Hi, [I'm sorry it took so many days before I finished this reply to Jobst's message. I'm also sorry there are a lot of other messages to which I've wanted to reply but haven't found time...]
Jobst H wrote: > Steve E wrote: >> Jobst's "immunity" is weaker than the "immunity from >> majority complaints" defined in my web pages (linked >> from <www.alumni.caltech.edu/~seppley>), which MAM >> satisfies but River does not. > > Of course beatpath or river or any other method designed > to elect a single winner don't satisfy that criterion > simply because they are not designed to construct a > social ordering but only to find a winner. However, > if you find it important to have a social ordering > in addition to the winner, you can easily take a tree > of maximal beatpaths (in case of the beatpath method) > or the tree-like river diagram (in case of the river > method) as this social ordering. Then your kind of > immunity is easily fulfilled. If you require orderings > to be complete (which I guess), you can easily complete > the tree-like ordering in whatever way to get a complete > ordering which fulfils your criterion. No, my "immunity from majority complaints" criterion (IMC) is more demanding than that. It can't be satisfied merely by extending a "single winner only" voting method so it also constructs a social ordering consistent with maximal strength beatpaths. The social ordering must be more self-consistent than that sometimes is. My web pages provide an example that shows BeatpathWinner is not immune even when extended to construct a social ordering. (See "Scenario A10" in appendix A of the "immunity from majority complaints" webpage linked to the www.alumni.caltech.edu/~seppley webpage.) > Some time ago, I sent you this reply: > When I called the criterion you refer to by the name > "immunity" some years ago, I surely didn't mean to > interfere with your terminology. I came to know your > excellent web pages only this year... So, it is only > a funny coincidence that we independently termed two > very similar criteria by almost the same name :-) > > Anyway, I don't think there is too much confusion. > First of all, I only termed it "immunity" in my > postings (and more lengthy "[weak or strong] immunity > from binary arguments" in the old paper of mine). > Secondly, as you say it can be considered a weaker > form of your "immunity from majority complaints". > Thirdly, my immunity is a property of *candidates* > given some preference profile, whereas your immunity > is a property of *rankings* which implies that > the top candidate is immune in my sense. But the reverse isn't true. When the top candidate is immune in Jobst's sense, it does not imply that some social ordering could be constructed (with the winner atop the social ordering, of course) that is immune from majority complaints in my sense. Jobst's immunity is significantly weaker, not just weaker in the mild technical sense of a social ordering not being required. > If we want to make it absolutely clear in the future, > we could stop calling a method itself immune and > instead say that a method "elects an immune candidate" > or "constructs an immune ranking", respectively. > What do you think? > > but I did not receive an answer yet :-) True, I didn't send an answer. Sorry about that. I haven't found the time to think about the terminology. It's clear now, though, that that discussion might have been premature since it appears from your writing that you have had a misconception about my IMC criterion, which is more demanding than you appear to have noticed. This is probably my fault since the descriptions of criteria in my home page are brief and don't always match the detailed descriptions linked on my other webpages. My IMC is so demanding that it can be satisfied only by MAM (and variations of MAM that differ from MAM only in tiebreaking. This is NOT just due to other voting methods being narrowly defined so they don't construct a social ordering. Even if those methods are extended so they construct a social ordering, they still fail IMC. BeatpathWinner also fails "immunity from second-place complaints" (I2C): Immunity from Second-Place Complaints (I2C) ------------------------------------------- Let w denote the alternative that wins. Let x denote the alternative that would win if w were deleted from all ballots. The number of voters who ranked x over w must not exceed the number of voters who ranked w over x. (I2C and some other related criteria are defined in my webpage that provides the detailed definition of IMC.) Note that I2C does not require the voting method to output a social ordering! (Instead, it implicitly refers to a natural iterative way to extend any single-winner voting method to produce a social ordering.) I2C is in the same spirit as IMC since it could be difficult to rebut the complaining majority when I2C is violated. Complaints from a majority who prefer a "2nd place" alternative over the winner would be the most dangerous, I think. Thus violations of I2C can be considered the most egregious way to fail IMC, which means satisfaction of I2C is at least as important as satisfaction of IMC. I do not know if River satisfies I2C. > You continued: >> But being a weaker property, perhaps it would be better >> to call his criterion "resistance to majority complaints" >> rather than "immunity from majority complaints." >> That would be in the same spirit as the use of >> that word in the "truncation resistance" criterion. > > I don't think it is in the same spirit since "truncation > resistance" refers to a strategy being applied at voting > time where "immunity from binary arguments" refers to > arguments being given after the election. Yes, in that sense truncation resistance differs from these immunity properties. But what I had in mind when I wrote "same spirit" was an analogy to a "truncation immunity" property stronger than truncation resistance. For instance, the truncation immunity satisfied by IRV: Truncation Immunity ------------------- Suppose x does not win given votes V. Let V' denote a collection of votes that is the same as V except some votes in V' are truncated somewhere below where they rank x. Then x must also not win given V'. All I meant was that, to me, the word resistance connotes something weaker than does the word immunity, so it might be a good name for a criterion weaker than my immunity criterion. >> Since the strength of pairwise "defeats" in Jobst's >> proposed compromise method is determined by "approval" >> rather than by preference, it's not obvious to me >> that that method satisfies all the criteria Jobst >> listed. Does it really satisfy either of the two >> immunity criteria, for instance? > > Immunity from binary arguments means the method elects > an immune candidate. A candidate x is immune when for > each defeat y>x, there is a sequence of defeats x>...>y > all having the same or larger *strength*. So, beatpath, > ranked pairs, and river all are immune no matter how > strength is defined, but the meaning of "immune" changes > with the meaning of "strength" of course! That's what I meant when I said I wasn't sure your (weaker) immunity criterion would be satisfied. One must redefine it in order to claim satisfaction, and that's not quite the same as satisfying the original criterion. The justification for one criterion won't necessarily hold for an amended criterion. > So, to be accurate, "immunity from majority complaints" > only implies "immunity from binary arguments" when > strength is defined as winning votes. I guess it will > be almost impossible to define a cardinal weighted > pairwise derivative which fulfils "immunity from majority > complaints" in the specific sense of your site. But when > we follow the idea of using cardinal information to > define defeat strength, then any appropriate definition > of immunity will also take this into account. My > motivation for imposing immunity is this: When there > is a number of people prefering some candidate y so > much to the winner x that they support an argument > to replace x by y, then there should be arguments > of the same kind leading back to x in order to be able > to show that the argument would be of no use to its > supporters. Now, when we distinguish between weak and > strong preferences, it seems natural to me to assume > that only those with a strong preference y>x will support > the argument to replace x by y, hence the definition of > immunity should then also refer to strong preferences > only. You may be right. But I'm thinking that since people are so familiar with "majority rule" that the main danger is from complaints based on a "majority rule" argument. Also, it may be that people with a mild preference for y>x will also tend to support the argument to replace x by y. Not just people with a strong preference. >> There's another "compromise" method that may be worth >> comparing to Jobst's, which I wrote about long ago >> when I defined the "sincere defense" criterion >> (which is stronger than minimal defense and >> Mike's SDSC.) > > How wonderful! Whenever I have some idea, I assume many > people must have had the same idea before :-) So what > is that method? I tried to find it in the archives > but didn't succeed... I searched too and couldn't find any messages about the sincere defense criterion. So a moment ago I posted a separate message with more info about sincere defense. >> My first impression of Jobst's proposed compromise method >> is that it satisfies sincere defense but that it would >> not be as robust as the methods I wrote about (that >> also satisfy minimal defense) in the case where a >> significant number of voters know their sincere order >> of preference but do not know where to strategically >> place the dividing line. > > When they don't know where to strategically place the line > but place it sincerely at the position of their strongest > pairwise preference, then that is just fine since we are > trying to keep voters from voting strategically, aren't > we? But perhaps I just don't understand what you mean > here... I suggest you read the message I just posted about sincere defense before continuing to read this... Note that if one of the voting methods I described in that separate message is used, for instance MAM suitably modified, the minimal defense strategy would still be effective to defeat "greater evil" candidates in the case when a majority fails to coordinate their use of the dividing line to defeat such candidates. For example, suppose Nader voters, who were part of the majority who preferred Gore over Bush, had voted "Nader / Gore > Bush," stubbornly refuse to rank Gore over the dividing line. It doesn't follow that they have only a weak preference for Gore over Bush, or that their preference for Gore over Bush is weaker than the preference for Bush over Gore of voters who vote Bush / Gore > Nader. In other words, what if two voters have these preference intensities: Z >>>> Y >>> X X >> Y > Z If the top voter places the dividing line at the point of strongest preference, between Z & Y, then she does not provide as much help to Y over X as she could. Yet her preference for Y over X is stronger than the preference for X over Y of the bottom voter, who would place the dividing line where it is strategically optimal. Attempting interpersonal comparisons of preference intensities given only expressions in ballots is too difficult for me! And I'm not certain it's the right goal anyway. Suppose selfish evil people tend to have preferences that are more intense than those of socially responsible people? I don't know that that's the case. But I'm putting my faith in the greater number of people who have some preference, including in that count the people who don't claim to have a strong preference. Here's my heuristic: The greater the number of people who say x is better than y, the more likely it is that x is better for society than y. Won't implementing that heuristic with a method such as MAM (or MAM with the dividing line, if the voters can tolerate its extra complexity) suffice to align the incentives of the politicians with the interests of the people, weeding out corruption, as well as any voting method could? (You can infer that I hope there is no need to "compromise" with advocates of Approval or with people who want to let voters express ratings.) By satisfying both (strong) sincere defense and my second "nearly equivalent" wording of minimal defense, that majority is given two ways to defeat the "greater evil" candidates, and the two ways are not mutually incompatible. Both strategies can be attempted. The compromise method that Jobst proposed does not satisfy my second wording of minimal defense, because his method measures the strength of a majority for x over y solely by the voters' placements of the dividing line: #Vx/y. Thus, in his method, a majority who rank x over y may be treated as smaller than #Vx>y, their actual size. Since I put my faith in the size of the larger majority rather than in the claimed intensities of a smaller majority (or minority), that seems unnecessarily risky. But I think I'm open to persuasion, if I see a good argument for heeding the smaller majority's "cheap talk" claim of stronger preferences. Perhaps I should summarize here an argument I've posted several times... Society has had a lot of experience with a voting method extremely similar to Approval: we vote "yes" or "no" on ballot propositions. (It's not identical to Approval since the option to continue the status quo is not explicitly on the ballot and wins if no proposition explicitly on the ballot earns more yeses than nos.) Conventional wisdom is that when two or more conflicting propositions are placed on the ballot, they can spoil each other's chances because some voters tend to vote no on compromise alternatives that they'd vote yes on if a more preferred alternative were not also on the ballot. (I cited an example from several years ago: The LA Times recommended voters vote yes to expand the LA City Council from 15 to 25 but no to expand the Council to 21, while simultaneously opining that either 25 or 21 would be better than 15. They explained their no on 21 by saying they didn't want the compromise to defeat their favorite, 25. But on election day, both lost. Would 21 have won if the Times had recommended yes on 21? If so, 25 was a spoiler.) Whether or not the conventional wisdom will be true of Approval, if the elite actors (politicians, rich donors, etc.) believe it's true then their incentive would be, as it is now, to avoid placing too many candidates on the ballot. In other words, we'd still have two big parties each nominating one candidate per office, and thus voters would not be given the chance to sort candidates having similar & winnable platforms from least corrupt to most corrupt. (Similar logic holds for simple cardinal rating schemes.) But Jobst's compromise method looks much better than Approval (neglecting its extra complexity) since it pays attention to the existence of majorities, differing from other good methods only in how it measures the strengths of the majorities (and the extra complexity of its dividing line). So it satisfies a lot of important majoritarian criteria, such as top cycle, and maybe it would suffice to bust the incentive that maintains the "only two viable candidates" system. --Steve ---- Election-methods mailing list - see http://electorama.com/em for list info
