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 n_mcntyr@yahoo.co.uk 27th April 2007

Linear orderings and permutations

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

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

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

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

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

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

Unrestricted domain

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

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

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

statement in intro vs body

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

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

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

Archiving

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

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

Non-imposition

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

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

Informal Proof

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

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

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

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

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

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

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

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

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

Doesn't "dictator" just reduce to "there must be at least one voter whose preferences are exactly the same as society" in this case? And why is that bad? No one person can say "_I_ am going to be the 'dictator'" in this case, because who happens to to be the "n:th" voter when "the switch is flipped" in the outcome is totally arbitrary.

External Links

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