Talk:Arrow's impossibility theorem: Difference between revisions
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:: [[User:Dr. Edmonds|Dr. Edmonds]], I find your selected quote unconvincing, for reasons that I described in [http://lists.electorama.com/pipermail/election-methods-electorama.com/2020-January/002408.html my January 9 response to "fdpk69p6uq"]. Note that "fdpk69p6uq" [http://lists.electorama.com/pipermail/election-methods-electorama.com/2020-January/002415.html replied to me on January 10] and then [http://lists.electorama.com/pipermail/election-methods-electorama.com/2020-January/002429.html 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. -- [[User:RobLa|RobLa]] ([[User talk: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. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 07:49, 19 March 2020 (UTC)
:::: [[User:RobLa|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. --[[User:Dr. Edmonds|Dr. Edmonds]] ([[User talk:Dr. Edmonds|talk]]) 21:30, 19 March 2020 (UTC)
::::: [[User:Dr. Edmonds|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 [http://lists.electorama.com/pipermail/election-methods-electorama.com/2020-January/002423.html 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. -- [[User:RobLa|RobLa]] ([[User talk: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. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 02:46, 20 March 2020 (UTC)
:::: [[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). -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 02:39, 20 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. [[User:Kristomun|Kristomun]] ([[User talk: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 [[User:Kristomun|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. --[[User:Dr. Edmonds|Dr. Edmonds]] ([[User talk: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. [[User:Kristomun|Kristomun]] ([[User talk:Kristomun|talk]]) 17:50, 20 March 2020 (UTC)
:::: [[User:Kristomun|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? --[[User:Dr. Edmonds|Dr. Edmonds]] ([[User talk: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. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk: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. [[User:Kristomun|Kristomun]] ([[User talk: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 [https://forum.electionscience.org/t/utilitarian-vs-majoritarian-in-single-winner/602 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]]. --[[User:Dr. Edmonds|Dr. Edmonds]] ([[User talk: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. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk: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. --[[User:Dr. Edmonds|Dr. Edmonds]] ([[User talk: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). [[User:Kristomun|Kristomun]] ([[User talk:Kristomun|talk]]) 10:29, 21 March 2020 (UTC)
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