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

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

     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 (talk • contribs) 20:36, 22 April 2015 (UTC)

@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)

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)

Arrow's theorem assumes the social ordering function takes a vector of rankings as an input. It then gives a single ranking as output. This definition excludes cardinal systems (which have a matrix of numbers as input).

> If voters are allowed to rank outside a specified range, they have infinite votes, thus theoretically every voter is the dictator.

A voting system is dictatorial if, for any possible combination of votes by citizens who are not dictators, the output is still always equal to the preferences of the dictator (which are held fixed); I think you mixed up the quantifier. Closed Limelike Curves (talk) 01:28, 5 April 2024 (UTC)

Dubious

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)

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)

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)

How does Gibbard's theorem "extend" Arrow's theorem?

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)

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)

"Unanimity criterion" listed at Redirects for discussion

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)

Arrow's impossibility vs. Condorcet paradox?

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)

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)

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)

Generalization

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)

Spoiler effects and IIA

This article, and the spoiler effect article, should carefully distinguish between an election method as such not changing its output when a candidate who didn't win is removed from all input ballots; and the stronger effect that a candidate deciding to enter the race where a particular election method is being used, and then losing, does not change who wins. It's a subtle distinction, but normalization issues can cause cardinal voting methods to fail the latter even though they pass the former, even if the voters use von Neumann-Morgenstern utilities.

It's a bit off topic for this article, but the article's phrasing should still make clear that the former definition is what's being talked about, and that it doesn't imply the latter.

The rank-order voting section of this page might be relevant. Wotwotwoot (talk) 13:10, 29 March 2024 (UTC)

I'm not sure that I understand the distinction. Suppose under a given voting system candidates A, B, and C are running and A wins, but B would have won if D joined. This seems to be your second ("stronger") type. But if we rephrase it as A, B, C, and D are running and B wins, but then if D drops out A wins, this seems to be your first type. - CRGreathouse (t | c) 19:19, 29 March 2024 (UTC)

If we're dealing with ordinal voting, then the two types are the same. But here's an example for Range voting:

Suppose that the three candidates are A, B, and C. We have three blocs of voters.

The first bloc has 31 voters who score A 10, B 5, and C 0.

The second bloc has 28 voters who score B 10, C 7, and A 0.

The third bloc has 27 voters who score C 10, A 3, and B 0.

In total, A has 391 points, B has 435 points, and C has 466 points. So C wins. Now if we remove A from the input data, then C still wins, because that doesn't change the other candidates' scores: Range passes IIA. But for that to translate to an immunity to spoilers, we must presume that in a counterfactual election where only B and C ran, the scores would be like this:

31 voters score B 5 and C 0

28 voters score B 10 and C 7

27 voters score C 10 and B 0

But it's quite possible that if only two candidates had run, the voters would have ranked their favorite 10 and their least favorite zero. In this other universe, A never ran, so the voters might not have been rating according to the same scale. And if the voters instead had voted their favorite maximum and least favorite minimum, we would have got:

59 voters score B 10 and C 0

27 voters score C 10 and B 0

which would have led B, not C to win. Hence A not running changes the winner from C to B, because the second bloc detests A so much that it makes C look better compared to B. In this stronger sense, elections conducted with Range might have spoilers, even though the method itself passes IIA when looked at in isolation.

The direction (going from the two-candidate election to three candidates or vice versa) doesn't play into it, just whether the voters have an "objective" scale they can rate the candidates on, in the sense that their scaling of scores of existing candidates don't change when candidates enter or leave. Wotwotwoot (talk) 21:33, 29 March 2024 (UTC)

I grant that there are normalization issues with cardinal voting systems, but Arrow is just about ordinal system, right? - CRGreathouse (t | c) 17:19, 30 March 2024 (UTC)

User:Mbmiller seems to think otherwise, but to my knowledge, Arrow is indeed just about ordinal systems. Since Arrow is inapplicable, cardinal methods are allowed to pass IIA and some do.

But currently parts of the article are phrased so as to equate passing IIA with eliminating spoilers (e.g. the section titled "Eliminating spoilers: Rated voting"). I think it would be useful to somehow make the distinction more clear so that "passing IIA" doesn't get interpreted as "spoiler immunity". Perhaps in a section describing the relation between IIA and spoiler immunity, or a reference to the spoiler effect article. Wotwotwoot (talk) 19:45, 30 March 2024 (UTC)

I've taken a stab at doing this. I think some of the problem lies in that information about different related effects (IIA, Arrow, and spoilers) are duplicated or belong to the wrong article, and some cleanup is needed. The "Non-Arrovian spoilers" section might be more suited to the spoiler effect article, for instance. Wotwotwoot (talk) 20:59, 30 March 2024 (UTC)

@Wotwotwoot I'm happy to mention tactical/strategic spoilers, but—

1. From what I can tell, the term spoiler typically refers to IIA failures. This aligns with the original use of the term, which is taken from sports: there, the term refers to one team eliminating another, despite not being eligible to win themselves. (This also seems like a much more common cause of IIA failures than tactical ballot-switching.) I think the average reader is going to be smart enough to understand that, if voters change their votes, this will change the results.
2. I'd like to keep discussions of tactical spoilers separate from Arrovian spoilers as much as possible; this includes keeping digressions into how tactical voting or human psychology can affect IIA below the fold, or ideally in a different article (with links coming from this page and footnotes).

Mostly I just want to focus on teaching readers one concept at a time and providing more detail in footnotes or linked articles. Trying to discuss both concepts at once is going to create a mess in the average reader's mind. Closed Limelike Curves (talk) 02:34, 2 April 2024 (UTC)

In particular, I really want to avoid the excessive detail that tends to plague Wikipedia voting systems articles. I remember trying to read through these articles maybe 10 years ago as a kid and I had no idea what any of it actually meant. (I've asked friends and family to try and tell me if they can understand what the page is saying; so far, this is the first draft they seem to actually understand, as long as they skip the proofs.) Closed Limelike Curves (talk) 03:09, 2 April 2024 (UTC)

If your concern is simplicity, I'm perfectly happy to remove everything that talks about non-straightforward (strategic) voting, or replacing it with a simple paragraph that says "clearly, under strategy all bets are off". However, that is not all I was doing. My main point is that a straightforward reading of "IIA = spoiler effect" conceals the fact that honest voters may cause a spoiler effect in cardinal methods by not operating on an absolute scale, so that the addition of losing candidates changes the scale they use, making someone else win.

It's important to note that this can happen even if we hold the voters to their von Neumann-Morgenstern utilities, because these are only unique up to a positive affine transformation. The spoilers thus don't need to be tactical spoilers, because even honest voters who don't care about how other people are voting can cause the winner of cardinal methods to change.

Arrow makes the same point in his book "Social Choice and Individual Values" in Chapter III, section 6, "The Summation of Utilities". So if a non-Arrovian spoiler would be relevant to this page, it would be this effect. But I think it would make more sense to just briefly say "there exist non-Arrovian spoilers", move the bulk of the material to the spoiler effect article, and let the Arrow's theorem article (this one) be focused on Arrow's theorem itself. That's why I deleted the material here. If we want to keep it simple, let the spoiler article discuss or describe what a spoiler means, and this article be about IIA and Arrow's theorem. Wotwotwoot (talk) 17:02, 2 April 2024 (UTC)

"honest voters may cause a spoiler effect in cardinal methods by not operating on an absolute scale, so that the addition of losing candidates changes the scale they use, making someone else win."

This is what I'm confused about. Why would a (rational and honest) voter's utility function change based on which candidates are running? The scale constant is, well, a constant. There's nowhere else in welfare economics where we allow the scale constant to vary when an agent's choices change; I can't claim that an economic intervention that causes utility to drop 10 points in my model is actually good, because the scale/units changed after I passed the policy. Closed Limelike Curves (talk) 15:36, 6 April 2024 (UTC)

Their internal utilities don’t change, it’s how they translate them to a cardinal ballot expression that does. Sen’s extended Arrow theorem’s assumes a property he calls “cardinality” which informally speaking means that the voter or the election system (based on whether we’re using unnormalized or normalized methods) treat positive linear transformations of the "true" utility function equally.

Another way to put this is that utilities of the von Neumann-Morgenstern type aren’t entirely determined. While B running doesn’t change how much a voter likes A, the voter is in both cases (B running or not) confronted with the problem of what honest ballot to cast. That is, the voter has to find some way to supply the missing information that vNM utilities do not provide.

Sen and Arrow then motivate that if the voter chooses to fill in this ambiguity in a way that depends on the candidates running, then we have IIA failure.

That doesn’t prove that they do. But it seems reasonable that a considerable number of voters would vote majority style in a two-candidate Approval election, no matter who the two candidates are: that they would rate their favorite max and their least favorite min. If so, they’re using relative scales, even if these are consistent with their vNM utilities. So the argument arises from the commonsense notion that that’s what voters in a two-candidate Approval election would do. It would seem strange for a voter to show up only to approve both candidates or neither.

There’s also a second argument: if voters use individually absolute scales, e.g. by anchoring 0% to the end of the world and 100% to a utopia, then we would expect to see a lot of ballots who approve of everybody simply because every candidate is better than a 50-50 gamble on either utopia or the end of the world.

Detailed Approval election breakdowns are rare, so it’s hard to say how many do approve of everybody. But the plausibility of the majority voting scenario, combined with that both Arrow and Sen have written about this problem, would make this information relevant, IMHO.

In addition, it seems that some cardinal designers either make relative voting explicit or try to directly counteract it. As an example of the former, the STAR Voting example ballot states "Give your favorite(s) five stars. Give your last choice(s) zero or leave blank". Of the latter, Balinski and Laraki attempt to deal with the problem in devising MJ.

As for the question "why don’t we see a concern for scales changing elsewhere?", I think that in part that’s due to the history of economics (see and the binary approach of pass/fail criteria in voting (I could give references, but this is already pretty long). The addition or removal of irrelevant alternatives is a bit difficult to bring into a questionnaire context, too. Wotwotwoot (talk) 18:28, 9 April 2024 (UTC)

Their internal utilities don’t change, it’s how they translate them to a cardinal ballot expression that does[...] While B running doesn’t change how much a voter likes A, the voter is in both cases (B running or not) confronted with the problem of what honest ballot to cast. That is, the voter has to find some way to supply the missing information that vNM utilities do not provide.

Ahh, OK, I see what you're arguing now. So your argument is voters need to establish some kind of common units? e.g. this problem of reported results when no scale is provided; voters need to figure out a way to choose which units to reply in. (The same way that, if I asked people to tell me how long something is, I'd have no idea if they meant it in millimeters, miles, light-years...)

When rating candidates, voters probably use a mix of approaches. But whatever they're doing, the empirical evidence is it works. If I had to take a guess, one common strategy is probably background comparisons: they're comparing the candidates to other, past winners or candidates (rather than each other). You can see this most clearly in races where voters are unhappy with the choices they have, like in 2016 or 2024. Tons of people say "I don't like Biden or Trump (or any other candidate)"; if I ask them to rate each one on a scale of "Strongly like, somewhat like, ..." or on a scale from 0 to 100, their answers reflect that. This doesn't make sense if voters can only establish a common scale by comparing candidates to each other, but it does make sense in the context of background comparisons: people are saying they're less enthusiastic about Biden in 2024 than Obama in 2012.

Alternatively, people could be approximating some kind of implicit utilitarian algorithm, which allows them to infer utilities from a combination of other people's ordinal preferences and their observed behavior over lotteries. Because we have a lot of experience interacting with other people, and often seeing the same ordinal preferences in many situations, we can infer their cardinal preferences with a high degree of accuracy (far better than could be inferred from rankings of just 5 or 6 candidates).

The Arrow/Sen argument would probably hold in one-shot elections with no common information being shared between voters except the ballots, but that sounds like an extremely unusual case. Closed Limelike Curves (talk) 17:51, 10 April 2024 (UTC)

That gets to the distinction that I referred to at the end of my previous reply: between the axiomatic criterion approach often used when describing election methods, and a more statistical approach favored elsewhere. The economic paper notwithstanding, a more pragmatic approach has been used in psychology and for analyzing Likert scale responses, questionnaires etc.

All that Arrow's theorem says is that there exist some situations where IIA must be violated. It doesn't say anything about how often these happen, as that would require knowing voter behavior. Perhaps the probability of a Condorcet cycle approaches zero exponentially fast in reality, say. Or perhaps voters do use a fixed anchor all the time, and not just most of the time, so that cardinal voting election outcomes indeed are independent of losers. But statistical reliability and validity can't answer such questions.

Since Arrow's theorem itself uses the axiomatic or deductive approach, it makes sense to stay with it. And in that context, Arrow and Sen say that if we can't ensure that voters use the same scale no matter who runs, we can't guarantee that the outcome is independent of losing candidates.

Whether and how much that applies in reality can of course be debated, just like the question of whether and how often sincere Condorcet cycles would occur. Personally, I think the chance of rated voters voting majority style when only two candidates are present is quite high, even though that's not proof.

But if we're using Arrow's axiomatic approach to back a statement that Condorcet only has minimal spoilers instead of no spoilers, we should also use Arrow's axiomatic approach to back a statement that elections with Range only has minimal spoilers instead of no spoilers. Cardinal elections shouldn't be able to get off on a technicality just because their method is asking for information that the voter can't supply with the absolute rigor that is necessary for the Arrow-style implications to hold. Wotwotwoot (talk) 16:07, 14 April 2024 (UTC)

But if we're using Arrow's axiomatic approach to back a statement that Condorcet only has minimal spoilers instead of no spoilers, we should also use Arrow's axiomatic approach to back a statement that elections with Range only has minimal spoilers instead of no spoilers. Cardinal elections shouldn't be able to get off on a technicality just because their method is asking for information that the voter can't supply with the absolute rigor that is necessary for the Arrow-style implications to hold.

The voter can't supply all of their utilities with perfect, absolute rigor, but this is a practical limitation, not a result about social choice. In the same way, we could say that ranked pairs fails the Condorcet criterion because sometimes people will miscount ballots, or make mistakes when filling them out ballots. I mean... sure, I guess that in some sense, you could claim no system satisfies the Condorcet criterion, and Condorcet methods just minimize the probability of a Condorcet failure. Then you could say ranked pairs is "getting off on a technicality" by making assumptions that will never hold in the real world. But that would be silly. It'd be like complaining my calculator gives wrong answers whenever I feed it the wrong numbers. If you give the social choice function the wrong utilities, the wrong answer will come out.

It's reasonable to mention interpersonal comparisons of utility are difficult and philosophically contentious, since we don't have direct access to other people's mental states. But people changing their preferences or utility functions in response to changing the set of options doesn't really work within the mathematical framework of welfare economics, mostly because as soon as you change the set of outcomes, you can never compare any policy or decision to any other.

A social welfare function is just a mathematical function that satisfies a few properties, and it has interpretations other than as a set of ballots (in fields like ethics, economics, decision theory, and multi-criteria optimization). Decisions where these algorithms depend on irrelevant alternatives are sometimes called spoiled as well. The mathematical core of Arrow's theorem is just that, if you apply the rank transformation to an aggregation function, the rank-order of your output will depend on irrelevant changes in preferences. –Maximum Limelihood Estimator 18:41, 14 April 2024 (UTC)

There's a critical difference, though. There exists a definition of a single honest vote for ranking - the one consistent with one's preferences. Such a definition doesn't exist for rated ballots because even von Neumann-Morgenstern utilities fail to pin down what the one correct expression for the vote is. Sen thus points out that under the ambiguity that results, we can't ensure IIA.

So when I say "their method is asking for information that the voter can't supply", I mean that literally. There's no way to go from vNM utilities to a rated ballot in an unambiguous way. That's why the analog to someone misfiling their ballot fails. Sen's theorem doesn't assume that there are random perturbations or errors, just that what attempts the voter makes to render their utility is impaired by the invariance to positive linear transforms. Wotwotwoot (talk) 19:07, 14 April 2024 (UTC)

To condense the point: with ranking we can set up theorems that take an unambiguous representation of the will of the voter and then shows that a method that makes use of this representation can pass (random ballot) or fail (minimax) IIA. But what unambiguous representation of the will of the voter can we make for rated voting? If we model the ambiguity by creating an equivalence class of sincere representations, then IIA failure returns. Wotwotwoot (talk) 19:36, 14 April 2024 (UTC)

Such a definition doesn't exist for rated ballots because even von Neumann-Morgenstern utilities fail to pin down what the one correct expression for the vote is.

Harsanyi and Vickrey showed the opposite; there's a specific lottery that pins down the comparison, which is preferences over the lottery of birth. If we ask someone how much money it would take for them to accept a 50% chance of swapping lives with another person, their answers imply interpersonal comparisons of utility.

Importantly, these don't imply the interpersonal comparison of utility would be the same across people, so that doesn't get us to a single "objectively correct" set of weights for interpersonal comparisons of utility. Two people with the same life might disagree about how much money they'd need to take that bet. But they do need some set of weights they use to compare their own utility against others', which has to be consistent across different situations. –Maximum Limelihood Estimator 02:49, 16 April 2024 (UTC)

With all the editing that has been going on, let me argue why the parts about generalizations deserve more space in this (Arrow's theorem) article, and why the latest edits in particular have decreased the quality of the article, because it seems like we got a bit sidetracked.

Arrow's theorem does not make a judgment of any nature on methods that are not ranked. It is relatively well known that rated methods pass IIA when the ballot data is held fixed, although this is not a consequence of Arrow's theorem as such.

There exist generalizations of Arrow's theorem that show how elections conducted with rated methods can fail IIA; and more generally, how methods that try to deal with the whole equivalence class of honest cardinal ballots under the assumption of no absolute scale, can fail.

Since Sen's result is a generalization of Arrow's theorem, it is relevant to an article that deals with Arrow's theorem. Furthermore it brings context to how the limited scope of Arrow's theorem have been expanded. The reference showing that Arrow informally made a similar point that with normalization serves as a verifiable background point of context, and shows that Arrow was considering the same problem, even if he didn't formalize it. Whether Arrow currently thinks that rated ballots are better than ranks has less relevance, because he does not directly refer to the point made there.

Further points could be made that authors of cardinal methods are aware of the ambiguity of rated ballots - in particular I'm thinking of things like mean-utility Approval thresholding guidelines - but that might be out of scope for an article about Arrow's in particular.

Cardinal and ordinal advocates may disagree about whether and how often the voters engage in this behavior. Vickrey's observation could be used to argue that the voters making use of it now have a point to fix their scales to, hence avoiding the problem.

But there are, to my knowledge, no sources showing what models voters empirically use (apart from a single section in an Approval paper).

Thus, due to the relevance of extensions of Arrow's theorem to the subject of the theorem and how it may be used, I think that there should be a point in the lead about how, though rated voting is not covered by Arrow's theorem, it is covered by generalizations of it. Reverting this point removes information that is relevant in an article about Arrow's theorem. Reducing, to a few sentences, the distinct point that consequentially there exist models where rated voting fails IIA (and ones where they pass) reduces the quality for a similar reason.

Better would be to have a section about generalizations as such, which the brief mention in the lead would serve as an introduction to.

The question of whether the voters actually have a way to unambiguously fill in a rated ballot, and about vNM on the one hand and solutions like Vickrey's on the other, like what we've been discussing above, could be detailed in that section. Or in some other article, depending on how relevant it is to Arrow's theorem as such. Wotwotwoot (talk) 18:07, 20 April 2024 (UTC)

Arrow claiming that rated voting doesn't have normalization-based IIA failures?

User:Closed Limelike Curves, I am responding because you have reverted my changes twice now. Last time you added the following quote to the CES podcast reference:

So I think Approval Voting is a little too coarse. I think if you had three or four candidates the incentives for this would be much less if you had three or four classes. There would be a tendency to approve candidates you don’t think very well of just to avoid somebody you think is a real catastrophe.

This is in reply to Aaron Hamlin asking if Approval would encourage the growth of third parties. The quote seems to be more related to strategy than to normalization problems. If it were related to normalization problems, it could just as easily be read the other way: "There would be a tendency to approve candidates you don't think very well of just to avoid somebody you think is a real catastrophe", implying that if the real catastrophe hadn't run, you wouldn't be approving those candidates you don't think very well of, hence a change in ballots would occur due to irrelevant candidates dropping out. So I would like to ask where you consider the quote, or the podcast reference in general, to imply that Arrow doesn't think Sen-type IIA failures will occur, i.e. that he "reversed his opinion later in life, coming to agree that scoring methods provided more useful information that make it possible for such systems to evade his theorem".

That sufficiently fine-grained cardinal ballots with sufficiently many candidates provide more ways to vote (more information) is not in contention, nor is that some cardinal methods like Range pass IIA when that extra data is held fixed. Indeed, the former is the reason Sen uses the wider concept of an SWFL and not just a plain SWF. Nor does the original Social Choice and Individual Welfare reference assume that data is being held fixed: Arrow directly refers to von Neumann-Morgenstern utilities and their invariance to positive linear transformations. Wotwotwoot (talk) 23:05, 19 April 2024 (UTC)

That’s correct. Yes. Now there’s another possible way of thinking about it, which is not included in my theorem. But we have some idea how strongly people feel. In other words, you might do something like saying each voter does not just give a ranking. But says, this is good. And this is not good. Or this is very good. And this is bad. So I have three or four classes. You have two classes is what you call Approval Voting. Just say some measures are satisfactory, and some aren’t. This gives more structure. And, in effect, say I approve and you approve, we sort of should count equally. So this gives more information than simply what I have asked for [... if] we don’t just rank the candidates. We say something like they’re good or bad or something. [...]

CES:But the system that you’re just referring to, Approval Voting, falls within a class called cardinal systems. So not within ranking systems. Dr. Arrow: And as I said, that in effect implies more information.

–Maximum Limelihood Estimator 02:57, 20 April 2024 (UTC)

Yes, Arrow does say that there is more information. That's why I said

That sufficiently fine-grained cardinal ballots with sufficiently many candidates provide more ways to vote (more information) is not in contention.

That much is clear. Sincere rated ballots based on vNM utilities give strength of preference, and ranks don't: that's the whole point. But he doesn't say nor does he imply that this invalidates Sen-type IIA.

We later have

But we have some idea how strongly people feel. In other words, you might do something like saying each voter does not just give a ranking. But says, this is good. And this is not good. Or this is very good.

Which implies again that there's a strength of preference ("not just give a ranking"), and that he sees a need for more ratings than just approve and disapprove, so that the voters may provide information about how strongly they feel about each candidate on some scale (good, very good, etc). But that is an orthogonal issue.

How fine-grained the rating resolution is doesn't itself validate or invalidate Sen-type IIA failure - unless you're thinking that Arrow, by using grade-like terms like "good" and "very good", is referring to Balinski and Laraki's MJ reasoning which is intended to reduce Sen-IIA type behavior. But that seems like somewhat of a reach. And were that the case, it's likely that he would have referred to MJ by name, given that he names Balinski elsewhere in the podcast.

In short, while the source shows that Arrow considers cardinal voting an improvement since such ballots can represent more information, Arrow does not specifically counter the point made in Social Choice and Individual Welfare. Wotwotwoot (talk) 17:31, 20 April 2024 (UTC)

I'm citing this to support the claim that Arrow, later in life, agreed cardinal voting is an improvement because it provides additional, meaningful information relative to ordinal ballots; and that this meaningful information allows such methods to evade his results. (Setting aside practical limitations like human psychology, which Balinski & Laraki take pains to minimize; I'll try and add your reference to their work back in.)

My issues with the previously-suggested edits:

1. The sentence raises a technical point that can only be properly discussed in the body (not the lede). The result is also only indirectly related to Arrow's theorem (enough to warrant mention in the article, but not the lede). Discussing these results in the lede serves to reinforce the common misconception that Arrow's theorem (either directly or in an only slightly-modified form) applies to cardinal methods.

2. The sentence makes it sound like the authors are proving another rigorous impossibility result, rather than raising a philosophical objection.

3. The sentence seems redundant, given the already-existing sections covering limitations caused by human psychology and philosophical disagreements about interpersonal utility comparisons.

4. The sentence makes it sound like rated methods are exempt under a technicality, or there's a similar result applying to rated methods under a different name (like how Satterthwaite technically doesn't apply to cardinal methods, but Gibbard's theorem proving only a slightly weaker result still does). Arrow, Vickrey, and Harsanyi would all disagree with the claim, and argue these kinds of numeric scores are meaningful in a way that allows score voting to avoid independence failures (up to practical failures). (Sen also might agree, since I've seen him argue elsewhere that interpersonal utility comparisons are possible.) Balinski and Laraki also showed cardinal utilities or grades proportional to aren't actually needed for IIA, so long as voters are allowed to rate candidates independently (median ratings only require ordinal information). –Maximum Limelihood Estimator 02:01, 21 April 2024 (UTC)

If you're using this citation to show that Arrow has changed his mind to that cardinal methods evade (i.e. aren't affected by) his own results, then it does not seem relevant to use this citation when discussing Sen's generalization. "Sen showed X, Arrow had an informal argument along these lines too, but later in life he changed his mind", makes it sound like the "later in life he changed his mind" part is relevant to the informal argument, which it isn't.

That's distinct from using the citation to reference that Arrow says that his own impossibility theorem does not apply to cardinal voting.

I now see you have removed the "informal argument" clause entirely, but I would like to restore it.

As for the rest, I've already given a summary above (under "Spoiler effects and IIA"), but I would like to add a few more points.

I am not proposing that the lead discuss the generalizations, only that it makes the reader aware that they exist. Actual discussion would take place in a separate section, not in the lead. If you'd like, we could add a contingency and say something like "generalizations exist that do apply to rated elections given additional assumptions".

My point is basically the converse of yours.

There are many places on Wikipedia where a footnote or caveat about rated voting says "the IIA result only holds if voters don't change their scales", or something to that effect. These exist because the consensus seems to be that some people do change their scales. The generalizations formalize the argument that if they do, then the broader election does fail IIA, giving a theoretical backing relevant to the theorem for what's being informally expressed in the caveats (as well as elsewhere, in Approval papers discussing how to vote, mean utility, etc.; or even right here on this page with CRGreathouse saying "I grant that there are normalization issues with cardinal voting systems").

So your concern is that discussing generalizations in the lead would risk people thinking that the standard Arrow's theorem applies to rated voting. Mine is that not doing so would risk them thinking that rated voting elections pass, just because the systems do when ratings are held fixed. The references to vNM utilities are intended to give a reasonably common theoretical model to explain such changes of scale. Responses could be dealt with in the section.

The reasons I gave under "Spoiler effects and IIA" give the theoretical relevance of the generalizations to Arrow's theorem. And the behavior seeming natural enough that there are caveats to this effect elsewhere indicate that it's practically relevant as well. Thus for two different reasons it deserves a more broad discussion than just a passing remark in the interpersonal comparison section. Wotwotwoot (talk) 11:17, 21 April 2024 (UTC)

These exist because the consensus seems to be that some people do change their scales. The generalizations formalize the argument that if they do, then the broader election does fail IIA, giving a theoretical backing relevant to the theorem for what's being informally expressed in the caveats (as well as elsewhere, in Approval papers discussing how to vote, mean utility, etc.; or even right here on this page with CRGreathouse saying "I grant that there are normalization issues with cardinal voting systems").

I'd grant there's an issue here in that some voters normalize their ballots, much like how behavioral economics has shown voters use a wide variety of heuristics to make even ordinal judgments (see decoy effect). In that case, no Smith-independent method actually satisfies ISDA: it's possible to introduce a new, strongly-dominated candidate who nevertheless changes the way voters rank other outcomes.

If we want to say "well, the system satisfies this axiom, but some voters act in a way that violates it", we'd have to apply that to every voting system and voting property, and every article's lede will quickly get very, very messy. (Also, we could no longer say that Condorcet methods uniquely minimize the rate of IIA failures.) –Maximum Limelihood Estimator 18:42, 22 April 2024 (UTC)