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Arrow's Theorem

67 点作者 infinity超过 10 年前

9 条评论

YokoZar超过 10 年前
Arrow&#x27;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&#x27;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&#x27;t exist if you are willing to accept that rock-paper-scissors situations can happen among people&#x27;s preferences.<p>I&#x27;m not being hyperbolic here. Number 1 and 2 are how people quickly use vague citations to Arrow&#x27;s Theorem to shut down talk about voting reform even when the status quo consists of provably terrible systems like plurality voting.<p>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&#x27;s theorem simply doesn&#x27;t apply. This is well known: that &quot;top loop&quot; is the Smith Set and every Condorcet method of voting satisfies it.<p>There&#x27;s also another interesting result: if voters have merely &quot;single-peaked preferences&quot;, such as opinions about where to set a volume knob, then Arrow&#x27;s theorem also doesn&#x27;t apply since there will be no rock-paper-scissors set of equally fair options.
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chimeracoder超过 10 年前
&gt; Arrow&#x27;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<p>The article alludes to one very important corollary, though IMHO it doesn&#x27;t explain it very well: Arrow&#x27;s theroem is like the CAP theorem - it seems to be a much stronger restriction than it is. In other words, if you&#x27;re willing to make just a few very straightforward assumptions (ie, compromises), you <i>can</i> create a system that appears to &quot;satisfy... rationality of their preferences&quot;.<p>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].<p>Let&#x27;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.<p>[0] <a href="https://en.wikipedia.org/wiki/Amartya_Sen" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Amartya_Sen</a><p>[1] <a href="https://en.wikipedia.org/wiki/Instant-runoff_voting" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Instant-runoff_voting</a><p>[2] The article does mention Sen, and alludes to this finding, but I don&#x27;t think it explains it very clearly.
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tunesmith超过 10 年前
&quot;Arrow&#x27;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.&quot;<p>The trick there is the &quot;apparently quite reasonable&quot; 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&#x27;s not quite so certain anymore that a &quot;perfect&quot; vote-counting method is impossible.
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Kryptor超过 10 年前
This theorem only applies to <i>ordinal</i> voting systems. Cardinal voting algorithms like score and approval voting escape this predicament.<p><a href="http://rangevoting.org/ArrowThm.html" rel="nofollow">http:&#x2F;&#x2F;rangevoting.org&#x2F;ArrowThm.html</a>
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theschwa超过 10 年前
Is there a good data structure to represent Ordinal Data? I frequently see pairwise count matrices used [1], but they don&#x27;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].<p>[1] <a href="http://en.wikipedia.org/wiki/Condorcet_method#Pairwise_counting_and_matrices" rel="nofollow">http:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Condorcet_method#Pairwise_count...</a> [2] <a href="http://en.wikipedia.org/wiki/Borda_count#Voting_and_counting" rel="nofollow">http:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Borda_count#Voting_and_counting</a>
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aaron-lebo超过 10 年前
William Poundstone&#x27;s <i>Gaming the Vote</i> is a great read that covers this and related topics.
yomritoyj超过 10 年前
Interestingly, the theorem is false if the number of voters is infinite <a href="http://blog.jyotirmoy.net/2013/10/arrows-impossibility-theorem-is-false.html" rel="nofollow">http:&#x2F;&#x2F;blog.jyotirmoy.net&#x2F;2013&#x2F;10&#x2F;arrows-impossibility-theor...</a>
jordanpg超过 10 年前
A fascinating result. I am interested in understanding how this can apply to decisions about consuming news.<p>Given some objective, like &quot;stay informed&quot; or &quot;make the best possible choice when I vote&quot;, and given certain problems like &quot;news is corrupted by advertising&quot; or &quot;news is dumbed down&quot;, is it at all plausible to achieve any of those goals in 1 hour of news consumption each day?<p>A naive application of this theorem suggests not... &quot;none, anyway, that satisfy certain apparently quite reasonable assumptions concerning the autonomy of the people and the rationality of their preferences.&quot;<p>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?
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Symmetry超过 10 年前
Always interesting, but given the median voter theorem is I&#x27;m not sure it&#x27;s a problem in practice.<p>[1]<a href="http://en.wikipedia.org/wiki/Median_voter_theorem" rel="nofollow">http:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Median_voter_theorem</a>
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