Talk:Arrow's impossibility theorem

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first sentence[edit]

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[edit]

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

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

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

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

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

@Mbmiller: Can you explain how it applies to range voting or approval voting? This also says that it doesn't: http://rangevoting.org/ArrowThm.html 71.167.64.83 (talk) 04:48, 2 September 2016 (UTC)[reply]
If voters are allowed to rank outside a specified range, they have infinite votes, thus theoretically every voter is the dictator. As such, ranking A≻B≻C requires scoring A higher than B, and B higher than C. If a voter finds A as 1.0, B less than 1.0, and C less than B, then to move to the order C≻A≻B requires the voter to reduce their score for A. The second axiom says the voters's preferences between X and Y must remain the same, thus so long as A has a score higher than B, the group's preference should be that A has a higher score than B; however, consider A=1.0, B=0.1, C=0.0, with the scores being A=10, B=9.9, C=0. If the ballot becomes C=1.0, A=0.8, B=0.08, then the new scores are A=9.8, B=9.88, C=1.0. In both cases, B is rated as 1/10 the welfare as A; and the order of preference remains A≻B. The irrelevant C is promoted, and the winner changes from A to B. John Moser (talk) 02:22, 9 February 2021 (UTC)[reply]

Dubious[edit]

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

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

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

Both of those are based on using Satterthwaite's version of the theorem. Gibbard's version is more general, and applies to any simple (one-step) multiplayer game type where the number of players, possible player actions, and possible outcomes are all finite and more than 2. (I say "game type" rather than "game", because each player's payoffs for each outcome must be allowed to vary.) It states that if there is a single optimum "honest" move for each player independent of what the other players do, then the game is a dictatorship. Thus, this version applies to any voting rule, whether rated, ranked, or other. Homunq () 18:23, 18 October 2016 (UTC)[reply]
The Range Voting web site is an advocacy Web site by a bunch of highly-invested, non-neutral players. It's like Shell Oil telling you fossil fuels don't cause global warming. John Moser (talk) 02:32, 9 February 2021 (UTC)[reply]

"Cardinal voting electoral systems convey more information than rank orders"[edit]

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

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

n! > 2^n for all n > 3

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

Refimprove template[edit]

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

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

How does Gibbard's theorem "extend" Arrow's theorem?[edit]

Near the top of the article:

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

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

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

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

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

"Unanimity criterion" listed at Redirects for discussion[edit]

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

Arrow's impossibility vs. Condorcet paradox?[edit]

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

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

2 years late but I think my recent overhaul should clarify—Condorcet's paradox is basically the cause of Arrow's theorem. Closed Limelike Curves (talk) 01:13, 26 March 2024 (UTC)[reply]
Without Condorcet's paradox, any tournament solution would satisfy IIA. (Very often, tournament solutions do satisfy IIA—definitely upwards of 90% of the time, probably more like 95%–99%.) Closed Limelike Curves (talk) 01:15, 26 March 2024 (UTC)[reply]

Generalization[edit]

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

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