Talk:Arrow's impossibility theorem

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)

---

I removed the link

• A lecture on Arrow's impossibility theorem

because it is dead.

• Thank you.

---

"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.

---

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. :)

---

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.

---

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.

---

"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)

---

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) .

first sentence

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

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

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)

From Dr. I.D.A.MacIntyre

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

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

--86.128.143.185

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

From Dr. I. D. A. MacIntyre.

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

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

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

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

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

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

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

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

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

Thirdly, and if this is what is getting up the other editos noses then leave just this out because it is most important that everyone stops being fooled about MAJORITY VOTING by Arrow's theorem and his Nobel driven prestige, anyone who thinks Arrow has a point has been led astray. If US academics and advisors believe he has then why do we bomb countries for not being demcracies?

And if no one does then why was the theorem taught unanswered to Iranian students here in the UK during the year of the fall of the Shah? If Iran is not a demcracy to your liking, I am speaking to the other editors, a good part of the reason is the theorems you are protecting, I can assure you. No one can be Janus faced about this. paricualrly not by suppressing solutions to the Theorem in a dictatorial way.

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

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

---

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

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

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

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

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

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

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

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)

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)

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)