# Talk:Arrow's impossibility theorem

## My removal of the electionmethods.org link

In the edit summary, Psephomancy added a link to an archived version of something posted on electionmethods.org:

...with the summary:

*did you just remove it because it's a dead link?*

Well, it was mainly that. The author (Russ Paielli) pulled the website down. It seems to me that we're not doing readers of this wiki much of service by offering an unannotated link to an archive of a webpage from 2005.

I agree with much of what that particular web page says. At some point, I'd like to perform a plagiarism-free restatement of some of the important arguments made on that page, rather than redirecting people to an old archived snapshot of a long defunct website. I think there was plenty of discussion about IIAC on the EM list when this page was first published. If I recall correctly, there was an essay or two on the old electionmethods.org that I contributed some prose to, but I can't remember which ones, and it *might* even be this essay. I'd have to dig up the very old email backups to find what I wrote.

In general, I've always been careful not to dismiss Arrow's result as irrelevant to my preferred method. As we've seen with w:Gibbard's theorem, the core principle of Arrow's theorem (i.e. there is no "perfect" voting system because many important criteria are mutually exclusive) holds true no matter the system. If I recall correctly, some of Russ's restatement of my writing implied I was dismissive of Arrow, and I didn't want to be associated with a dubious effort to declare Arrow's theorem irrelevant.

So thanks for digging up the archive link to that page. A little bit of annotation around that link would be wonderful. Hopefully an active Electowiki editor can perform the plagiarism-free restatement I was hoping for. -- RobLa (talk) 05:04, 8 January 2020 (UTC)

## EPOV on Arrow and cardinal methods

Just a bit ago, I posted a message to the EM list about Arrow's theorem and cardinal methods. I'll quote a little bit of the message here:

Many Score voting[1] activists claim that cardinal methods somehow dodge Arrow's theorem. It seems to me that *all* voting systems (not a mere subset) are subject to some form of impossibility problem. Arrow's impossibility theorem deserved great acclaim for subjecting all mainstream voting systems of the 1950s to mathematical rigor, and it's clear that his 1950 paper and 1951 book profoundly influenced economics and game theory for the better. His 1972 Nobel prize was well deserved. It seems that it has become fashionable to find loopholes in Arrow's original formulation and declare the loopholes important. Even if the loopholes exist, talking up those loopholes doesn't seem compelling, given the subsequent work by other theorists broaden the scope beyond Arrow's version.

I expand on these points quite a bit in my email. The full message is archived here: "**(EM) Arrow's theorem and cardinal voting systems**". I'm eager to read what the membership there thinks on this topic, since that group of people has frequently been successful at changing my mind on a particular topic. I'm also eager to read your response to my message (privately or publicly, via whatever mode of communication you prefer). -- RobLa (talk) 23:47, 9 January 2020 (UTC)

- I would not say you are wrong but there are a few things you are missing. Most importantly is that Gibbard-Satterthwaite theorem is not Arrows theorem. Yes they are related but Gibbard-Satterthwaite is way more general. Arrows theorem talks about the specific criteria which will be failed. The importance of arrows theorem is that it shows that Ordinal systems must have one of the bad issues. Gibbard-Satterthwaite shows that all systems has some issue with strategy but it may not be a particularly bad issue. I realize that it is tempting to say all systems have issues but I think that is a misguided narrative because it implies that all issues are equally bad. All systems have trade-offs and some of the trade-offs are not worth it. The question is about what is the optimal balance. Most use Arrow as a way to say that Ordinal systems will not be monotonic and monotonicity is too important to trade away when cardinal systems exist. The narrative "Who cares if IRV is nonmonotonic? Arrows theorem shows that all systems will be nonmonotonic" is pushed by FairVote. It is wrong. Monotonicity is a core concept to fair voting. --Dr. Edmonds (talk) 00:26, 10 January 2020 (UTC)
- Dr. Edmonds, I think we agree on many things. For example, monotonicity is an important (and underrated) criterion for electoral systems. Furthermore, all systems have trade-offs, and some of the trade-offs are not worth it (we agree on that, too). But, I believe all rhetorical strategies also have trade-offs, and some are not worth it. Trying to jump through a loophole in Arrow's theorem just complicates the discussion in an unhelpful way. The EM list discussion in January (that stretched into February) didn't come to a consensus, but generally was sympathetic to my original January 9 email. I particularly appreciated Forest Simmons' response on January 13, where he improves on my "no perfect car" metaphor.
*Even if*cardinal methods are not subject to Arrow's theorem, there are plenty of other theorems that close the loophole described. If you still hold this position, could you make your case on EM list? -- RobLa (talk) 22:56, 16 March 2020 (UTC)

- Dr. Edmonds, I think we agree on many things. For example, monotonicity is an important (and underrated) criterion for electoral systems. Furthermore, all systems have trade-offs, and some of the trade-offs are not worth it (we agree on that, too). But, I believe all rhetorical strategies also have trade-offs, and some are not worth it. Trying to jump through a loophole in Arrow's theorem just complicates the discussion in an unhelpful way. The EM list discussion in January (that stretched into February) didn't come to a consensus, but generally was sympathetic to my original January 9 email. I particularly appreciated Forest Simmons' response on January 13, where he improves on my "no perfect car" metaphor.

- If I may, I'd like to point out that one of the main reasons imo that it is interesting to say cardinal methods aren't affected by Arrow's Theorem is because this implies cardinal methods pass IIA. That is an argument some cardinal advocates make in favor of their methods (that this implies their voting methods are free of the spoiler effect, since candidates can enter and drop out of the race without changing the result). While that can be vigorously debated (for example, if any voter changes their scores for candidates who are present both before and after some candidates enter or drop out of the race, then cardinal methods fail IIA), it seems only right to put something on this wiki that helps the debate to happen. BetterVotingAdvocacy (talk) 23:22, 16 March 2020 (UTC)

- RobLa I just can't get behind your argument. You admit that cardinal methods are not subject to Arrow's theorem. Your argument seems to be that since they are subject to other similar theorems we should say that it is subject to Arrow's. That is totally illogical. We need to be precise in what applies to what. I do not understand your motivation for wanting to do this. --Dr. Edmonds (talk) 03:07, 17 March 2020 (UTC)

## Finding EPOV on Arrow

*This is a continuation of "EPOV on Arrow and cardinal methods"*

Dr. Edmonds, could you do me a favor? Could you do one of the following:

- a. find someone that agrees with you in the January 2020 list of respondents to my "Arrow's theorem and cardinal voting systems" email, and let me know which person I need to provide a better response to?
- b. join the EM list, and state your objection to my January email on that list?
- c. propose some other off-wiki mechanism to convince me that your position isn't controversial?
- d. just concede that we need to agree to disagree, and that because I disagree with your point of view (and many smart people on EM list seem to agree with me), the idea that "Arrow doesn't apply to cardinal methods" is controversial (e.g. I believe it does, you believe it doesn't)

I'm willing to concede that I may be wrong. But the membership of EM list wasn't able to convince me I'm wrong, and believe me: they have a lot of practice telling me that I'm wrong, and some of them have been doing it for 24 years. It's a tough crowd. I *might* be willing to change venues, but given the robust response I received in January, I feel pretty confident saying "the EM list has spoken", and I don't see the value of trying to overrule the discussion by forum shopping.

I'm already willing to concede that I'm not going to convince you that I'm correct. I'm asking that we agree to disagree, and consider the EPOV to be "Some people believe Arrow doesn't apply to cardinal methods" rather than flat out insisting that "Arrow doesn't apply to cardinal methods".

Could you choose one of the options above? Thanks! -- RobLa (talk) 04:00, 17 March 2020 (UTC)

- I am not going to read that forum. It is too hard to follow the flow to get any information out. It seems all you want is proof that Arrow's theorem does not apply to cardinal methods. I will choose your option C. How about a quote from arrow saying exactly that?

Now there’s another possible way of thinking about it, which is not included in my theorem … [E]ach 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 … This changes the nature of voting.

- That quote is taken from here. Read section 5.3 and 5.4 for the details. They get into some of the extensions and how people have tried to build a cardinal version of Arrows theorem. At this point there are two possibilities you could have to resist this. 1) You dispute what Stanford and Arrow himself are saying. If this is the case then we will have to agree to disagree. 2) You admit that Arrow's actual theorem does not cover cardinal systems but know cardinal systems are covers by OTHER theorems. You want to call all theorems "Arrows theorem" so as to hide the fact that Ordinal systems are more flawed than cardinal systems. If this is the case then you have a clear biased and malicious intent. --Dr. Edmonds (talk) 20:59, 17 March 2020 (UTC)

- Dr. Edmonds, quick point: STAR is covered by Arrow's Theorem. This is because it satisfies the majority criterion in the two-candidate case and thus fails IIA. So I recommend creating some terminology to cover only Approval and Score (and I think Majority Judgement also?) to ensure people don't get misled into thinking all cardinal methods are not covered by Arrow's Theorem. BetterVotingAdvocacy (talk) 18:29, 18 March 2020 (UTC)

- BetterVotingAdvocacy Semantics are important here. Arrow theorem was only for Ordinal systems when originally written. There have been extensions which would apply to majoritarian cardinal systems like STAR. Each of these extensions needs to be considered separately since they all have different assumptions. We should add sections to the arrow's theorem page for each and clearly spell out the differences in axioms to arrrow's theorem. There may be a way to write down a generalized arrows theorem which encompasses all extensions but I have never seen it. The formulation on the electo wiki page is the one for Ordinal systems only. It does not apply to score. --Dr. Edmonds (talk) 18:51, 18 March 2020 (UTC)

- Dr. Edmonds, I find your selected quote unconvincing, for reasons that I described in my January 9 response to "fdpk69p6uq". Note that "fdpk69p6uq" replied to me on January 10 and then Kristofer Munsterhjelm replied to both of us on January 14. It may be worth bringing User:Kristomun into this conversation, but regardless, that particular interview didn't persuade me. -- RobLa (talk) 07:15, 19 March 2020 (UTC)
- I think my vote here would be simple: we should let a mention of the possible/likely evasion of cardinal methods of Arrow's Theorem go to the top of the cardinal voting methods page, and then include a disclaimer "see below for controversy and discussion" linking to a more detailed section section. It doesn't seem right to not have this be very close to the top, since it's one of the major vectors of comparison between ranked and rated methods, and one of the biggest reasons someone would consider abandoning traditional voting theory results. Also, RobLa, I'm still interested in hearing why you took out the sentence saying "all pure cardinal methods pass the participation criterion", since you presumably don't have an objection to that. BetterVotingAdvocacy (talk) 07:49, 19 March 2020 (UTC)

- RobLa I have read through much of thead thread and it does not give me much hope. There are several people there telling you that you are wrong in several different ways. Perhaps their error was being too charitable in their method of telling you this. You are not wrong that all systems have problems. You are wrong that Arrow's theorem applies to all systems. You will not accept the statements from Arrow or authoritative compendiums either. In all the time I have been studying this I have never heard of anybody but you who thinks Arrow's theorem applies to Cardinal methods. I understand that it is nuanced but there is no real trick. I think you are stuck in a bit of cognitive dissonance since you seem to understand how unrestricted domain is not defined in a way which includes score but still want to include score in the theorem. I suspect your motivations are biased to want to suppress that cardinal systems have this clear advantage over ordinal systems. What concerns me here is that you have a lot of power over electowiki as you are the moderator. Those adding content are unable to convince you but you have final veto power. You are using this to force us to change your mind but you are not willing to let it be changed despite a ton of evidence. I am not sure how to resolve this situation. --Dr. Edmonds (talk) 21:30, 19 March 2020 (UTC)
- Dr. Edmonds, thank you for reading the thread. Let's start with our respective biases. You are stating my bias in an uncharitable way: "
*to suppress that cardinal systems have this clear advantage over ordinal systems*". I'll admit to having a bias, but allow me to restate it: "*It seems to me that *all* voting systems (not a mere subset) are subject to some form of impossibility problem.*". That's a direct quote from my January 9 email. My fear, confirmed by your attempt to state my bias, is that you believe that the inapplicability of the universality criterion to cardinal methods is somehow confers superiority of cardinal systems over ordinal systems. Even if I were to concede that cardinal methods aren't subject to Arrow's theorem, it wouldn't change my belief that cardinal methods fail other important criteria, and are subject to other important impossibility theorems. Moreover, in my January 12 response to Jim Faran, I expanded on why I believed that universality criterion is important in assessing the fairness of a system. I'll respond to your point about my moderator role in my subsequent response to BetterVotingAdvocacy. -- RobLa (talk) 02:13, 20 March 2020 (UTC)- RobLa, I think it may be worth adding a point in response to yours. The goal of a wiki, imo, should be to capture all the sides of a debate to a good degree somewhere; otherwise, there is no resource left where readers can find unbiased information (or even, perhaps, know that there are any disputes about a certain thing). So in that regard, it is absolutely a good idea for us to document what different people say about cardinal methods in relation to Arrow's Theorem, why, and what each argument implies for the quality of the systems. BetterVotingAdvocacy (talk) 02:46, 20 March 2020 (UTC)

- Dr. Edmonds, thank you for reading the thread. Let's start with our respective biases. You are stating my bias in an uncharitable way: "
- User:BetterVotingAdvocacy I think I more-or-less agree with your proposal. The "see below" bit is implied, so it may not be needed; everything in the summary should be covered in more detail in the latter part of the article. I removed the bit about the participation criterion because of the context that it was stated; it implied that cardinal methods don't violate any important criteria and minimized the significance of universality criterion. We don't have to have an exhaustive list of all criteria for the reader of the summary, but I don't want to leave the impression that cardinal methods pass all criteria. We still need to trim the introduction by quite a bit; we may need a general "criteria passed by cardinal methods" section (or some other similar name). -- RobLa (talk) 02:39, 20 March 2020 (UTC)

- RobLa I have read through much of thead thread and it does not give me much hope. There are several people there telling you that you are wrong in several different ways. Perhaps their error was being too charitable in their method of telling you this. You are not wrong that all systems have problems. You are wrong that Arrow's theorem applies to all systems. You will not accept the statements from Arrow or authoritative compendiums either. In all the time I have been studying this I have never heard of anybody but you who thinks Arrow's theorem applies to Cardinal methods. I understand that it is nuanced but there is no real trick. I think you are stuck in a bit of cognitive dissonance since you seem to understand how unrestricted domain is not defined in a way which includes score but still want to include score in the theorem. I suspect your motivations are biased to want to suppress that cardinal systems have this clear advantage over ordinal systems. What concerns me here is that you have a lot of power over electowiki as you are the moderator. Those adding content are unable to convince you but you have final veto power. You are using this to force us to change your mind but you are not willing to let it be changed despite a ton of evidence. I am not sure how to resolve this situation. --Dr. Edmonds (talk) 21:30, 19 March 2020 (UTC)

- Trying to think of my own position, I think mine is that under reasonable assumptions, cardinal voting violates IIA, and that this can be proven by arguments that lie very close to Arrow's theorem. (Basically, majority plus something at least as powerful as ranked universal domain violates IIA.) However, you can't use the literal Arrow's theorem, because Arrow's definition of universal domain
*restricts*the voting method to be ordinal. On the one hand, people who say "Arrow doesn't apply to Range, so we can have IIA" are strictly speaking right. But unless the ratings are independently calibrated (as my EM post refers to), you get an IIA violation. "Arrow's theorem doesn't apply" simply says that the exact theorem can't be used on cardinal methods, but it doesn't prove that the method avoids IIA failure. There's a more general theorem hiding somewhere, but Arrow's is not it. Kristomun (talk) 12:26, 20 March 2020 (UTC)

- The Stanford article I linked, mentions some of the extensions. I think we should add sections to the page detailing these. It is important to say that these are not Arrow's theorem but extensions. They all have their own assumptions and limitations. As Kristomun says they come down to " majority plus something at least as powerful as ranked universal domain violates IIA". This means Score passes and so do several multimember score systems. It is really a restriction on majoritarian systems so maybe the title of the section should be "extensions to majoritarian systems". User:Psephomancy added a change to the page to clear up the wording and he added some references. --Dr. Edmonds (talk) 16:03, 20 March 2020 (UTC)

- However, as I'd reiterate, an important point is that Range only passes if people don't calibrate their scales relatively. In the pizza election, they (presumably) have an absolute scale, but any normalization that reduces a two-candidate elction to a majority vote makes the procedure (plus implicit normalization) fail IIA. This point is probably stronger against Approval than Range: an absolute scale calibration implies that there can be voters in an Approval election who would approve every candidate or none of them, something which is very hard to imagine would happen in a real election. This is reminiscent of the Approval/Range "manual DSV" sleight of hand that I've talked about on EM. Kristomun (talk) 17:50, 20 March 2020 (UTC)

- Kristomun To be clear a relative scale is when you put your favourite(s) to MAX_SCORE and everybody you do not like to 0, right? And your claim is that there is an extension of Arrow's theorem which would apply to Score voting if that was true. I would think this is always true so I would be very interested in such a proof. Do you have a reference? --Dr. Edmonds (talk) 22:27, 20 March 2020 (UTC)

- I think Kristomun is talking about something like a 3-candidate Condorcet cycle, where no matter who Score elects, if one candidate drops out, then if voters normalize between the two remaining candidates, then you get majority rule. BetterVotingAdvocacy (talk) 22:47, 20 March 2020 (UTC)

- Pretty much. By "relative scale" I mean in the sense of Balinski and Laraki: an absolute scale is where the grades (or ratings) mean the same thing for everybody, a relative one is where what ratings you provide depend on what candidates are running. I don't know of a published proof, but it seems obvious, just follow User:BetterVotingAdvocacy's suggestion: you set up an election where there's a Condorcet cycle and no matter who wins, if another candidate drops out then winner loses the resulting majority election. Kristomun (talk) 23:25, 20 March 2020 (UTC)

- OK I understand. This is the sort of thing that systems like Distributed Voting try to get around. I have a bit of a rant on this here. Score has a built in assumption that candidates will not be added and removed. In any case, I do not think this is really related to Arrow's theorem directly so can we all agree that Arrow's theorem does not apply to score? This other stuff is interesting though. Perhaps somebody wants to add some explanation to the Voting paradox page. All the theorems are tied together in some way and they are all important. I did not know till vary recently that Balinski–Young theorem extends to multi-member systems. It implies that all Monroe type systems fail something like participation. Furthermore, there are Multimember systems where the score does have a much more absolute scale. The most obvious is Sequentially Spent Score. --Dr. Edmonds (talk) 01:53, 21 March 2020 (UTC)

- Quote from Dr. Edmonds: "Score has a built in assumption that candidates will not be added and removed." With such an assumption, it may be possible to make many ranked methods evade Arrow's Theorem as well. BetterVotingAdvocacy (talk) 02:07, 21 March 2020 (UTC)

- Well if you can formalize and prove that then there is a Nobel Prize in it for you. Social choice theory generally assumes that the choices come as part of the problem. This is one criticism of the whole field. We are going pretty far off topic. Lets move this to a forum. --Dr. Edmonds (talk) 05:22, 21 March 2020 (UTC)

- Yes, let's take it to the election-methods list, then we can refer to the thread from here. I'd just say, in conclusion, that I think it's possible to phrase this in an Arrovian context, and that E-M style voting theory is already outside of social choice if what you're saying is true (consider e.g. Tideman's independence of clones criterion). Kristomun (talk) 10:29, 21 March 2020 (UTC)