, 3 October 2009 (UTC)

Formally they are equivalent, because formally A *is* defined (in Arrow's book) as 1..n. But I don't see a good reason to keep the claim. CRGreathouse (t | c) 06:29, 30 December 2009 (UTC)

Unrestricted domain

The article says that unrestricted domain entails that The social welfare function should account for all preferences among all voters. Whatever the intent, this sounds like it is saying that the function must depend on every voter's preferences, i.e., it cannot ignore any voters. As I understand it, unrestricted domain says nothing of the sort. A function with unrestricted domain can ignore any or all of the voters. The problem is even worse at unrestricted domain. 68.239.116.212 (talk) 18:09, 6 December 2009 (UTC)

Unrestricted domain means that the social welfare function associates an outcome to every possible sequence of votes from voters. A restricted domain might consider only votes which met certain criteria, such as all candidates except the most-preferred being tied on each ballot. Feel free to edit the wording. CRGreathouse (t | c) 01:42, 10 December 2009 (UTC)

: Why is there no mention of unrestricted domain in the bullet points "In short, no "fair" voting system can satisfy these three criteria" ? 3:35, 8 May 2010 (UTC) —Preceding unsigned comment added by 115.70.109.186 (talk)

Archiving

Does anyone object to me setting up automatic archiving for this page using MiszaBot? Unless otherwise agreed, I would set it to archive threads that have been inactive for 30 days and keep the last ten threads.--Oneiros (talk) 19:27, 9 January 2010 (UTC)

 Done--Oneiros (talk) 00:45, 13 January 2010 (UTC)

Non-imposition

This means that the social welfare function is surjective: It has an unrestricted target space.

Can we say that it has an unrestricted range? That makes a nice parallel with "unrestricted domain". 68.239.116.212 (talk) 02:56, 12 January 2010 (UTC)

I have never heard that term. By the range of a function $f:A\to B$, we usually mean B (though some people mean the image $f(A)$).

If we follow that usage, "unrestricted range" is always satisfied since the range is B even when f(B) contains only one element.

statement in intro vs body

The intro section lists the IIA condition as If every voter prefers X over Y, then adding Z to the slate won't change the group's preference of X over Y. The body lists the IIA condition as what sounds to me like If every voter's preference between X and Y remains unchanged, then adding Z to the slate won't change the group's preference of X over Y. Is my paraphrase correct for the version in the body, and is the version in the intro sufficient for the theorem? -- user:Dan_Wylie-Sears_2 (not logged in because I'm away from home and don't have my password with me) 98.243.221.202 (talk) 05:55, 30 December 2009 (UTC) (yes, that's me --Dan Wylie-Sears 2 (talk) 21:17, 1 January 2010 (UTC))

I think that the statement in the intro is an example of IIA, and not sufficient for the proof. Let me ponder this. CRGreathouse (t | c) 06:33, 30 December 2009 (UTC)

I don't think it should be phrased in terms of "adding Z to the slate". In this context IIA is a condition on a function whose domain is rankings of a fixed number of possibilities. Adding something to the slate would require a new function with a different domain. I think it should say that the order of X and Y in the group ranking depends only on the individual relative rankings of X and Y. 68.239.116.212 (talk) 03:44, 2 January 2010 (UTC)

It's misleading to use the expression like "adding Z" in the explanation of IIA. I have modified the wording.--Theorist2 (talk) 06:15, 8 October 2010 (UTC)

See Talk:Arrow's_impossibility_theorem#Confusion_about_IIA_.28independence_of_irrelevant_alternatives.29.--Theorist2 (talk) 07:16, 12 October 2010 (UTC)

Informal Proof

I may be only one person, but I was thoroughly confused by the informal proof. Someone either needs to

a) rewrite this article for the Simple English Wikipedia, or

b) make the informal proof on this page easier to understand.

☠ QuackOfaThousandSuns (Talk) ☠ 03:14, 27 January 2010 (UTC)

'Dictator' is clearly not what it means in common English.

The term 'dictator' doesn't have its plain-language meaning. And the claim that there can be 'only one dictator' is misleading also. To see why is simple. Suppose that every ordering is voted for by at least two people (as is natural for a large population voting on a small number of options). The theorem seems to show that if a voter changes from ordering X to ordering Y, the result changes. But there are many people who voted ordering X. Thus, there are in fact many dictators - everyone who voted a certain way. There is also no way to know who are the dictators before voting has started. This is not what we normally mean by 'dictator'. 87.127.16.175 (talk) —Preceding undated comment added 12:06, 26 February 2010 (UTC).

No. A dictator, for this theorem, is a voter whose choice will be reflected in the societal choice for *any* arrangement of votes. If there are C candidates and v voters, then a dictator is a voter who gets her way in all $(C!)^{v}$ possible votes.

CRGreathouse (t | c) 04:05, 4 June 2010 (UTC)

The sketch proof given in this article appears to prove that $\exists (R_{1},\ldots ,R_{i-1},R_{i+1},\ldots ,R_{n})$ such that $\forall R_{i},F(R_{1},\ldots ,R_{n})=R_{i}$. The implications of this are certainly not the same as $\forall (R_{1},\ldots ,R_{n}),F(R_{1},\ldots ,R_{n})=R_{i}$. As such, $R_{i}$ should not be called dictator. Birkett (talk) 16:35, 22 August 2010 (UTC)

Part two of the proof is not well written, but it starts with an arbitrary profile p1 in which voter n prefers A to C and shows that society prefers A to C. (In the last part where "person n flipped A and C", it considers an arbitrary profile p2 in which n prefers C to A.) Profiles 2 and 3 are constructed from p1 by only changing the position of B in each voter's preference. The position of B is irrelevant to the social ordering of A and C.--Theorist2 (talk) 23:26, 20 November 2010 (UTC)

Doesn't "dictator" just reduce to "there must be at least one voter whose preferences are exactly the same as society" in this case? And why is that bad? No one person can say "_I_ am going to be the 'dictator'" in this case, because who happens to to be the "n:th" voter when "the switch is flipped" in the outcome is totally arbitrary. —Preceding unsigned comment added by 208.80.119.3 (talk) 23:30, 10 September 2010 (UTC)

No. See, e.g., Taylor 2005 for an explanation. CRGreathouse (t | c) 21:57, 10 October 2010 (UTC)

Just saying "No. See <some paper which I neglect to link to>." is not a counterargument. Please give a link to the paper and refer to what part of the paper is relevant. As far as I can tell, the definition of "dictator" used in this proof is wholly unrelated to its English definition. The mere existence of a tiebreaker does not imply dictatorship. Clement Cherlin (talk) 01:54, 20 November 2010 (UTC).

CRGreathouse is right. Taylor, Alan D. (2005) is a textbook mentioned in a footnote. Actually the definition given in the article is sufficient. If you don't understand the definition (e.g., if you do not know the difference between "there exists a voter such that for all profiles" and "for all profiles there exists a voter"), go to college and learn elementary logic first. Arrow's theorem is a mathematical theorem and "dictator" is just a term used for a voter that has a certain mathematical property. Do not expect that a mathematically defined notion has the same meaning as a similarly spelled word in the everyday language.

Having said that, Arrow's "dictator" is even more problematic than real dictators defined in a dictionary. (It is not just a tiebreaker or pivotal voter!!) If an aggregation rule has a dictator, then (assuming the dictator is never indifferent between two alternatives) the only effective input to the rule is the dictator's preference. The others' preferences are totally ignored. Maybe one can call him an "extreme dictator."--Theorist2 (talk) 23:26, 20 November 2010 (UTC)

External Links

At the risk of self-citing, as I wrote the article referred to below, can I ask if there is any interest in including the following proof in the "External Links" section of this Wikipedia article? This proof requires no knowledge of maths, nor of proofs, and so it is accessible to anyone who can read. The article is available via JSTOR. For Wikipedians reviewing this post who do not have access to JSTOR, it is also available in a slightly less well-produced form (some of the figures are a bit blurry) from the URL below.
Paul Hansen (2002) “Another graphical proof of Arrow’s Impossibility Theorem”, Journal of Economic Education 33, 217-35
http://www.business.otago.ac.nz/econ/Personal/PH/P%20Hansen%20Proof%20of%20Arrow%27s%20Imposs%20Theorem.pdf 139.80.81.58 (talk) 05:47, 27 March 2010 (UTC)

Confusion about IIA (independence of irrelevant alternatives)

From User_talk:CRGreathouse#Arrow.27s_impossibility_theorem

I have never imagined that I have to explain Arrow's IIA to an expert mathematician interested in social choice (CRGreathouse) or a designer of an nice voting method who has also published in Social Choice and Welfare (Markus Schulze), but that is not too surprising, given the fact that Arrow himself provided an explanation that does not match his own formulation of IIA. Let me clarify the problem.

First thing to note is that "IIA" is used in different meanings in different contexts. When stating Arrow's theorem, the relevant IIA is that for Arrow's theorem, not the "standard" one, whatever that may refer to. In the context of Arrow's theorem, it is simply not correct to say "the standard formulation of IIA is adding nonwinning candidates to the slate." I really wonder which book they read!

The most popular definition of IIA in this context is as follows: an aggregation rule (e.g., social welfare function) f is Pairwise Independent if for any profiles $p=(R_{1},\ldots ,R_{n})$, $p'=(R'_{1},\ldots ,R'_{n})$ of preferences and for any alternatives x, y, if $R_{i}\cap \{x,y\}^{2}=R'_{i}\cap \{x,y\}^{2}$ for all i, then $f(p)\cap \{x,y\}^{2}=f(p')\cap \{x,y\}^{2}$. This is the definition of Arrow's IIA adopted in the contxt of Arrow's theorem in most textbooks (Austen-Smith and Banks, 1999, page 27; Gaertner, 2009, page 20; Mas-Colell, Whinston, Green, 1995, page 794; Nitzan, 2010, page 40; Tayor, 2005, page 18; see also Arrow, 1963, page 28 and Sen, 1970, page 37). Observe that you do not add any alternative in this formulation, since the set of alternatives is fixed. Also, note that this is a condition involving two profiles.

Arrow, (1963, page 27) and Sen (1970, page 41) gave a different formulation, but it is not directly applicable to aggregation rules, since it is a condition for rules that can deal with different agendas (subsets of alternatives). Their formulation applies, e,g., to what Austen-Smith and Banks (1999, page 49) call "collective choice rules." A collective choice rule is a mapping C that maps each pair of profile p and adenda S to a subset C(p,S) of S. Arrow's and Sen's formulation can be restated: A collective choice rule C is IIA if for any profiles $p=(R_{1},\ldots ,R_{n})$, $p'=(R'_{1},\ldots ,R'_{n})$ of preferences and for any agenda S, if $R_{i}\cap S^{2}=R'_{i}\cap S^{2}$ for all i, then $C(p,S)=C(p',S)$. This condition implies Pairwise Independence, if f(p) is suitably defined from C(p, {x,y}). Note that this condition involves two profiles and one agenda.

Let me comment on the following misunderstanding of IIA: "If every voter's preferences between X and Y remain unchanged when Z is added to the slate, then the group's preference between X and Y will also remain unchanged." Let me formulate this condition: a collective choice rule C is S-G IIA (I can't recall the common name) if for for any profiles $p=(R_{1},\ldots ,R_{n})$, $p'=(R'_{1},\ldots ,R'_{n})$ of preferences and for any x, y, z, if $R_{i}\cap \{x,y\}^{2}=R'_{i}\cap \{x,y\}^{2}$ for all i and $C(p,\{x,y\})=\{x\}$, then $y\notin C(p',\{x,y,z\})$. While this is an interesting condition, it is not the same as Arrow's or Sen's formulation. In particular, it involves two agenda {x, y} and {x, y, z}. but Arrow's and Sen's formulation involves just one agenda S. Indeed, the following example violates S-G IIA but satisfies theirs: Give names a_1, a_2, a_3, ...to all alternatives. Define a collective choice rule C by C(p,S)={a_i}, where a_i is the first element (the one with the least index) in S if S contains even number of alternatives, and a_i is the last element in S otherwise. (If you want to obtain a ordering, then (supposing i<j) let a_i be better than a_j if S has an even number of alternatives; let a_j be better than a_i if S has an odd number of alternatives.) It is possible that my formulation of S-G IIA above is not the same as what S&G intended. If so, then their English is probably not simple enough for international audience.

In short, Arrow's IIA or an aggregation rule has nothing to do with addition or deletion of alternatives. While I did say "even if voters' preferences between other pairs like X and Z, Y and Z, or Z and Z' change", I was not talking about adding an element to an agenda or deleting one from an agenda. Those are simply elements belonging to the set of alternatives.

I am thinking of adding a short example to clarify the notion of IIA to the article on Arrow's impossibility theorem. It is okay to move it to the article on Independence of irrelevant alternatives later, but in view of the fact that this notion confuses sophisticated editors like them and that what one means by "IIA" critically depends on the context, it probably makes sense if such an example appears on the article for Arrow's theorem. --Theorist2 (talk) 07:04, 12 October 2010 (UTC)

For a good general treatment, try Taylor (2005); for a discussion of different types of IIA, Paramesh Ray (1973) is the classic. Certainly there are many different formulations in the literature.

The traditional explanation of the reason for IIA comes, of course, from Arrow:

Suppose that an election is held, with a certain number of candidates in the field, each individual filing his list of preferences, and then one of the candidates dies. Surely the social choice should be made by taking each of the individual's preference lists, blotting out completely the dead candidate's name, and considering only the orderings of the remaining names in going through the procedure of determining the winner. (Arrow 1950, p. 337)

The particular form of IIA he uses is:

Let $R_{1}$, $R_{2}$, and $R'_{1}$, $R'_{2}$ be two sets of individual orderings. If, for both individuals i and for all x and y in a given set of alternatives S, $xR_{i}y$ if and only if $xR'_{i}y$, then the social choice made from S is the same whether the individual orderings are $R_{1}$, $R_{2}$, or $R'_{1}$, $R'_{2}$. (Independence of irrelevant alternatives.)

I'll go into more detail later if I have a chance.

CRGreathouse (t | c) 13:11, 12 October 2010 (UTC)

I have read pages 16-20 of Taylor, the part dealing with Arrow's theorem. Thanks for suggesting Ray (1973). I'll get it later, but the first page on JSTOR suggests that Arrow's explanation cased a confusion. In short, these sources only strengthen my position. Arrow's explanation that you cited is the one that caused much misunderstanding. It is probably okay as a motivation, but it is clearly not a close description of the IIA condition. Arrow's condition that you cited is the same as the one that I defined above for collective choice rules (except the number of individuals). I have added an example in the article on Independence_of_irrelevant_alternatives and reverted the article on Arrow's theorem.--Theorist2 (talk) 14:05, 12 October 2010 (UTC)

Intro References Range Voting Article

Judging by the back-and-forth commits, Theorist2 appears to object to the "Many voting methods" paragraph reference from a article/professor favoring range voting (making it the first reference). Other known exceptions are approval voting and equal-ranked systems. To my satisfaction the wording is broad enough to include these. Does anyone else think that it is skewed toward range-voting? --Osndok (talk) 00:17, 20 October 2010 (UTC)

Present wording:

There are several voting systems which can be considered to satisfy the spirit of these requirements, but which fail to satisfy universality because they do not require voters "to rank candidates in order of preference". The section #Other possibilities overviews various attempts to overcome Arrow's conclusion.

I'd change 'attempts' to 'ways' --Osndok (talk) 00:39, 20 October 2010 (UTC)

Theorist2 suggests:

An obvious way out of the impossibility is to combine cardinal utility and a weakened notion of independence. But there are others. The section #Other possibilities overviews various attempts to overcome Arrow's negative conclusion.

The source mentions voting systems and Arrow's requirements, not "a way out of impossibility" & weakening.

I take exception at "negative conclusion" & "weakening" wording, and suspect that this is one-or-more person's sacred cow. --Osndok (talk) 00:39, 20 October 2010 (UTC)

The source (which is not a referred article) is "Arrow's "impossibility" theorem – how can range voting accomplish the impossible?" Obviously, it is about a way to overcome Arrow's impossibility. And the answer it gives is cardinal utility based rules such as Range voting.

"Which can be considered to satisfy the spirit of these requirements" is rather vague. The source is more specific: stay away from voting systems based on rank-order ballots (that is, use cardinal utility). Also, IIA it mentions is a weakened one.

"Which fail to satisfy universality" is flatly wrong, unless there are too few labels. It's not universality that is violated, but the definition of social welfare function. So, better delete this sentence.--Theorist2 (talk) 01:14, 20 October 2010 (UTC)

Agreed. I'm not sure where "universality" popped up. Probably b/c it is the first bullet-point item to mention ordering. I suggest you fix it :) [rather than deleting it] --Osndok (talk) 03:22, 20 October 2010 (UTC)

This is no longer relevant, but let me point out why I say it is not universality that is violated.

Those who say Range voting violates universality probably think this way: Universality requires the domain consist of profiles of complete transitive preferences. Since Rv has a different domain (which does not consists of preference profiles), it violates universality. Well, now I see why they insist.

Well, the standard understanding of universality is this: Universality is an unrestricted domain condition. We declare the domain consisting of complete transitive preferences" universal". If the domain is restricted to a proper subset, then the function is not universal. But no matter how we restrict the domain consisting of preference profiles, we cannot obtain a domain consisting of lists of cardinal utility. It's just a mathematically different object. The universality condition simply does not apply. What is violated is not universality, but the very definition of a social welfare function.--Theorist2 (talk) 04:56, 20 October 2010 (UTC)

I'm also puzzled by the expression "because they do not require voters to rank candidates in order of preference". If scores are given to each point, then ranking is determined. The rules ask voters to rank and do more. Cut this sentence at least.--Theorist2 (talk) 01:52, 20 October 2010 (UTC)

This referenced quote is attempting to convey several facts. Principally that several voting systems exist which lie outside of the rather-strict rank-order system (e.g. that allow cardinal measurement, incomplete ranking, or equal ranking), but also that some voting systems (like range voting in particular) *can* let voters "rank" there results (although it is not substantive to Arrow's theorem). How can this better be conveyed? --Osndok (talk) 03:15, 20 October 2010 (UTC)

For that, just use the word "cardinal utility". By the way, Arrow's framework rules out cardinal measurement, but it does not rule out incomplete ranking or equal ranking. Since the framework allows incomplete or intransitive preferences, they are explicitly listed as axioms. So "all conceivable rules (that are based on complete, ordered preferences) within one unified framework." should be "all conceivable rules (that are based on preferences) within one unified framework." This point is clear from the "other possibilities" discussion. There, preferences not satisfying e.g., transitivity are discussed within Arrow's framework. As for "the earlier one[clarification needed] in voting theory, in which rules were investigated one by one", all contributions (by Borda, Condorcet, Lewis, etc) to voting theory were such that rules were investigated one by one. I think it's clear, but if not, could you express that idea in simple english? —Preceding unsigned comment added by Theorist2 (talk • contribs) 03:51, 20 October 2010 (UTC)

I am quite sure you are simply in err at this one point; b/c Arrow's theorem does rule out equal ranking. Otherwise it would be readily and easily disprovable by a simple approval-voting-conversion counter-example. wrt the clarification, if you mean an earlier arrow theory it should be said so ("his earlier theorem", not "the earlier one"); that's all, maybe it is missing context in a casual reading. --70.252.4.113 (talk) 16:19, 20 October 2010 (UTC)

Sorry, I do not know what "b/c" means or I do not understand what you are saying. But I can say this: I never said that Arrow's theorem rules out equal ranking. I said that Arrow's framework does not rule out equal ranking! This is easily shown from the fact that Arrow assumes in the universality condition that preferences are complete and transitive. Complete transitive binary relations of course allows indifference between different alternatives. Note that R is complete if for any x, y, we have xRy or yRx. If both are satisfied, then x and y are indifferent. Note also that the definition of a social welfare function in the article rules out indifference just for simplicity.--Theorist2 (talk) 13:04, 23 October 2010 (UTC)

The introduction of this article has an inappropriate and unbalanced "shout out" to a fringe in the voting reform advocacy world. For now, I left the references to Warren Smith's work, etc. (though I think this does not belong in the introduction - is it even peer-reviewed published work?), but moved this a bit lower down and gave a more balanced treatment. The prior language that asserted that voting methods such as Range, by using cardinal scores, can satisfy the "spirit" of the criteria was unbalanced. To add balance I added the fact that Arrow expressly rejected the use of cardinal scores, and focused on ordinal social welfare ranking. Cardinal scores are expressly NOT in the "spirit" of Arrow's work. I would prefer that this whole concept be removed from the introduction and moved to the "other" section below. Tbouricius (talk) 17:34, 30 October 2010 (UTC)

I did delete the citation to Warrren Smith from the Introduction earlier [10], but as far as I know, one editor disliked it. See 13:47, 23 October 2010 for my strategic response to that. As far as social choice theory goes, this situation is very unbalanced from the purely academic point of view, too. He probably gives 99 points for his position, so let me give 99 points for mine and why not decide by range voting.  ;-) --Theorist2 (talk) 22:41, 30 October 2010 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Arrow's impossibility theorem/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

I think that the section presenting the theorem's proof should, for the sake of completeness, mention, that all conclusions can be easily shown to be applicable to voting systems offering more than three options. Dienekis (talk) 01:59, 7 March 2008 (UTC)

Last edited at 01:59, 7 March 2008 (UTC). Substituted at 14:16, 1 May 2016 (UTC)

Explaining why there cannot be two dictators

The article clearly explains why there cannot be three (or more) dictators, but then leaves it to a jump of logic of the reader to know why there cannot be two dictators, and hence only must be one. 202.36.179.66 (talk) 09:07, 27 October 2010 (UTC)

This part of the proof is also insufficient. I haven't looked at the source, but it would go as follows:

• voter i dictates between A and B (i.e., i is a local dictator: if i prefers A to B, so does the society; if i prefers B to A, so does the society);
• voter j dictates between B and C;
• voter k dictates between A and C.

If i, j, k are all distinct, consider the A-C pair and obtain a contradiction (as is shown). If one of i, j, k is different and the other two are equal, assume i=j wlog and obtain a contradiction (consider the profile in which i=j prefers A to B to C and k prefers C to A to B). It follows that i=j=k. --Theorist2 (talk) 13:39, 21 November 2010 (UTC)

Arrow and free markets

Anonymous editor at 173.60.203.172 added this in a recent edit: "It is important to reflect on the implications here: Arrow's Impossibility Theorem implies that politics does not have the capability of satisfying some notion of the general will. In other words, only a market can satisfy all the infinite competing desires." This is uncited opinion which I don't believe Arrow ever stated or even implied, so I plan to remove it. -- RobLa (talk) 04:50, 21 December 2010 (UTC)

Agreed. Tom Ruen (talk) 04:52, 21 December 2010 (UTC)

I also second the deletion. Actually Arrow states "[voting and the market mechanism] being regarded as a special cases of the more general category of collective social choice" (1963, page 5). So he considered that his theorem covers (or at least he wanted it to cover) the market decisions, too. (I do not think the theorem establishes that far, but later research by others show more or less that Arrow's theorem can be extended to economic environments. So the competitive market is not an exception.) In any case, a more neutral view is mentioned in the Section "Interpretations of the theorem".--Theorist2 (talk) 07:45, 21 December 2010 (UTC)

which Pareto criterion

Pareto principle was recently linked; I doubt that's what the author of the phrase intended. On another hand, I'm not sure that Pareto efficiency is meant either, so I won't change it there. —Tamfang (talk) 04:18, 25 June 2011 (UTC)

Yes, it's Pareto efficiency, in particular the weak version. CRGreathouse (t | c) 16:35, 25 June 2011 (UTC)

voting vs. swf

The article presents the theorem as a voting result. But isn't AIT more general then that? In the sense that it says that no social welfare function which satisfies the stated criteria exists - in other words, it's not just that a voting system can't do something, it's that there is no way to aggregate individual preferences into a social preference.Volunteer Marek (talk) 20:23, 1 January 2012 (UTC)

This is a good point, Volunteer Marek. I think the issue is that Arrow's theorem is a theorem of mathematics that can be interpreted in terms of voting, or in terms of preference aggregation. i.e. the determination of a social preference ordering from individual preference orderings. I would call the latter interpretation "social choice", subject to a relatively insubstantial caveat mentioned below. I would not say it is more general, just different. I think the difference between these two interpretations should be made clear. The main issue is whether you take each individual preference ordering to be a ranked list of alternatives submitted by that individual, which is then used in a voting procedure, or to be that individual's actual preferences over the alternatives. Different mathematical requirements on the choice rule/ voting system may be intuitively appealing on these two different interpretations (this is indeed mentioned in the article).

There is perhaps a minor caveat: It may be, though, that the "social choice" language is usually applied to a slightly different formulation, in which there is a "choice function" that for each subset X of alternatives, says which subset of X consists of acceptable choices for society. I think there is likely to be an equivalence between this formulation and the one where the output is social preference ordering (the choice function will return, on input X, the subset of X consisting of alternatives maximal with respect to the social preference ordering restricted to X), and there will be a way of recovering a ranking from a choice function defined on all subsets of alternatives as well....). This strikes me as a fine point to be avoided if possible, but perhaps discussed under the heading of Social Choice Theory or Preference Aggregation, and just mentioned in passing (and cited) in this article.

I may do some editing based on the above observations, though probably not before taking some more time to think about it, and doing some more research. I also have other minor issues regarding whether the individual and/or social preference orderings are assumed to be strict or not, which I think should be made crystal clear in the article. MorphismOfDoom (talk) 17:31, 2 November 2012 (UTC)

Volunteer Marek is right: Arrow's theorem shows that "there is no way to aggregate individual preferences [orderings] into a social preference [ordering]". However, as my added brackets and emphasis suggest, that is less restrictive than it sounds like. For instance, there are multiple ways to aggregate individual utilities into social utilities. Because this result only talks about preferences, its main application is to voting. Thus it would be nice to have a passage explaining that it could be applicable in other situations, but I find the focus on voting, especially in the intro, to be perfectly appropriate. Homunq (࿓) 11:10, 5 November 2012 (UTC)

On the name "impossibility theorem"

EDIT: So this got deleted last time, but I never got a reply out of the person who did it, so I'm simply re-posting it again.

Where does the term "impossibility theorem" come from? In both Arrow's 1950 paper "A difficulty in the concept of social welfare" and the 1951 book "Social choice and individual values" the only terminology I see is "The General Possibility Theorem for Social Welfare Functions". Could someone direct me to the cause of why everywhere this is called the "impossibility" theorem? Drozdyuk (talk) 20:45, 3 December 2012 (UTC) (originally posted 25 April 2012)

IIA Range Voting Counter-example

The footnote concerning IIA versus range voting (with the 9 & 1 comparison) seems irrelevant. What is it trying to demonstrate? I recommend it be removed. --Osndok (talk) 00:17, 20 October 2010 (UTC)

I write "Whether such a claim is correct depends on how each condition is reformulated." As I wrote in User talk:Osndok, the footnote demonstrates that Rv violates Arrow's IIA. Of course, Rv satisfies a weakened IIA. So, it depends on how IIA is formulated.--Theorist2 (talk) 00:54, 20 October 2010 (UTC)

As discussed, I see that the example you have given is translating a range-voting example into rank-order results. I think we agree that range voting does not apply to Arrow's theorem, as the example could just as easily be: one voter scores all the candidates the same, therefore range voting does not produce an ordered result, therefore it "violates" a precondition. It still does not seem relevant to me. --Osndok (talk) 02:59, 20 October 2010 (UTC)

If you define a social welfare function so that it excludes indifference, then it means such a case is ignored because it is deemed uninteresting, not the point, excluded for simplification, etc. Why do you consider the uninteresting case in which all alternatives have the same score? Of course, it does violate the domain condition. But you can redefine the domain of a swf so that indifference is allowed. Then, the same score alternatives are treated as indifferent. If you think the same score case is important, then just use the latter definition. Don't use the former uninteresting definition. There, we regard IIA as important. We care if Rv satisfies it. The footnote is obviously relevant since it clarifies that the assertion that Rv satisfies all conditions depends on how those conditions are defined. Without the footnote, the reader might think Rv satisfies Arrow's original IIA.--Theorist2 (talk) 05:19, 20 October 2010 (UTC)

What is at issue here is that range voting (by-definition) satisfies "general IIA". The only reason it does not satisfy "Arrow's IIA" is because you must translate it into ranks. Range voting does not fall under the scope of Arrow's theorem, so why should this be included? As best I can see, you are adding a new statement; that, "range-voting-when-translated-into-ranks does not satisfy part of arrow's theorem" (it is original research, and [IMO] not relevant). --Osndok (talk) 16:28, 20 October 2010 (UTC)

Why should this be included?---Because, a "solution" to Arrow's impossibility is already included. As you say, Range voting does not fall under the scope of Arrow's theorem. But it is already cited as an example of a rule that "can be considered to satisfy the spirit of" Arrow's conditions. I think it is best to delete the citation (after all, very few professional works mention Rv). But doing so would not be very effective, since someone will add the same thing later anyway. A compromise solution is to retain the citation to Range voting, but clarify what it means for Rv to satisfy the conditions. For most readers, it is enough to know that "Whether such a claim is correct depends on how each condition is reformulated." For someone who cares about Rv, (supposing academic sincerity) it is important to know which formulation of IIA Rv violates. (If the footnote is deleted, then they will request the statement "Whether such a claim is correct depends on how each condition is reformulated" be removed, because it is unfounded. The result is that most reader will incorrectly think Rv satisfies all of Arrow's conditions. I think that is not a desirable situation.)

Let's not hide the fact that proposed solutions like Rv do not actually satisfy all of Arrow's conditions. By being academically sincere, I think more professionals will begin to link to the article and contribute to it. That should be good news to the supporters of Rv in the long run.--Theorist2 (talk) 13:47, 23 October 2010 (UTC)

I have replaced the footnote that Osndok complained about. There was nothing original in it, but it seems easier just to mention the well known fact from the well known source (Sen) than to find an exact source supporting the particular example in the footnote. The problem resolved.--Theorist2 (talk) 09:55, 24 October 2010 (UTC)

Can I take this issue up again? Range voting does not satisfy Arrow's IIA or Samuelson's cardinal version of IIA - and nor does any mechanism using cardinal utility. Kalai and Schmeidler (1979) demonstrate this rather clearly. Every cardinal preference is also an ordinal preference - since it expresses a ranking over outcomes. Hence, mechanisms that make use of cardinality are still subject to Arrow's theorem. (There is nothing in Arrow's theorem that requires preferences to be ordinal-but-not-cardinal.) Here is a counter-example: Suppose there are 3 outcomes - A,B,C and 3 agents (with utilities u,v,w, respectively). The cardinal preferences are in profile 1 are given by: {u1(A,B,C,D)=(9,3,0,5), v1(A,B,C,D)=(0,2,1,3) and w1(A,B,C,D)=(8,3,2,1)} whilst preferences in profile 2 are given by: {u2=(3,1,0,9), v2=(0,10,5,6) and w2=(8,3,2,1)]. By Range voting, the social preference according to the first profile is A>D>B>C, whilst the social preference by the second profile is D>B>A>C. Note that w is the same in both profiles, and u and v are cardinally equivalent over the subset {A,B,C} in both profiles - i.e. u1(x)=3*u2(x) and v1(x)=0.2*v2(x). Then by IIA, Range voting should rank A,B and C in the same way - but it doesn't. Finally, the "weakened" notion of IIA in footnote 28 is surely not helpful. Consider two preference profiles in which all but agent 1 have the exact same utility and agent 1's utility is different only in that his utility in the second profile is twice his utility in the first profile. Clearly the two profiles are identical - even agent 1's preferences are exactly the same. You would hope that the social choice axioms would say that the social choice function must choose the same social ranking under both profiles. But the weakened notion of IIA does not require this. It is weak indeed! --Gparames (talk) 21:15, 25 March 2013 (UTC)

Every cardinal utility expresses an ordinal preference, but in translating a utility function into a preference, you lose certain information. A social welfare function in Arrow's sense cannot use that lost information. In other words, Arrow's theorem does require preferences to be ordinal-but-not-cardinal. The definition of a social welfare function does that. Mechanisms (like Range voting) that make use of cardinality are generally not a social welfare function in Arrow's sense. So to deal with such mechanisms, you need to redefine IIA. Provided that IIA is defined so that only ordinal information is taken into account (as you like), it is correct to say Range voting violates IIA. However, many people prefer defining IIA in the weaker sense, where cardinal information is also taken into account. That way, the redefined IIA can reflect the strength of preference, treating (u(x), u(y), v(x), v(y)) = (1,0,0,10) and (10,0,0,1) differently.Theorist2 (talk) 00:09, 26 March 2013 (UTC)

Another problem here is the article claims that the Gibbard–Satterthwaite theorem applies to range (score) voting. But the Gibbard–Satterthwaite page says it only applies to voting systems “where each voter ranks all candidates in order of preference”. In other words, G-S only applies to ordinal ranking systems, which means it does not apply to cardinal rating systems like score voting. Qaanol (talk) 02:23, 2 February 2014 (UTC)

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Dr. Hillinger's comment on this article

Dr. Hillinger has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

Arrow's theorem is about the agggregation of orderings. No voting procedure in actual use is of this form. They are all in one way or other cardinal. The importance of Arrow's theorem is therefore in my opinion much overrated.

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

We believe Dr. Hillinger has expertise on the topic of this article, since he has published relevant scholarly research:

• Reference : Hillinger, Claude, 2004. "Utilitarian Collective Choice and Voting," Discussion Papers in Economics 473, University of Munich, Department of Economics.

ExpertIdeasBot (talk) 17:35, 26 July 2016 (UTC)

In what sense are ordinal voting systems actually cardinal? 71.167.64.83 (talk) 04:46, 2 September 2016 (UTC)

"Cardinal voting electoral systems convey more information than rank orders"

A ranked ballot with n options can be filled out in n! different ways

An approval ballot (a type of cardinal voting) can be filled out in 2^n different ways

n! > 2^n for all n > 3

So the statement "cardinal voting electoral systems convey more information than rank orders" is not correct in all cases — Preceding unsigned comment added by 202.36.29.253 (talk) 07:16, 8 June 2017 (UTC)

Refimprove template

There are several sections that are entirely unreferenced, including "Pairwise voting" and Approaches investigating functions of preference profiles (which also has essaylike language), other sections have large amounts of unsourced content "Social choice instead of social preference" - please don't remove maintenance templates added by other editors. Seraphim System ^(talk) 05:44, 5 February 2018 (UTC)

Now removed by tagger. Jonpatterns (talk) 19:19, 9 May 2018 (UTC)

How does Gibbard's theorem "extend" Arrow's theorem?

Near the top of the article:

"Cardinal voting electoral systems are not covered by the theorem, as they convey more information than rank orders. However, Gibbard's theorem extends Arrow's theorem for that case."

What exactly does this mean? The two theorems seem to be unrelated. I note that there is a link in the text that leads to a section later in the article. The section references Gibbard's theorem once:

"Note that Arrow's theorem does not apply to single-winner methods such as these, but Gibbard's theorem still does: no non-defective electoral system is fully strategy-free, so the informal dictum that "no electoral system is perfect" still has a mathematical basis."

Is this the claim that is being extended by Gibbard's theorem? If so, then what does "strategy-free" mean in this context, and how does it relate to Arrow's theorem? I think I understand what strategy-free means in the context of Gibbard's theorem, but I don't understand how that meaning is relevant here. Douglas Cantrell (talk) 03:32, 20 August 2020 (UTC)

I now notice that the article on Gibbard's theorem contains the following claim:

"Gibbard's theorem can be proven using Arrow's impossibility theorem."

This seems impossible, unless something beyond Arrow's theorem is used in the proof, because Arrow's theorem does not apply to cardinal voting systems. But supposing the claim is true, the language in this article seems confusing. A corollary might prove something about a new domain, but I would not say that it extends the original theorem for the new domain. That phrasing seems ambiguous. Douglas Cantrell (talk) 03:59, 20 August 2020 (UTC)

"Unanimity criterion" listed at Redirects for discussion

A discussion is taking place to address the redirect Unanimity criterion. The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 September 16#Unanimity criterion until a consensus is reached, and readers of this page are welcome to contribute to the discussion. _signed, Rosguill ^talk 15:34, 16 September 2020 (UTC)

Dubious

This article states that Arrow doesn't apply to range systems, but that Gibbard–Satterthwaite still does.

Yet Gibbard–Satterthwaite theorem says right in the first sentence that it only applies to ranked systems, and the range voting website also says that Gibbard–Satterthwaite doesn't apply to it. http://rangevoting.org/GibbSat.html

"Wait a minute. Doesn't Range voting (in the ≤3-candidate case) satisfy all GS criteria, accomplishing the "impossible"?! Huh? ... How can this be? The explanation is simple. The Gibbard-Satterthwaite theorem only applies to rank-order-ballot voting systems." 71.167.64.83 (talk) 04:43, 2 September 2016 (UTC)

Both of those are based on using Satterthwaite's version of the theorem. Gibbard's version is more general, and applies to any simple (one-step) multiplayer game type where the number of players, possible player actions, and possible outcomes are all finite and more than 2. (I say "game type" rather than "game", because each player's payoffs for each outcome must be allowed to vary.) It states that if there is a single optimum "honest" move for each player independent of what the other players do, then the game is a dictatorship. Thus, this version applies to any voting rule, whether rated, ranked, or other. Homunq (࿓) 18:23, 18 October 2016 (UTC)

The Range Voting web site is an advocacy Web site by a bunch of highly-invested, non-neutral players. It's like Shell Oil telling you fossil fuels don't cause global warming. John Moser (talk) 02:32, 9 February 2021 (UTC)

Generalization

I came across this paper by Saharon Shelah: https://arxiv.org/pdf/math/0112213.pdf

I don't understand it at all, but maybe it is relevant here. 2601:648:8200:990:0:0:0:F1B9 (talk) 03:38, 2 February 2023 (UTC)

Arrow's impossibility vs. Condorcet paradox?

What's the difference between Arrow's impossibility theorem and the Condorcet paradox?

Arrow discusses this on pp. 109-111 of Kenneth Arrow, Social Choice and Individual Values, Wikidata Q4227976. Sadly, I don't perceive I have the time to study this well enough to write something sensible about that to add to this article, but I think such an addition would improve the value of this article. Similarly, I think the article on the Condorcet paradox would benefit from a companion mention of Arrow's impossibility theorem. I hope someone else more familiar than I am with these issues will find the time to make such additions. Thanks, DavidMCEddy (talk) 06:54, 25 December 2021 (UTC)

2 years late but I think my recent overhaul should clarify—Condorcet's paradox is basically the cause of Arrow's theorem. Closed Limelike Curves (talk) 01:13, 26 March 2024 (UTC)

Without Condorcet's paradox, any tournament solution would satisfy IIA. (Very often, tournament solutions do satisfy IIA—definitely upwards of 90% of the time, probably more like 95%–99%.) Closed Limelike Curves (talk) 01:15, 26 March 2024 (UTC)

Rank-order voting and the definition of Arrow's theorem

     Arrow did not define his theorem in terms of rank-order voting. The theorem is about choosing an aggregate preference hierarchy from the collection of preferences that exist among the members of a social group. It is not necessary for preferences to be expressed as ordinal rankings for the theorem to apply. You cannot get around the Impossibility Theorem by replacing individuals' rank orderings with something else. None of Arrow's four criteria require that information about individual preferences come in the form of ranks, and it is not possible for any system, based on ranks or not, to satisfy all four of Arrow's criteria. For example, contrary to claims in the current draft of this article, Arrow's theorem applies to range voting and approval voting.

     I am concerned that individuals striving to promote certain voting systems have badly damaged this article. I think it was less flawed three or more years ago, before the unsupported claims about rank-order voting were added. Redefining the Theorem in terms of rank-order voting was a serious error committed by someone who presented no evidence that the previous definition was wrong or that his/her definition in terms of rank-order voting was right.

     Furthermore, as currently written, the article might lead readers to believe, falsely, that rank-order voting methods have a special flaw, revealed by Arrow, and because of that, we ought to avoid such methods. — Preceding unsigned comment added by Mbmiller (talk • contribs) 20:36, 22 April 2015 (UTC)

@Mbmiller: Can you explain how it applies to range voting or approval voting? This also says that it doesn't: http://rangevoting.org/ArrowThm.html 71.167.64.83 (talk) 04:48, 2 September 2016 (UTC)

If voters are allowed to rank outside a specified range, they have infinite votes, thus theoretically every voter is the dictator. As such, ranking A≻B≻C requires scoring A higher than B, and B higher than C. If a voter finds A as 1.0, B less than 1.0, and C less than B, then to move to the order C≻A≻B requires the voter to reduce their score for A. The second axiom says the voters's preferences between X and Y must remain the same, thus so long as A has a score higher than B, the group's preference should be that A has a higher score than B; however, consider A=1.0, B=0.1, C=0.0, with the scores being A=10, B=9.9, C=0. If the ballot becomes C=1.0, A=0.8, B=0.08, then the new scores are A=9.8, B=9.88, C=1.0. In both cases, B is rated as 1/10 the welfare as A; and the order of preference remains A≻B. The irrelevant C is promoted, and the winner changes from A to B. John Moser (talk) 02:22, 9 February 2021 (UTC)

Arrow's theorem assumes the social ordering function takes a vector of rankings as an input. It then gives a single ranking as output. This definition excludes cardinal systems (which have a matrix of numbers as input).

> If voters are allowed to rank outside a specified range, they have infinite votes, thus theoretically every voter is the dictator.

A voting system is dictatorial if, for any possible combination of votes by citizens who are not dictators, the output is still always equal to the preferences of the dictator (which are held fixed); I think you mixed up the quantifier. Closed Limelike Curves (talk) 01:28, 5 April 2024 (UTC)

Spoiler effects and IIA

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)

I'm not sure that I understand the distinction. Suppose under a given voting system candidates A, B, and C are running and A wins, but B would have won if D joined. This seems to be your second ("stronger") type. But if we rephrase it as A, B, C, and D are running and B wins, but then if D drops out A wins, this seems to be your first type. - CRGreathouse (t | c) 19:19, 29 March 2024 (UTC)

If we're dealing with ordinal voting, then the two types are the same. But here's an example for Range voting:

Suppose that the three candidates are A, B, and C. We have three blocs of voters.

The first bloc has 31 voters who score A 10, B 5, and C 0.

The second bloc has 28 voters who score B 10, C 7, and A 0.

The third bloc has 27 voters who score C 10, A 3, and B 0.

In total, A has 391 points, B has 435 points, and C has 466 points. So C wins. Now if we remove A from the input data, then C still wins, because that doesn't change the other candidates' scores: Range passes IIA. But for that to translate to an immunity to spoilers, we must presume that in a counterfactual election where only B and C ran, the scores would be like this:

31 voters score B 5 and C 0

28 voters score B 10 and C 7

27 voters score C 10 and B 0

But it's quite possible that if only two candidates had run, the voters would have ranked their favorite 10 and their least favorite zero. In this other universe, A never ran, so the voters might not have been rating according to the same scale. And if the voters instead had voted their favorite maximum and least favorite minimum, we would have got:

59 voters score B 10 and C 0

27 voters score C 10 and B 0

which would have led B, not C to win. Hence A not running changes the winner from C to B, because the second bloc detests A so much that it makes C look better compared to B. In this stronger sense, elections conducted with Range might have spoilers, even though the method itself passes IIA when looked at in isolation.

The direction (going from the two-candidate election to three candidates or vice versa) doesn't play into it, just whether the voters have an "objective" scale they can rate the candidates on, in the sense that their scaling of scores of existing candidates don't change when candidates enter or leave. Wotwotwoot (talk) 21:33, 29 March 2024 (UTC)

I grant that there are normalization issues with cardinal voting systems, but Arrow is just about ordinal system, right? - CRGreathouse (t | c) 17:19, 30 March 2024 (UTC)

User:Mbmiller seems to think otherwise, but to my knowledge, Arrow is indeed just about ordinal systems. Since Arrow is inapplicable, cardinal methods are allowed to pass IIA and some do.

But currently parts of the article are phrased so as to equate passing IIA with eliminating spoilers (e.g. the section titled "Eliminating spoilers: Rated voting"). I think it would be useful to somehow make the distinction more clear so that "passing IIA" doesn't get interpreted as "spoiler immunity". Perhaps in a section describing the relation between IIA and spoiler immunity, or a reference to the spoiler effect article. Wotwotwoot (talk) 19:45, 30 March 2024 (UTC)

I've taken a stab at doing this. I think some of the problem lies in that information about different related effects (IIA, Arrow, and spoilers) are duplicated or belong to the wrong article, and some cleanup is needed. The "Non-Arrovian spoilers" section might be more suited to the spoiler effect article, for instance. Wotwotwoot (talk) 20:59, 30 March 2024 (UTC)

@Wotwotwoot I'm happy to mention tactical/strategic spoilers, but—

1. From what I can tell, the term spoiler typically refers to IIA failures. This aligns with the original use of the term, which is taken from sports: there, the term refers to one team eliminating another, despite not being eligible to win themselves. (This also seems like a much more common cause of IIA failures than tactical ballot-switching.) I think the average reader is going to be smart enough to understand that, if voters change their votes, this will change the results.
2. I'd like to keep discussions of tactical spoilers separate from Arrovian spoilers as much as possible; this includes keeping digressions into how tactical voting or human psychology can affect IIA below the fold, or ideally in a different article (with links coming from this page and footnotes).

Mostly I just want to focus on teaching readers one concept at a time and providing more detail in footnotes or linked articles. Trying to discuss both concepts at once is going to create a mess in the average reader's mind. Closed Limelike Curves (talk) 02:34, 2 April 2024 (UTC)

In particular, I really want to avoid the excessive detail that tends to plague Wikipedia voting systems articles. I remember trying to read through these articles maybe 10 years ago as a kid and I had no idea what any of it actually meant. (I've asked friends and family to try and tell me if they can understand what the page is saying; so far, this is the first draft they seem to actually understand, as long as they skip the proofs.) Closed Limelike Curves (talk) 03:09, 2 April 2024 (UTC)

If your concern is simplicity, I'm perfectly happy to remove everything that talks about non-straightforward (strategic) voting, or replacing it with a simple paragraph that says "clearly, under strategy all bets are off". However, that is not all I was doing. My main point is that a straightforward reading of "IIA = spoiler effect" conceals the fact that honest voters may cause a spoiler effect in cardinal methods by not operating on an absolute scale, so that the addition of losing candidates changes the scale they use, making someone else win.

It's important to note that this can happen even if we hold the voters to their von Neumann-Morgenstern utilities, because these are only unique up to a positive affine transformation. The spoilers thus don't need to be tactical spoilers, because even honest voters who don't care about how other people are voting can cause the winner of cardinal methods to change.

Arrow makes the same point in his book "Social Choice and Individual Values" in Chapter III, section 6, "The Summation of Utilities". So if a non-Arrovian spoiler would be relevant to this page, it would be this effect. But I think it would make more sense to just briefly say "there exist non-Arrovian spoilers", move the bulk of the material to the spoiler effect article, and let the Arrow's theorem article (this one) be focused on Arrow's theorem itself. That's why I deleted the material here. If we want to keep it simple, let the spoiler article discuss or describe what a spoiler means, and this article be about IIA and Arrow's theorem. Wotwotwoot (talk) 17:02, 2 April 2024 (UTC)

"honest voters may cause a spoiler effect in cardinal methods by not operating on an absolute scale, so that the addition of losing candidates changes the scale they use, making someone else win."

This is what I'm confused about. Why would a (rational and honest) voter's utility function change based on which candidates are running? The scale constant is, well, a constant. There's nowhere else in welfare economics where we allow the scale constant to vary when an agent's choices change; I can't claim that an economic intervention that causes utility to drop 10 points in my model is actually good, because the scale/units changed after I passed the policy. Closed Limelike Curves (talk) 15:36, 6 April 2024 (UTC)

Their internal utilities don’t change, it’s how they translate them to a cardinal ballot expression that does. Sen’s extended Arrow theorem’s assumes a property he calls “cardinality” which informally speaking means that the voter or the election system (based on whether we’re using unnormalized or normalized methods) treat positive linear transformations of the "true" utility function equally.

Another way to put this is that utilities of the von Neumann-Morgenstern type aren’t entirely determined. While B running doesn’t change how much a voter likes A, the voter is in both cases (B running or not) confronted with the problem of what honest ballot to cast. That is, the voter has to find some way to supply the missing information that vNM utilities do not provide.

Sen and Arrow then motivate that if the voter chooses to fill in this ambiguity in a way that depends on the candidates running, then we have IIA failure.

That doesn’t prove that they do. But it seems reasonable that a considerable number of voters would vote majority style in a two-candidate Approval election, no matter who the two candidates are: that they would rate their favorite max and their least favorite min. If so, they’re using relative scales, even if these are consistent with their vNM utilities. So the argument arises from the commonsense notion that that’s what voters in a two-candidate Approval election would do. It would seem strange for a voter to show up only to approve both candidates or neither.

There’s also a second argument: if voters use individually absolute scales, e.g. by anchoring 0% to the end of the world and 100% to a utopia, then we would expect to see a lot of ballots who approve of everybody simply because every candidate is better than a 50-50 gamble on either utopia or the end of the world.

Detailed Approval election breakdowns are rare, so it’s hard to say how many do approve of everybody. But the plausibility of the majority voting scenario, combined with that both Arrow and Sen have written about this problem, would make this information relevant, IMHO.

In addition, it seems that some cardinal designers either make relative voting explicit or try to directly counteract it. As an example of the former, the STAR Voting example ballot states "Give your favorite(s) five stars. Give your last choice(s) zero or leave blank". Of the latter, Balinski and Laraki attempt to deal with the problem in devising MJ.

As for the question "why don’t we see a concern for scales changing elsewhere?", I think that in part that’s due to the history of economics (see and the binary approach of pass/fail criteria in voting (I could give references, but this is already pretty long). The addition or removal of irrelevant alternatives is a bit difficult to bring into a questionnaire context, too. Wotwotwoot (talk) 18:28, 9 April 2024 (UTC)

Their internal utilities don’t change, it’s how they translate them to a cardinal ballot expression that does[...] While B running doesn’t change how much a voter likes A, the voter is in both cases (B running or not) confronted with the problem of what honest ballot to cast. That is, the voter has to find some way to supply the missing information that vNM utilities do not provide.

Ahh, OK, I see what you're arguing now. So your argument is voters need to establish some kind of common units? e.g. this problem of reported results when no scale is provided; voters need to figure out a way to choose which units to reply in. (The same way that, if I asked people to tell me how long something is, I'd have no idea if they meant it in millimeters, miles, light-years...)

When rating candidates, voters probably use a mix of approaches. But whatever they're doing, the empirical evidence is it works. If I had to take a guess, one common strategy is probably background comparisons: they're comparing the candidates to other, past winners or candidates (rather than each other). You can see this most clearly in races where voters are unhappy with the choices they have, like in 2016 or 2024. Tons of people say "I don't like Biden or Trump (or any other candidate)"; if I ask them to rate each one on a scale of "Strongly like, somewhat like, ..." or on a scale from 0 to 100, their answers reflect that. This doesn't make sense if voters can only establish a common scale by comparing candidates to each other, but it does make sense in the context of background comparisons: people are saying they're less enthusiastic about Biden in 2024 than Obama in 2012.

Alternatively, people could be approximating some kind of implicit utilitarian algorithm, which allows them to infer utilities from a combination of other people's ordinal preferences and their observed behavior over lotteries. Because we have a lot of experience interacting with other people, and often seeing the same ordinal preferences in many situations, we can infer their cardinal preferences with a high degree of accuracy (far better than could be inferred from rankings of just 5 or 6 candidates).

The Arrow/Sen argument would probably hold in one-shot elections with no common information being shared between voters except the ballots, but that sounds like an extremely unusual case. Closed Limelike Curves (talk) 17:51, 10 April 2024 (UTC)

That gets to the distinction that I referred to at the end of my previous reply: between the axiomatic criterion approach often used when describing election methods, and a more statistical approach favored elsewhere. The economic paper notwithstanding, a more pragmatic approach has been used in psychology and for analyzing Likert scale responses, questionnaires etc.

All that Arrow's theorem says is that there exist some situations where IIA must be violated. It doesn't say anything about how often these happen, as that would require knowing voter behavior. Perhaps the probability of a Condorcet cycle approaches zero exponentially fast in reality, say. Or perhaps voters do use a fixed anchor all the time, and not just most of the time, so that cardinal voting election outcomes indeed are independent of losers. But statistical reliability and validity can't answer such questions.

Since Arrow's theorem itself uses the axiomatic or deductive approach, it makes sense to stay with it. And in that context, Arrow and Sen say that if we can't ensure that voters use the same scale no matter who runs, we can't guarantee that the outcome is independent of losing candidates.

Whether and how much that applies in reality can of course be debated, just like the question of whether and how often sincere Condorcet cycles would occur. Personally, I think the chance of rated voters voting majority style when only two candidates are present is quite high, even though that's not proof.

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. Wotwotwoot (talk) 16:07, 14 April 2024 (UTC)

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.

The voter can't supply all of their utilities with perfect, absolute rigor, but this is a practical limitation, not a result about social choice. In the same way, we could say that ranked pairs fails the Condorcet criterion because sometimes people will miscount ballots, or make mistakes when filling them out ballots. I mean... sure, I guess that in some sense, you could claim no system satisfies the Condorcet criterion, and Condorcet methods just minimize the probability of a Condorcet failure. Then you could say ranked pairs is "getting off on a technicality" by making assumptions that will never hold in the real world. But that would be silly. It'd be like complaining my calculator gives wrong answers whenever I feed it the wrong numbers. If you give the social choice function the wrong utilities, the wrong answer will come out.

It's reasonable to mention interpersonal comparisons of utility are difficult and philosophically contentious, since we don't have direct access to other people's mental states. But people changing their preferences or utility functions in response to changing the set of options doesn't really work within the mathematical framework of welfare economics, mostly because as soon as you change the set of outcomes, you can never compare any policy or decision to any other.

A social welfare function is just a mathematical function that satisfies a few properties, and it has interpretations other than as a set of ballots (in fields like ethics, economics, decision theory, and multi-criteria optimization). Decisions where these algorithms depend on irrelevant alternatives are sometimes called spoiled as well. The mathematical core of Arrow's theorem is just that, if you apply the rank transformation to an aggregation function, the rank-order of your output will depend on irrelevant changes in preferences. –Maximum Limelihood Estimator 18:41, 14 April 2024 (UTC)

There's a critical difference, though. There exists a definition of a single honest vote for ranking - the one consistent with one's preferences. 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. Sen thus points out that under the ambiguity that results, we can't ensure IIA.

So when I say "their method is asking for information that the voter can't supply", I mean that literally. There's no way to go from vNM utilities to a rated ballot in an unambiguous way. That's why the analog to someone misfiling their ballot fails. Sen's theorem doesn't assume that there are random perturbations or errors, just that what attempts the voter makes to render their utility is impaired by the invariance to positive linear transforms. Wotwotwoot (talk) 19:07, 14 April 2024 (UTC)

To condense the point: with ranking we can set up theorems that take an unambiguous representation of the will of the voter and then shows that a method that makes use of this representation can pass (random ballot) or fail (minimax) IIA. But what unambiguous representation of the will of the voter can we make for rated voting? If we model the ambiguity by creating an equivalence class of sincere representations, then IIA failure returns. Wotwotwoot (talk) 19:36, 14 April 2024 (UTC)

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.

Harsanyi and Vickrey showed the opposite; there's a specific lottery that pins down the comparison, which is preferences over the lottery of birth. If we ask someone how much money it would take for them to accept a 50% chance of swapping lives with another person, their answers imply interpersonal comparisons of utility.

Importantly, these don't imply the interpersonal comparison of utility would be the same across people, so that doesn't get us to a single "objectively correct" set of weights for interpersonal comparisons of utility. Two people with the same life might disagree about how much money they'd need to take that bet. But they do need some set of weights they use to compare their own utility against others', which has to be consistent across different situations. –Maximum Limelihood Estimator 02:49, 16 April 2024 (UTC)

With all the editing that has been going on, let me argue why the parts about generalizations deserve more space in this (Arrow's theorem) article, and why the latest edits in particular have decreased the quality of the article, because it seems like we got a bit sidetracked.

Arrow's theorem does not make a judgment of any nature on methods that are not ranked. It is relatively well known that rated methods pass IIA when the ballot data is held fixed, although this is not a consequence of Arrow's theorem as such.

There exist generalizations of Arrow's theorem that show how elections conducted with rated methods can fail IIA; and more generally, how methods that try to deal with the whole equivalence class of honest cardinal ballots under the assumption of no absolute scale, can fail.

Since Sen's result is a generalization of Arrow's theorem, it is relevant to an article that deals with Arrow's theorem. Furthermore it brings context to how the limited scope of Arrow's theorem have been expanded. The reference showing that Arrow informally made a similar point that with normalization serves as a verifiable background point of context, and shows that Arrow was considering the same problem, even if he didn't formalize it. Whether Arrow currently thinks that rated ballots are better than ranks has less relevance, because he does not directly refer to the point made there.

Further points could be made that authors of cardinal methods are aware of the ambiguity of rated ballots - in particular I'm thinking of things like mean-utility Approval thresholding guidelines - but that might be out of scope for an article about Arrow's in particular.

Cardinal and ordinal advocates may disagree about whether and how often the voters engage in this behavior. Vickrey's observation could be used to argue that the voters making use of it now have a point to fix their scales to, hence avoiding the problem.

But there are, to my knowledge, no sources showing what models voters empirically use (apart from a single section in an Approval paper).

Thus, due to the relevance of extensions of Arrow's theorem to the subject of the theorem and how it may be used, I think that there should be a point in the lead about how, though rated voting is not covered by Arrow's theorem, it is covered by generalizations of it. Reverting this point removes information that is relevant in an article about Arrow's theorem. Reducing, to a few sentences, the distinct point that consequentially there exist models where rated voting fails IIA (and ones where they pass) reduces the quality for a similar reason.

Better would be to have a section about generalizations as such, which the brief mention in the lead would serve as an introduction to.

The question of whether the voters actually have a way to unambiguously fill in a rated ballot, and about vNM on the one hand and solutions like Vickrey's on the other, like what we've been discussing above, could be detailed in that section. Or in some other article, depending on how relevant it is to Arrow's theorem as such. Wotwotwoot (talk) 18:07, 20 April 2024 (UTC)