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Arrow's Theorem
- aaron-lebo 12y agoWilliam Poundstone's Gaming the Vote is a great read that covers this and related topics.
- jordanpg 12y agoA fascinating result. I am interested in understanding how this can apply to decisions about consuming news. Given some objective, like "stay informed" or "make the best possible choice when I vote", and given certain problems like "news is corrupted by advertising" or "news is dumbed down", is it at all plausible to achieve any of those goals in 1 hour of news consumption each day? A naive application of this theorem suggests not... "none, anyway, that satisfy certain apparently quite reasonable assumptions concerning the autonomy of the people and the rationality of their preferences." In other words, might it be the case that there is little intrinsic value in consuming the news piecemeal, in the way that most of us do?
- aaron-lebo 12y agoI'm not sure you can expand the theorem to deal with that issue. Arrow's theorem essentially says even if you as an individual or society as a whole had perfect news sources, there is no voting method (edit: no ordinal voting method), or method of choice (other than dictatorship, which is bad for obvious reasons), which can accurately register preferences. All voting systems fail at some point, though some are better than others. To be cynical, in the context of voting, consume news however you want to because to some extent it doesn't matter.
- yongjik 12y agoNOPE. It just means that those "apparently quite reasonable assumptions" weren't reasonable after all and must be relaxed a bit according to a precise definition. You just took a mathematical theorem and went into a completely unrelated tangent. It's as if someone says there's no point in keeping development schedule because temporal order is relative to the observer. Please don't do that.
- jordanpg 12y ago"Say there are some alternatives to choose among. They could be policies, public projects, candidates in an election, distributions of income and labour requirements among the members of a society, or just about anything else. There are some people whose preferences will inform this choice, and the question is: which procedures are there for deriving, from what is known or can be found out about their preferences, a collective or “social” ordering of the alternatives from better to worse?" Keep reading. This one is more than just math.
- tunesmith 12y ago"Arrow's theorem says there are no such procedures whatsoever—none, anyway, that satisfy certain apparently quite reasonable assumptions concerning the autonomy of the people and the rationality of their preferences." The trick there is the "apparently quite reasonable" part. Far too many people take it as a given that those assumptions are sacred, which leads to a form of worship about this theorem. The IIAC in particular is problematic, and if that is relaxed, it's not quite so certain anymore that a "perfect" vote-counting method is impossible.
- baddox 12y agoI'm curious which of the assumptions you think are unreasonable.
- tunesmith 12y agoIf you have a Smith Set, you don't want to just arbitrarily remove one of its candidate to resolve the "tie". Similarly, if an election has a Condorcet Winner, and adding a candidate creates a Smith Set (thus potentially violating IIA if a certain candidate is chosen in a "tiebreaker"), this is not necessarily a bad thing. If that happens it means that people always legitimately had those views, and were not able to properly express their views by choosing among the more limited number of candidates.
- baddox 12y agoRegarding your second sentence, who says that would be a bad thing? I'm a little confused about the situation you're describing. Most analysis of voting systems I've seen isn't concerned with the fact that valid candidates may be left off the ballot. That's obviously an important consideration in real elections, and if there are primary elections the choice of voting system there obviously matters, but in the final election it doesn't seem particularly relevant.
- chimeracoder 12y ago> Arrow's theorem says there are no such procedures whatsoever—none, anyway, that satisfy certain apparently quite reasonable assumptions concerning the autonomy of the people and the rationality of their preferences The article alludes to one very important corollary, though IMHO it doesn't explain it very well: Arrow's theroem is like the CAP theorem - it seems to be a much stronger restriction than it is. In other words, if you're willing to make just a few very straightforward assumptions (ie, compromises), you can create a system that appears to "satisfy... rationality of their preferences". Nobel laureate Amartya Sen[0] has demonstrated that if you assume that there are certain rankings of preferences that are rare enough to be ignored altogether, then instant-runoff voting[1] will in fact satisfy all the constraints of the Impossibility Theorem[2]. Let's use the 2000 US Presidential election as an example. There were three main candidates in Florida (Bush, Gore, Nader), for a total of 6 rankings. While Nader played a spoiler role, Nader and Gore shared more of a platform than Nader and Bush did. So it is very reasonable to assume that there are few people who would have voted for Nader over Bush, but Bush over Gore. This reasoning allows us to eliminate a number of those 6 rankings - and more importantly, enough rankings that the criteria of the Impossibility Theorem are likely to hold. [0] https://en.wikipedia.org/wiki/Amartya_Sen https://en.wikipedia.org/wiki/Amartya_Sen [1] https://en.wikipedia.org/wiki/Instant-runoff_voting https://en.wikipedia.org/wiki/Instant-runoff_voting [2] The article does mention Sen, and alludes to this finding, but I don't think it explains it very clearly.
- trhway 12y ago> Arrow's theroem is like the CAP theorem accepting dictatorship in the former or non-availability in the later gives you consistency. Reminds about USSR - high consistency (at it least it was initially) with very limited availability of salami (all the way until the last day of USSR).
- tunesmith 12y agoThis is probably just being pedantic, but the Nader->Bush->Gore ranking was commonly preferred by Nader supporters, not because they preferred Bush's policies to Gore's, but because they thought it would teach society the error of their ways in the long run or something. Also because they hated Nader being labeled a spoiler candidate, so it was probably something of a spiteful response.
- YokoZar 12y agoArrow's theorem has to be one of the most overhyped and overplayed theorems in existence. Here, for instance, are three major things what it doesn't say: 1) That all voting systems are equivalent in any sort of meaningful sense 2) That the current voting system you are using is not awful 3) That a perfect class of voting system doesn't exist if you are willing to accept that rock-paper-scissors situations can happen among people's preferences. I'm not being hyperbolic here. Number 1 and 2 are how people quickly use vague citations to Arrow's Theorem to shut down talk about voting reform even when the status quo consists of provably terrible systems like plurality voting. Number 3 is the true result that if you relax the rather overly-strongly defined IIA criteria with a much more-reasonable criteria -- that the winner must remain among the top rock-paper-scissors loop of the voters -- then Arrow's theorem simply doesn't apply. This is well known: that "top loop" is the Smith Set and every Condorcet method of voting satisfies it. There's also another interesting result: if voters have merely "single-peaked preferences", such as opinions about where to set a volume knob, then Arrow's theorem also doesn't apply since there will be no rock-paper-scissors set of equally fair options.
- aaron-lebo 12y agoNumber 3 is a valid criticism, but criticizing the theorem on the basis of people using it to support the status quo is like criticizing water because some people drown. His theorem says nothing about the way things should be, but rather about how things are. I mentioned Poundstone's Gaming the Vote in another post, but that book in particular discusses the implications of the theorem and analyzes better alternatives to current systems.
- Retric 12y agoThe problem is people assume it means more than it does. It's like someone using Godel's incompleteness theorem to argue that math is pointless.
- robbrown451 12y agoWell he said it was overhyped, which isn't so much a criticism of the theorem, but a criticism of the hypers.
- Symmetry 12y agoAlways interesting, but given the median voter theorem is I'm not sure it's a problem in practice. [1]http://en.wikipedia.org/wiki/Median_voter_theorem http://en.wikipedia.org/wiki/Median_voter_theorem
- eru 12y agoThat only applies for one dimensional politics, and, if plurality voting is used, a two party system.
- Kryptor 12y agoThis theorem only applies to ordinal voting systems. Cardinal voting algorithms like score and approval voting escape this predicament. http://rangevoting.org/ArrowThm.html http://rangevoting.org/ArrowThm.html
- robbrown451 12y agoAnd they have far bigger problems of their own, compared to condorcet methods. Score is the worst....any rational voter will just vote with a 0% or 100% for all candidates. Those who vote "sincerely" are at a huge disadvantage. Approval gives a big advantage to those who can best predict how others will vote. Yuk.
- Retric 12y agoNot nessisarily, suppose you like X your OK with Y and you hate Z. Do you vote 100% on Y or 0% on Y.
- baddox 12y agoIt would depend on whether X or Y is more likely to win the election (according to polls). You can hurt X by voting 100% on Y.
- Retric 12y agoYour specific vote is only important when the election is to close to call. If X is up by 80% in the polls you can probably stay home and nothing happens.
- robbrown451 12y agoYeah as baddox says, you'd still be smart to vote 0% or 100%, as long as you can make a better than random pick at who is the leader and runner up. That's because voting in "black and white" is the same as Approval, and as I said, Approval gives an advantage to those who can guess better who are the leaders. Making it especially annoying in elections (such as local ones) where you have no idea who the front runners are.
- theschwa 12y agoIs there a good data structure to represent Ordinal Data? I frequently see pairwise count matrices used [1], but they don't seem to completely capture all of the information. For example, the number of times a choice was first, second, third, or last such as would be necessary for the Borda method [2]. [1] http://en.wikipedia.org/wiki/Condorcet_method#Pairwise_counting_and_matrices http://en.wikipedia.org/wiki/Condorcet_method#Pairwise_count... [2] http://en.wikipedia.org/wiki/Borda_count#Voting_and_counting http://en.wikipedia.org/wiki/Borda_count#Voting_and_counting
- rspeer 12y agoWell, Condorcet and Borda do fundamentally different things with people's preferences. There's no real way to simplify it beyond "here's a set containing the rank order of every voter" if you don't know in advance what election method will be used. In fact, you might argue that the rank order is insufficient, and you also need to know whether the voter would approve of each option in an approval vote, or 0-10 rankings for range voting, or whatever. Condorcet cares most of all about people's preferences between pairs of options, so that's why you'd summarize it as a matrix made out of those pairwise preferences. Borda cares most of all how many times something is "first place", "second place", and so on, and cares nothing about the pairwise rankings; Borda advocates claim that pairwise rankings will just confuse the issue. (Opinion: I happen to believe that Borda is garbage for anything where politics is involved. It might be okay for the kind of contests where it's in use, like the baseball MVP or Eurovision, except those are probably secretly political too.)
- yomritoyj 12y agoInterestingly, the theorem is false if the number of voters is infinite http://blog.jyotirmoy.net/2013/10/arrows-impossibility-theorem-is-false.html http://blog.jyotirmoy.net/2013/10/arrows-impossibility-theor...