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

The main problem with this article

… is that it presents the theorem as being exclusively about voting systems. But that’s neither how Arrow or standard texts on the subject characterize it. Arrow himself, in his famous paper, right of the bat mentions market exchange as an example of aggregation of individual preferences into social outcome. Look at how Stanford Encyclopedia of Philosophy approaches it [1]. The word “voting” doesn’t appear until the fourth paragraph in the very specific context of Condorcet’s Paradox. The whole point - as SEoP makes abundantly clear - of the impossibility theorem is that it’s NOT JUST voting (specifically majority voting) that is subject to anomalies like that of Condorcet, but *social choice* in general.

Presenting this subject as just about voting is both misleading to the reader and does quite an injustice to a very important, even fundememtal, result. Volunteer Marek 00:56, 23 May 2024 (UTC)

If you'd like to add more discussion of the social choice perspective, be my guest! There's a close relationship between voting and social choice—Arrow often referred to his theorem as being about either "social choice" or "voting" interchangeably—but I focused on voting because it's more concrete and easier to understand. –Sincerely, A Lime 18:22, 28 May 2024 (UTC)

Voting is just one way that society can make choices. Market exchange is another. The point of the theorem is to treat social choice at a highly general level.

I appreciate that different folks come to this subject from different backgrounds. At the same time we need to be aware of that and not let these backgrounds skew the presentation of the subject. The current problem is that the present version is SOOOOO skewed towards a particular version that it would truly be a great task to rewrite it appropriately. Volunteer Marek 04:57, 9 June 2024 (UTC)

I think I've improved on this.

Although, thinking about it more, it seems to me like Arrow's theorem—unlike other theorems of social choice—is in practice limited to voting. Markets etc. rarely (if ever) rely on pure ranking data; there might be a few situations where monetary transfers are prohibited like organ-matching, but generally social choice involves comparisons of utility. –Sincerely, A Lime 16:39, 10 June 2024 (UTC)

I think this article is much more clear for talking about voting in the lead instead of immediately plunging into the phrase "aggregation of individual preferences into social outcome". I am sympathetic to OP's view, but we must remember that Wikipedia has a pretty different readership than SEP. Mathwriter2718 (talk) 14:17, 18 July 2024 (UTC)

Non-nullity defined incorrectly?

As described here, a null voting system would be one that has an a priori ordering of all candidates, and always returns that ordering regardless of the votes. But there are other voting systems that do not meet this definition but still obey IIA. Here's one:

• Use an a priori weak order of the candidates, in which (among the entire field of potential candidates) each candidate has at most one other candidate with whom they are tied.
• Return a linear extension of this weak order, resolving ties between pairs of tied candidates by majority vote.

For a natural example of this, consider a voting system that always chooses the majority winner between the candidates from two major parties, and then lists the third parties in alphabetical order. There can be no spoilers, because they cannot affect the majority-vote tie-breaking system and nothing can affect the other comparisons. On the other hand, there are plenty of pairs of candidates for whom the voters are ignored. I think maybe the correct formulation of non-nullity is: for every two candidates, both outcomes are possible. —David Eppstein (talk) 08:58, 9 June 2024 (UTC)

The redefinition you proposed seems to be Wilson's weakened form of the citizen sovereignty (onto) condition, which he drops in the last section, but I think your counterexample is correct (which means I'm missing a condition somewhere). Closed Limelike Curves (talk) 18:42, 12 June 2024 (UTC)

Score voting

I am bothered by this article's blatant advocacy of score voting both in the lead and in the "Eliminating IIA failures: Rated voting" section, for three reasons:

1. it is off-topic.
2. I am not convinced it is neutrally presented.
3. In my personal experience (having seen this system in action in certain polarized committee votes) it is a very bad system, not because it can be gamed (all systems can be gamed) but because it is so blatantly obvious that it can be gamed as to put any honest participants at a severe disadvantage. Participants willing to game the system devolve to approval voting, honest participants spread their scores among different candidates, and the approval voters win. If you're going to enforce that voters spread their scores more uniformly you might as well just use Borda, and if you're going to allow approval voting then just use approval voting and put all voters on a more equal footing.
4. Our coverage of this gaming issue dishonestly mixes the two by talking about it as a voting system but then using sources such as Harsanyi that talk about aggregating utility (without opportunity for voters to misprepresent their preferences) rather than scored voting.

—David Eppstein (talk) 19:44, 9 June 2024 (UTC)

I've improved the lead, and will work on improving the rated voting section later. –Sincerely, A Lime 01:29, 10 June 2024 (UTC)

ArXiv Pre-prints are not Reliable Sources

Currently FN10: Holliday, Wesley H.; Pacuit, Eric (2023-02-11), Stable Voting, arXiv:2108.00542, retrieved 2024-03-11 is a link to this arXiv page which does not show a publication. This cannot be considered a reliable source as anyone can post there. Czarking0 (talk) 00:12, 19 June 2024 (UTC)

Per WP:ARXIV, Arxiv reprints are allowed/considered reliable if published by subject matter experts. That said you can also find a publication here:

https://link.springer.com/article/10.1007/s10602-022-09383-9 Closed Limelike Curves (talk) 14:18, 19 June 2024 (UTC)

I upgraded the arXiv item to the journal version. XOR'easter (talk) 02:05, 3 October 2024 (UTC)

Huh, I thought I'd fixed that already, but I guess not. Thank you! :) – Closed Limelike Curves (talk) 17:25, 3 October 2024 (UTC)

Statement of the theorem, NPOV, and sourcing

I am concerned about how the theorem is stated.

1. The lead says that:

No rank-based procedure for collective decision-making can behave rationally or coherently. Specifically, any such rule violates independence of irrelevant alternatives.

This is highly problematic because a) one might not think that IAA is required for rationality or coherentness, b) there are other assumptions in the theorem statement besides IAA. It would be much more accurate to say that ranked-choice collective decision-making procedures cannot simultaneously satisfy several axioms that we intuitively think fair systems should satisfy. We ought to be careful to take an WP:NPOV and avoid making a definitive judgement about whether IAA is required for "rationality" or "coherentness".

2. The theorem statement in this article appears to say:

Total ordering + non-dictatorship + IAA implies contradiction.

It cites Wilson to support this. However, Wilson's paper does not support this!!! The assumptions are different.

3. The section "Intuitive argument (voting)" uses one source, Iain McLean's paper, to support several claims. However, these claims are more hyperbolic than they have a right to be. For example, it says "many authors" take a certain stance, and cites only that McLean takes this stance. It also says:

Given these assumptions, the existence of the voting paradox is enough to show the impossibility of rational behavior for ranked-choice voting.

which I again believe is an NPOV problem.

I think these problems must be resolved before this article can be considered a Good Article. Mathwriter2718 (talk) 14:10, 18 July 2024 (UTC)

So, I'll first mention on the topic of rationality/coherence that in decision and social choice theory, these have a specific meaning, given by the von Neumann–Morgenstern axioms (including IIA); I've tried making that more clear by linking to them. IIA is considered a requirement for rationality because violating it implies your behavior will be self-contradictory (see spoiler effect) and opens you up to Dutch books.

On Wilson, could you explain to me how I've been misunderstanding him? I thought I was missing something but the paper says he drops the assumption that the function is onto. Closed Limelike Curves (talk) 17:58, 18 July 2024 (UTC)

1. You probably know more about this than I do, but my impression is that the von Neumann-Morgenstern independence axiom should not be thought of as "equivalent to" the IIA axiom. If nothing else, the von Neumann–Morgenstern axioms are about individuals, and IIA is about societal aggregation. Even if the idea is the same, the mathematical content and context are quite different, no? There could be other analogues of von Neumann-Morgenstern independence that also seem reasonable to require. For example, the relevant SEP page (linked below) has more than one non-equivalent formulation of IIA. Now, when I read IIA, it seems like a really strong assumption compared to von Neumann-Morgenstern independence. My impression is that some authors resolve the Arrow dilemma by rejecting that IIA is required for rationality, but I've never heard of someone rejecting von Neumann-Morgenstern independence. (To be clear, I'm not endorsing or rejecting this view, I'm just saying what I believe to be the case in the field.) For some evidence besides just my impression, the relevant SEP page ([[2]]) discusses IIA as if it needs justification. For example:

Gerry Mackie (2003) argues that there has been equivocation on the notion of irrelevance. It is true that we often take nonfeasible alternatives to be irrelevant. That presumably is why, in elections, we do not ordinarily put the names of dead people on ballots, along with those of the live candidates. But [IIA] also excludes from consideration information on preferences for alternatives that, in an ordinary sense, are relevant. An example illustrates Mackie’s point. George W. Bush, Al Gore, and Ralph Nader ran in the United States presidential election of 2000. Say we want to know whether there was a social preference for Gore above Bush. [IIA] requires that this question be answerable independently of whether the people preferred either of them to, say, Abraham Lincoln, or preferred George Washington to Lincoln. This seems right. Neither Lincoln nor Washington ran for President that year. They were, intuitively, irrelevant alternatives. But [IIA] also requires that the ranking of Gore with respect to Bush should be independent of voters’ preferences for Nader, and this does not seem right because he was on the ballot and, in the ordinary sense, he was a relevant alternative to them. Certainly Arrow’s observability criterion does not rule out using information on preferences for Nader. They were as observable as any in that election.

2. Maybe I'm the one misunderstanding, so I'll explain my reading of Wilson and we can discuss. I assume the theorem you are referencing is Theorem 3: Every social welfare function is either null or dictatorial. Take any set $S$ and call the set of preferences on $S$ (complete and transitive binary relations) $\Pi$. Then to Wilson, a "social welfare function" is any map $f:\Pi ^{n}\to \Pi$ satisfying two axioms:

I (IIA). If $R,R'\in \Pi ^{n}$ agree on a subset, then $f(R),f(R')$ also agree on that subset.

II. If $x,y\in S$, there exists $R\in \Pi ^{n}$ such that $xf(R)y$.

To summarize, my reading is this: the section Arrow's_impossibility_theorem#Formal statement says

IIA + non-dictatorship $\implies$ contradiction

but Wilson says

IIA + II + non-null + non-dictatorship $\implies$ contradiction.

Mathwriter2718 (talk) 18:56, 18 July 2024 (UTC)

To be clear, I'm not inherently against using the word "rational" or "coherent" to refer to principles such as von Neumann Morgenstern independence that it is widely accepted a rational agent must obey. Instead, I am questioning whether it is really widely-accepted that any social aggregation function violating IIA is incoherent or irrational. Mathwriter2718 (talk) 19:07, 18 July 2024 (UTC)

There's some ambiguity here in what we mean by "rejecting IIA". First, for every widely-accepted axiom there's some fringe philosopher willing to argue against it (same for VNM's IIA).

Second, if it's impossible to behave completely rationally (because you don't have cardinal information), violating IIA becomes second-best and therefore "rational" in a sense. (Assuming you care about >1 person's welfare). If you decide you want to reconstruct the utility function from the orderings, you have to give up IIA. e.g. if you have two ballots, with the first ranking A > 24 candidates > Z, and the second ranking A > Z > 24 candidates, we can't logically prove the 1st prefers A > Z more strongly than the 2nd, but we could reasonably infer it by looking at all of the "irrelevant" alternatives sandwiched between A and Z in the first one. But it would still be better to have the actual utilities for each candidate, so we don't have to use heuristics like that. David Pearce has a wonderful discussion here. Closed Limelike Curves (talk) 20:50, 18 July 2024 (UTC)

I found Pearce's discussion of Gorgias's "On Nonexistence" very amusing. Anyway, by this point maybe we could just find a reputable citation about whether or not IIA is viewed as a necessary condition for coherence/fairness, or just as a possible condition for coherence/fairness one might reject. (To be honest, I find the use of "rational" to refer to a social aggregation function and not an agent a bit strange.)

I am interested to know if you agree or disagree with my reading of this article and of Wilson. Mathwriter2718 (talk) 22:19, 18 July 2024 (UTC)

I believe Wilson says in Section (not theorem) 3 that he drops the assumption of citizen sovereignty (that the SCF is onto), but I'm actually a bit confused, because I'm not sure what he replaces it with. Closed Limelike Curves (talk) 22:51, 18 July 2024 (UTC)

I looked pretty hard at the article again today. I found new discrepancies. A) Wilson talks of complete and transitive binary relations (which he calls preferences and Wikipedia calls total preorders), but the article talks of total orders, which are antisymmetric total preorders. B) Wilson's requirement of non-dictatorship also requires that there is no "inverse dictator" whose preferences are always the exact opposite of those of the function. C) Wilson is extremely fussy about exactly what assumptions imply exactly what conclusions. The article theorem says that stuff implies IIA is violated, but neither Wilson nor the arguments on this page take that logical path. Wilson himself takes the path of IIA and surjectivity implies either null or dictator.

Wilson says in the abstract quite clearly that he drops surjectivity and still proves Arrow's theorem. However, his only relevant theorem (Theorem 5) is simply not the promised result. Perhaps if you do WP:OR, you can see how Theorem 5 gets you the desired result. But I think the prudent thing to do is to not say in this article that you can drop surjectivity. Mathwriter2718 (talk) 13:30, 19 July 2024 (UTC)

Speaking of WP:OR, the proofs of Arrow's result on this page are apparently "simplified versions" of proofs in the literature. I'm not sure if this "simplification" is OR or not. Mathwriter2718 (talk) 13:33, 19 July 2024 (UTC)

@Closed Limelike Curves I agree with many of the changes in your recent edit. But there are some I disagree with, including some reverts you made of my edits.

1. The Arrow quote in the lead: I removed the link to Condorcet paradox because there is already a link to it only a few sentences ago, and it's not clear to me that Arrow was even talking about the Condorcet paradox. Seems more likely he was talking about IIA violations.
2. Removing Voting paradox from See also: this is just a redirect to Condorcet paradox, which is already in the See also.
3. Neutrality does not imply Non-imposition: the null voting method that is indifferent between all alternatives is neutral but not surjective.
4. I am not so sure about calling neutrality a "free and fair election". To me, "free and fair election" means more about how the election is administered, whether or not some candidates are arrested, whether or not everyone in society is allowed to vote, etc. The lead for free and fair elections supports this view:

A free and fair election is defined by political scientist Robert Dahl as an election in which "coercion is comparatively uncommon". A free and fair election involves political freedoms and fair processes leading up to the vote, a fair count of eligible voters who cast a ballot, a lack of electoral fraud or voter suppression, and acceptance of election results by all parties. An election may partially meet international standards for free and fair elections, or may meet some standards but not others.

A social choice function on the other hand doesn't even need to be an election. I feel less sure about calling anonymity "one vote, one value". The slogan "one vote, one value" seems to me to imply that anonymity is somehow counting up votes, when it really just requires the function to treat each voter the same, and the function a priori might not have a natural interpretation in terms of voting. But the page for one man, one vote says it is about "equal representation", which feels right on point with what anonymity is.

It looks from that edit like you agreed with me that it is prudent to not drop surjectivity. In that case, I think I should add the surjectivity requirement to the formal statement (it is the only requirement in the Non-degenerate systems section that is not in the formal statement). Mathwriter2718 (talk) 13:32, 20 July 2024 (UTC)

On the name of the surjectivity requirement: is there a source in the literature that calls this "Non-imposition"? As I'm sure you already know, Wilson just calls this a "weaker version of Arrow's condition of Citizen's Sovereignty", which is not super helpful. I would really like to not come up with our own name for this, but it seems like we have to. I feel like "weak Citizen's Sovereignty" or "surjectivity" are both names that are minimally new, so those are the ones I support at this moment. Mathwriter2718 (talk) 13:39, 20 July 2024 (UTC)

One last concern similar to the one for "non-imposition": is there a source in the literature that defines the term "non-degenerate ranked choice voting systems" as ones satisfying every Arrow hypothesis except for IIA? I couldn't find this term in Wilson or Arrow. I worry it may be an invention of Wikipedia. Mathwriter2718 (talk) 14:02, 20 July 2024 (UTC)

I'm using "degenerate" to mean dictatorships or externally-imposed outcomes, which are kind of like voting rules, but not really. Closed Limelike Curves (talk) 01:42, 21 July 2024 (UTC)

I understand that that is how this page uses the term, but making up a new meaning of a term and presenting it on Wikipedia violates WP:Forum. If this term is not used in the wild outside of Wikipedia or sources citing Wikipedia to describe voting systems, it absolutely cannot be used here. Mathwriter2718 (talk) 03:15, 21 July 2024 (UTC)

Gibbard calls these social choice functions "trivial" here. Closed Limelike Curves (talk) 16:14, 21 July 2024 (UTC)

Gibbard's definition of "trivial" is "dictatorial or two alternatives", which is quite different from "non-imposition + dictatorial". Mathwriter2718 (talk) 01:30, 22 July 2024 (UTC)

Gibbard groups "non-imposition" with "dictatorial", as "outcome must depend on at least 2 players' actions". I'm not sure if Gibbard is describing duples as trivial as well, but either way it's fine; I'm not saying that dictatorships are the only trivial voting rule, just that they're a kind of trivial/degenerate voting rule. (It's also fine because duples are also exempt from Arrow's theorem (they pass IIA trivially—there's no irrelevant alternative to affect the results). Gibbard's conditions here are actually the same as Arrow's because he's using Arrow's theorem as a lemma; it turns out strategyproofness requires IIA. Closed Limelike Curves (talk) 01:58, 22 July 2024 (UTC)

@Mathwriter2718 I think I preferred the original presentation better (grouping nondictatorship and nonimposition as mild background conditions defining voting). That makes it clearer that Arrow's theorem isn't about making tradeoffs between different properties (a misconception I encounter very often). It's just about the impossibility of rational social choice with ordinal rules. Closed Limelike Curves (talk) 02:33, 23 July 2024 (UTC)

Can we compromise on the current version of how the Background section treats this issue as per your most recent edit? To be honest, I still have some concerns about OR and neutrality. But I think the current version as per your most recent edit is acceptable to me, and I think avoiding an edit war is more important than addressing my concerns on that issue.

The mathematical theorem and its proof really do just only say that there exist no functions satisfying all of those properties. I don't think seeing it this way is a misconception. It being "about the impossibility of rational social choice with ordinal rules" is a valid interpretation, but it's not part of the mathematical theorem itself, its about how we interpret it. As I mentioned, some economists reject that IIA is required for "rationality" in this context. You call them fringe, but I am not convinced this is true, and if it is, I would want to know just how fringe. I feel nervous to use Wikipedia, which is in common perception very neutral and reputable, and the word "theorem", which indicates a mathematical truth that we are forced to accept, no matter what, to describe an interpretation (even if accepted by 90% of economists, but disputed by 10%) which has dramatic consequences for politics and public dialogue. The lead of this article subscribes very heavily to your viewpoint on this. Mathwriter2718 (talk) 20:28, 25 July 2024 (UTC)

Right, I see; I think this version is pretty good and I don't want to start an edit war either, but if you feel uncomfortable with it I'm happy to make edits.

I've never heard of an economist who disputes IIA as an axiom for rational choice, but tons of people have confused empirical disputes in behavioral economics about whether it applies to observed human behavior with disputes on its rationality. Like Pearce noted, quite a few economists interpreted Arrow's theorem as a mathematical proof of moral nihilism, because of how important they consider IIA to rational choice. This led a couple philosophers to try and reject IIA, since they misunderstood Arrow's theorem as saying it was either that or nihilism.

An explanation of why economists all accept IIA can be found over at money pump: Offer someone a choice between A, B, and C, and say they pick A. Now offer them the opportunity to switch from A to B if they pay you epsilon dollars (i.e. their choices are A or B - epsilon). They will accept. Offer them the opportunity to switch from B to either A or C, for a fee (choices are A - epsilon, B - epsilon, or C - epsilon); they will choose A - epsilon.

Repeat. The result is a series of decisions that someone claims all made them better-off, yet clearly they're worse-off at the end of the procedure, i.e. This person's preferences are self-contradictory. Closed Limelike Curves (talk) 22:47, 25 July 2024 (UTC)

I thought that the money pump argument is supposed to show us why preferences should be transitive. I'm not sure what it has to do with independence. If it does have to do with independence, surely it is about VNM independence and not Arrovian IIA. I think we should be careful to distinguish between VNM independence and Arrovian IIA; they are very different assumptions mathematically, and it isn't clear that they have entirely the same interpretation. I posted above that the SEP page on Arrow's theorem is much more neutral than this page is on whether Arrovian IIA is justified. A different SEP page, the one on voting methods (https://plato.stanford.edu/entries/voting-methods/), goes further:

[Arrovian IIA] is a very strong property that has been extensively criticized (see Gaertner, 2006, for pointers to the relevant literature, and Cato, 2014, for a discussion of generalizations of this property).

In fact, the article from Pearce you sent ferociously argues that Arrovian IIA should not be interpreted as being required for rationality:

Rather than satisfaction of IIA being a badge of rationality, it is evidence of irrationality.

Reinterpretation of Arrow’s Theorem. If you insist on throwing away critical ordinal information, bad things will happen.

By the "critical ordinal information", he means the information that IIA requires you throw out. In fact, Pearce lists a very large number of IIA critics:

Hildreth (1953) was an early critic of IIA. Arrow objected to the use of preference profile information to make rankings based on interpersonal comparisons, and therefore imposed IIA (Arrow, 1950, pg. 342). Hildreth pointed out that if you write down a nondictatorial social choice function, you have already used preference profile information to make rankings based on interpersonal comparisons; if this is forbidden, there can be no acceptable nondictatorial social welfare functions, and one doesn’t need Arrow’s Theorem to prove it. For other critical perspectives on IIA, see for example Rothenberg (1961), Gibbard (1968/2014), Hansson (1973), Mayston (1974), Bailey (1979), Pazner (1979), Lehtinen (2007), Fleurbaey and Maniquet (2008) and Coakley (2016). Many of these, notably including Rothenberg (1961), propose weakenings of IIA, as do Young (1976) and Maskin (2020). Not all of them are aware of the others’ work. Lehtinen (2007) is more concerned with strategic issues, but his title is on target: “Farewell to IIA”. Arrow himself gradually softened his insistence on IIA: see his remarks in Arrow (1967, pg. 19).

Mathwriter2718 (talk) 23:11, 25 July 2024 (UTC)

I thought that the money pump argument is supposed to show us why preferences should be transitive. I'm not sure what it has to do with independence. If it does have to do with independence, surely it is about VNM independence and not Arrovian IIA.

Dropping Arrovian IIA leads to intransitivity if you can vary the set of options under consideration; IIRC Arrow describes his theorem as a proof of intransitivity in his thesis. In some cases you have A > B > C, but that means removing B as an option gives you C > A, so A > C > A ⇒ contradiction.

Pearce's point (and the point of the researchers he cites) is that IIA violations are second best if you don't have all the relevant (cardinal) information, but you're still committed to making some kind of social choice. In that case, dropping IIA is the least-irrational option (because the "irrelevant" alternatives provide information about cardinal preferences).

I don't think any of these papers dispute IIA should apply to rational choice in the individual case (because, as mentioned, it implies intransitive preferences in the dynamic setting of >1 decision); but if for some reason we don't have access to cardinal information, we have to look for the second-best.

The citation to Arrow mentions this (Arrow describes IIA as a rational choice condition). Closed Limelike Curves (talk) 19:05, 28 July 2024 (UTC)

Sorry for the large volume of posts, but one last thing I just spotted: non-imposition/weak Citizen's Sovereignty/surjectivity is currently defined as "it is possible for any candidate to win", but this is a weaker statement than surjectivity. Mathwriter2718 (talk) 14:07, 20 July 2024 (UTC)

Citation added for the term nonimposition. Closed Limelike Curves (talk) 01:56, 21 July 2024 (UTC)

Your citation for nonimposition defines it differently than either the social choice function being surjective (which is the actual condition Wilson requires) or "it is possible for any candidate to win" (which is what you define nonimposition to mean on the article). Mathwriter2718 (talk) 03:28, 21 July 2024 (UTC)

Striking through my previous comment. My apologies. The non-imposition from your citation defines it the same as Wilson does. However, the characterization in the article of non-imposition is still flawed. I will fix it today. Mathwriter2718 (talk) 14:07, 21 July 2024 (UTC)

I very much agree with these NPOV concerns. The suggested edit in #1, or something along those lines, seems good to me. Gumshoe2 (talk) 18:47, 18 July 2024 (UTC)

@Closed Limelike Curves: I think the current version of the article sufficiently addresses all of the concerns that I had (except for the lead, which I think could be improved but is basically fine). However, I know that you disagreed with some of my concerns, so I want to invite you to voice any concerns you have about the current state of the article, or any changes that I have made that you think should be reverted.

Regarding the lede: as a matter of writing style, it's poor form to use everyday words like "rationally" in the opening line and trust that a link to a different article will clarify that a technical meaning is intended. Indeed, turning "behave rationally" into a link to decision theory is an Easter egg. Throwing a technical endnote into the middle of the first sentence is also a problem. Moreover, pulling one assumption up front when the SEP considers the possibility of dropping each one is out of line with NPOV. XOR'easter (talk) 01:34, 3 October 2024 (UTC)

Hi XOR'easter, thank you very much for your edits! Your contributions have definitely improved the article. :)

Do you know if there's a tool to automatically merge duplicate citations? I'm unsure if/how some should be merged, since some cite different quotes from the same source.

SEP does happen to discuss dropping every assumption of rational choice, but SEP has a bit of an issue in that, because it's a philosophy publication, it has to list anything a philosopher has said about the topic. This means dialatheism and its equivalents (cyclic preferences like A > B > C > A) are technically on the table, since philosophers have defended them; this feels like a bit of a silly caveat to put at the top of the article.

Once you drop that, and note that Wilson's proof shows you don't need to assume Pareto efficiency, Arrow's theorem can be restated as:

Say you have a group that makes a decision using some function ${\mathcal {F}}$, which maps a tuple of rankings to a single ranking. Then, except for the trivial cases—a group of size 0 or 1 makes the decision—you can't satisfy IIA.

I think this is the easiest framing for readers, because it doesn't include digressions into several unnecessary assumptions or trivial cases. Many (most?) discussions of Arrow's theorem put IIA front-and-center because of this. I can provide sources of other people presenting it this way, if you're worried about NPOV. – Closed Limelike Curves (talk) 17:20, 3 October 2024 (UTC)

I don't know of a tool for automatically merging duplicate citations. In this case, I wouldn't trust an automated method anyway, because it'd have to handle cases like citations to different portions of the same document. And hoo boy, over-reliance on automated tools has burned the project badly in the past.

I'm not eager to second-guess an encyclopedia article by saying that it's obviously being non-selective. If a reliable source covers an aspect of the topic, that's a reason for us to cover that aspect of the topic. (There's no "vibes" exception to WP:NOR.) There could well be grounds to emphasize one assumption over the others, but we should be clear that that's what's happening, using language like, "Most importantly, Arrow assumed..." or "The crucial premise is that...". XOR'easter (talk) 18:39, 3 October 2024 (UTC)

I think that sounds like a good compromise; I'll add that. – Closed Limelike Curves (talk) 00:00, 4 October 2024 (UTC)

Perhaps a short paragraph about the assumptions could be slotted into the lede, so that it would go (1) overview and context, (2) assumptions, (3) implications, (4) rated voting and cardinal utilities. XOR'easter (talk) 15:02, 4 October 2024 (UTC)

I have to say, I'm not a fan of dropping "rational choice" into the opening line and trusting that making it a wikilink will clarify its meaning. This article has to be approachable for people who don't instinctively treat "rational" as synonymous with "impervious to being Dutch-booked" or something like that. XOR'easter (talk) 21:18, 8 October 2024 (UTC)

Rebooting GA nom

Confused as to what's left to address before GA status (cc @Mathwriter2718); I've started a new thread since the last one was getting unwieldy.

• Talk:Arrow's impossibility theorem/GA2

Closed Limelike Curves (talk) 02:37, 13 August 2024 (UTC)

@Closed Limelike Curves The article is already had a problem, judging from the tag. Dedhert.Jr (talk) 12:43, 5 October 2024 (UTC)

Yes, having a maintenance banner that is unquestionably still valid is grounds to quick-fail an article. XOR'easter (talk) 21:20, 8 October 2024 (UTC)

The word forced in the intro is not very clear, and nothing in the main text of the article makes it more clear. Moreover, it links to a page, Condorcet paradox, which doesn't explicate the term either (or contain it at all). In general, linking to a page whose title is completely different from the text of the link is a sign that something needs to be reworked. Here, I think, it is hard for anyone who does not already know what the text is trying to say to get a meaning from it. Likewise, the article drops the technical term non-degenerate into the text twice, first as a parenthetical qualification in the opening paragraph and then, without elaboration, in "Basic assumptions". Rather than explaining what a "degenerate" rule would be, the article links to the page on the general concept degeneracy (mathematics), which is mostly about Euclidean geometry and says nothing about voting systems. A savvy reader might guess at what degeneracy might mean in this context, but they shouldn't have to. XOR'easter (talk) 21:46, 12 October 2024 (UTC)

Another matter of vernacular versus technical terminology: many readers will likely find the line runoffs and other perverse voting rules strangely judgmental. (For example, residents of the United States might well look at elections in France, Brazil, etc., and say that whatever happens, it's gotta be less perverse than the Electoral College.) Here, of course, perverse is being used in a technical sense. But should it be so used here? Arrow's original paper doesn't do that. The secondary source, Doron and Kronick (1977), uses perverse in the title but calls it nonnegative association in the text. They say that some writers refer to this condition as the "Non-Perversity" condition, and they observe that other authors use monotonicity instead. On the available evidence, it would be better for us to say something like, the monotonicity assumption, also known as non-perversity, and to use other non-monotonic instead of other perverse. XOR'easter (talk) 22:09, 13 October 2024 (UTC)

Switched to negative responsiveness—you're right that "perversity" sounds judgmental, but on the plus side, it's a word everyone recognizes that accurately conveys the gut reaction a mathematician would have to hearing this. Monotonicity is a very dry, technical-sounding word that very few people are likely to understand.

Negative responsiveness strikes me as a decent compromise. – Closed Limelike Curves (talk) 22:11, 19 October 2024 (UTC)

Political scientists have found such cycles to be fairly rare, likely in the range of a few percent... A few percent of what? The cited source is more equivocal in its conclusions than the summary here: The opinion that instances of Condorcet’s paradox are empirically infrequent or rare is not based on empirical evidence. Only an infinitesimally small fraction of the many committee decision making processes in daily life have been observed. Moreover, the evidence collected in 265 elections shows 25 times the occurrence of the paradox, which gives a frequency percentage of 9.4 %. Clearly, this percentage cannot be neglected. However, the conclusion that the paradox “is all around us” as, e.g., Riker (1980, 1982) and others wrote, cannot be held either. Surely, there are strong theoretical arguments furnished by the theory of spatial majority games for the frequent existence of the paradox. However, the empirical evidence collected so far is casual and mainly ad hoc. This evidence is insufficient either to confirm or to refute the statement that the paradox is empirically relevant. It would be better to report the kind of election which that analysis studied — mainly large elections, up to the national scale, rather than boards of directors and the like — and to give the actual percentage, rather than judging whether that percentage is only "a few". XOR'easter (talk) 16:55, 14 October 2024 (UTC)

Makes some sense, but it's kinda complicated; Condorcet paradox#Empirical studies has more info. Deemen's estimate of 9.4% was the highest one I found, whereas most other estimates are under 2%. This is may be because he includes a broader set of elections (e.g. parliamentary votes) or because Deemen is less systematic in collecting data which could create a notability bias. (If van Deemen's paper was the only one cited here, that's a mistake on my part. I think I cited the other papers elsewhere.) – Closed Limelike Curves (talk) 21:04, 14 October 2024 (UTC)

"A few percent" sounds too high for some of the figures quoted at Condorcet paradox#Empirical studies (e.g., 0.4%), while also sounding too low for Deemen's number. Of course, anything like "how many is 'several'?" will be a matter of taste, but by that same token, we shouldn't risk giving readers the wrong impression by using words that suggest different ranges to different people. XOR'easter (talk) 22:08, 15 October 2024 (UTC)

Yeah, that makes sense. I'll try and think of some better way to describe this, thanks! – Closed Limelike Curves (talk) 22:03, 19 October 2024 (UTC)

This article needs to be substantially more technical and substantially less political

Arrow's Impossibility Theorem is a mathematical result subject to the rigor of any other mathematical result. As such this article should look more like e.g. Analysis of Boolean functions (which in fact contains a backlink to this article) than it should look like a POV editorial about how to decide what a "good" voting rule is.

https://arxiv.org/abs/2008.08451 will be a good source. It contains insightful and technical discussion of various forms of IIA. Affinepplan (talk) 17:05, 21 October 2024 (UTC)

Just expanding that reference out for convenience: Holliday, Wesley H.; Pacuit, Eric (2021). "Axioms for Defeat in Democratic Elections". Journal of Theoretical Politics. 33 (4): 475–524. arXiv:2008.08451. doi:10.1177/09516298211043236.

Can you point to specific passages that in your view are off-topic and/or overly editorial? XOR'easter (talk) 21:36, 21 October 2024 (UTC)

• satisfy the requirements of rational choice theory.

"Rational choice theory" is a model of behavior and does not have "requirements"

• [...] are highly sensitive to spoilers, particularly in situations where they are not forced.

Probably true in the case of plurality, probably not true in the case of IRV, but in either case is unrelated soapboxing

• the entirety of footnote 18

a random monologue criticizing "modern economic theory" 's use of ordinal utility has no place in an article about a specific theorem

• The goal of social choice theory is to identify a social choice rule, a mathematical function that determines which of two outcomes or options is better, according to all members of a society

no, the goal of social choice theory is to study and understand social choice. not to determine what mechanism is normatively "better"

• any non-degenerate (i.e. actually usable)

Either define degenerate (mathematically), or leave just the link as is. but "usable" is a subjective term editorialized in.

• While Arrow considered it an obvious requirement of any proposed social choice rule, ranked-choice voting (RCV) fails this condition.

this would need citation specifically to the fact that Kenneth Arrow considered this "obvious" ... except it has no place in this article anyway. it's just more soapboxing against IRV. While it is certainly true that IRV is not positively responsive, that information belongs in an article about IRV or about positive responsiveness, not in an article about Arrow's Theorem.

• Among the most important axioms of rational choice

again, the word "important" here does not belong in technical writing. and as the paper I linked goes into detail about, there are multiple (very subtly different) definitions of IIA. it would be good for Wikipedia to have that information somewhere rather than relying on this vague definition here.

• Arrow's theorem shows that if a society wishes to make decisions while avoiding such self-contradictions, it cannot use methods that discard cardinal information.

it shows no such thing. for example a society with dichotomous preferences would be doing just fine. again, this is more politically-motivated editorializing (the political motive being advocacy for election reform to use "cardinal" rules such as Approval, Score, or STAR)

• Condorcet's example is already enough to see the impossibility of a fair ranked voting system, given stronger conditions for fairness

again, "fair" is not an appropriate word here. a technical article about a technical subject should strive to be literal and precise without imposing human interpretation onto the analysis. a better word would be "always consistent with a majority against any alternative" or of course simply (though maybe too tautologically) "Condorcet"

• Attempts at dealing with the effects of Arrow's theorem take one of two approaches: either accepting his rule and searching for the least spoiler-prone methods, or dropping his assumption of ranked voting to focus on studying rated voting rules.

this shows the author's hand at being part of an amateur community that obsessively generates an ever-growing list of "proposals" and small tweaks on existing election rules in attempts to "fix" Arrow's impossibility. I equate these folk to [3]trisectors. the academic perspective would simply treat this theorem like any other, and build on it, and study relaxations and generalizations etc. etc.

• lastly `Meaningfulness of cardinal information`

is just wholly entirely unrelated and this section should be deleted wholesale (or at the bare minimum, moved to an independent article) Affinepplan (talk) 21:56, 21 October 2024 (UTC)

Thanks for all that. I don't know when I'll have time to address these points more systematically, but I should say now that "satisfy the requirements of rational choice theory" was a replacement [4] for the phrasing that I had tried, "a mathematical standard of rational behavior codified by decision theory" [5]. My attempt prior to that had been "no method of obtaining a collective result from the preferences of multiple individuals can simultaneously satisfy all of a certain set of seemingly simple and reasonable conditions" [6]. I think that latter option conveys how the conditions are indeed often presented as being, well, seemingly simple and reasonable. The middle option is shorter and more bluelink-dependent, but maybe it avoids the concern you raise here. XOR'easter (talk) 22:07, 21 October 2024 (UTC)

of those I think the last is the best. the statement should either be fully technical & accurate or a lay summary. as is, it is imposing normative conclusions onto entire fields of research (or otherwise non-normative mathematical models) Affinepplan (talk) 22:45, 21 October 2024 (UTC)

I like the last one best, too. In this case, given the range of potential audiences for the article, I believe a lay summary is the best way to go for the opening line. XOR'easter (talk) 06:14, 22 October 2024 (UTC)

the section "Meaningfulness of cardinal information" should be removed

this is not related to Arrow's Impossibility Theorem and is just election related pseudoscience soapboxing. I will remove in 3 days time if there is no protest. Affinepplan (talk) 17:08, 21 October 2024 (UTC)

Some of that section looks on topic (e.g., Arrow originally agreed with these positions and rejected cardinal utility...), while other parts might be better suited to an article about IIA in particular. None of the references in the paragraph beginning In psychometrics, there is a near-universal scientific consensus... mention Arrow's theorem, and one of them predates it by decades, so that whole passage is WP:SYNTH here. Perhaps a trim would be more prudent than a wholesale chop. XOR'easter (talk) 21:49, 21 October 2024 (UTC)