5 ms·

Arrow's Impossibility Theorem is one of the great results in decision theory. This is an area where Kahneman and Tversky made major contributions. Voting is i

Arrow's Impossibility Theorem is one of the great results in decision theory. This is an area where Kahneman and Tversky made major contributions. Voting is important in cases where multiple criteria need to be optimally fused into a single decision. The above link came from page talking about Arrow's Impossibility Theorem: https://mises.org/wire/arrows-impossibility-theorem-exposes-big-problem-democracy?utm_source=Mises+Institute+Subscriptions&utm_campaign=b6b76242be-EMAIL_CAMPAIGN_9_21_2018_9_59_COPY_01&utm_medium=email&utm_term=0_8b52b2e1c0-b6b76242be-227985629 https://mises.org/wire/arrows-impossibility-theorem-exposes-... ... I'm not that happy with the scary tone of the article but the summary of the theorem is a useful start for what follows... Interestingly, the IIA result (Arrow's postulate 3, on that page), was shown experimentally by Khaneman and discussed in _The Undoing Project_ to be a irrational behavior experimentally exhibited by humans making a decision. In this experiment prisoner's were asked to choose between food items and then a third, irrelevant food choice, was offered. Non-transitive choices would appear in a significant fraction of the prisoners. Reading the Mises.org article reminds me why humans might change their decision given irrelevant information (Arrow postulate 3). Breaking Arrow postulate 3 might be necessitated by using a voting algorithm that prefers being anti-dictatorial (Arrow postulate 1) and assuring that any globally preferred choice is selected over any globally less-preferred choice (Arrow postulate 2). Y It has been shown that a voting scheme can preserve Arrow postulates 1 and 2 only if one allows irrelevant choices to change voting behavior (e.g. strategic voting in our current first-past-the-post, plurality, voting system). Please consider Borda Count which is said to be close to optimal but still can be 'gamed' by strategic collusive voting. See: https://en.wikipedia.org/wiki/Borda_count https://en.wikipedia.org/wiki/Borda_count Also please consider Ranked Choice. In ranked choice there are rare cases where globally preferred choice lose to a globally less-preferred choice (Arrow postulate 2). See: https://en.wikipedia.org/wiki/Ranked_voting https://en.wikipedia.org/wiki/Ranked_voting Interestingly, Ranked Choice tends to retain incumbents in popular elections because name-recognition tends to produce many valuable runner-up votes. This is why some think a certain Senator from Maine survives challenges by well-funded opponents.

obelos 5y ago
Using cardinal (score/range) methods instead of ordinal ones avoids the constraints of Arrow. It only applies to ordinal methods.
- rfreytag 5y ago
  Perhaps obelos is referring to: https://en.wikipedia.org/wiki/Cardinal_voting https://en.wikipedia.org/wiki/Cardinal_voting
  - obelos 5y ago
    Correct. Methods that use “rating” instead of “ranking.”
- chisquared 5y ago
  I’m not sure what you mean by “avoiding constraints”, but Arrow’s theorem very much applies to cardinal methods too. Typically these methods (e.g. the Borda count) do not satisfy IIA. Cardinal methods induce an order, and once a method creates an order, anything said about ordinal methods applies.
  - obelos 5y ago
    I'm no mathematician, but Arrow himself said he was speaking of ordinal methods: “And in my theorem I’m assuming that the information is a ranking. Each voter can say of any two candidates, I prefer this one to this one.”[1] That's not to say that cardinal methods don't have failure modes, but the particular set of interdependent failures and how pathologically one or more failure of them can appear is not described generally by Arrow's theorem. How can a ballot capturing cardinal values be reduced to ordinal ones? I don't understand what you're saying here. In an election using a cardinal method, the slate of candidates can be ordered ultimately when summing results, but that's not the same information as the collective mass of ballot data. 1. https://electionscience.org/commentary-analysis/voting-theory-remembering-kenneth-arrow-and-his-impossibility-theorem/ https://electionscience.org/commentary-analysis/voting-theor...
YokoZar 5y ago
You don't need to bring irrational individual behavior into it: you can get rock-paper-scissors group cycles even when every voter has a well-ordered list of preferences. As a basic example, imagine voters prefer whoever is closer to them on some 2-dimensional politics. 3 candidates form a triangle. Draw the altitudes to form six regions, and put a voter in alternating ones. That's a rock-paper-scissors cycle, with very simple rational voters.

obelos 5y ago

Using cardinal (score/range) methods instead of ordinal ones avoids the constraints of Arrow. It only applies to ordinal methods.

rfreytag 5y ago
Perhaps obelos is referring to: https://en.wikipedia.org/wiki/Cardinal_voting https://en.wikipedia.org/wiki/Cardinal_voting
- obelos 5y ago
  Correct. Methods that use “rating” instead of “ranking.”
chisquared 5y ago
I’m not sure what you mean by “avoiding constraints”, but Arrow’s theorem very much applies to cardinal methods too. Typically these methods (e.g. the Borda count) do not satisfy IIA. Cardinal methods induce an order, and once a method creates an order, anything said about ordinal methods applies.
- obelos 5y ago
  I'm no mathematician, but Arrow himself said he was speaking of ordinal methods: “And in my theorem I’m assuming that the information is a ranking. Each voter can say of any two candidates, I prefer this one to this one.”[1] That's not to say that cardinal methods don't have failure modes, but the particular set of interdependent failures and how pathologically one or more failure of them can appear is not described generally by Arrow's theorem. How can a ballot capturing cardinal values be reduced to ordinal ones? I don't understand what you're saying here. In an election using a cardinal method, the slate of candidates can be ordered ultimately when summing results, but that's not the same information as the collective mass of ballot data. 1. https://electionscience.org/commentary-analysis/voting-theory-remembering-kenneth-arrow-and-his-impossibility-theorem/ https://electionscience.org/commentary-analysis/voting-theor...

YokoZar 5y ago

You don't need to bring irrational individual behavior into it: you can get rock-paper-scissors group cycles even when every voter has a well-ordered list of preferences. As a basic example, imagine voters prefer whoever is closer to them on some 2-dimensional politics. 3 candidates form a triangle. Draw the altitudes to form six regions, and put a voter in alternating ones. That's a rock-paper-scissors cycle, with very simple rational voters.