Talk:Arrow's impossibility theorem

first sentence

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

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

From Dr. I.D.A.MacIntyre

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

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

--86.128.143.185

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

From Dr. I. D. A. MacIntyre.

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

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

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

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

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

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

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

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

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

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

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

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

---

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 [email protected] 27th April 2007

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)

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)

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 [1], 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)

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)