6 ms·

Neither the blog post kindly submitted here nor the comments (so far) nor the Wikipedia article mentioned in one comment mention the impossibility theorem prove

Neither the blog post kindly submitted here nor the comments (so far) nor the Wikipedia article mentioned in one comment mention the impossibility theorem proved by Kenneth Arrow,[1] which shows that we can't build a perfect voting system to take into account preferences among three or more candidates. As a voter in a state with as many as ten candidates on the ballot in a typical presidential election, and some amazingly close three-way statewide elections,[2] I'm surprised that the Arrow paradox (Arrow impossibility theorem) isn't more widely known. I learned about it in an article in Scientific American back in the 1970s. My state has three "major parties" that have automatic ballot access for nominated candidates, and I expect that this year we may add one more political party to the list of major parties by the rules established in state law. If voters have more than two choices, odd results can happen in elections. AFTER EDIT: A comment posted while I was typing the first version of this comment mentions "range voting" as a response to the Arrow paradox. A commentary by an economist[3] and a problem set by a mathematician[4] may suggest to thoughtful readers here some issues to ponder while discussing whether or not range voting provides better trade-offs than our current voting system, and whether it indeed escapes the theorem proved by Arrow. [1] https://en.wikipedia.org/wiki/Arrow's_impossibility_theorem https://en.wikipedia.org/wiki/Arrow's_impossibility_theorem http://www.math.ucla.edu/~tao/arrow.pdf http://www.math.ucla.edu/~tao/arrow.pdf [2] http://latimesblogs.latimes.com/washington/2008/11/franken-coleman.html http://latimesblogs.latimes.com/washington/2008/11/franken-c... http://www.politico.com/news/stories/0609/24383.html http://www.politico.com/news/stories/0609/24383.html http://www.amazon.com/This-Is-Not-Florida-Minnesota/dp/0816670382 http://www.amazon.com/This-Is-Not-Florida-Minnesota/dp/08166... [3] http://marginalrevolution.com/marginalrevolution/2009/11/range-voting.html http://marginalrevolution.com/marginalrevolution/2009/11/ran... [4] http://www.math.cornell.edu/~mec/Summer2008/anema/approval.html http://www.math.cornell.edu/~mec/Summer2008/anema/approval.h...

_dps 13y ago
I will add to your list of references an excellent book, "Basic Geometry of Voting" by D. Saari [1]. It covers all major alternative voting systems in an accessible mathematical framework that clearly highlights their tradeoffs (necessitated by Arrow's theorem) in simple diagrams. I strongly recommend this book for anyone interested in multi-participant decision systems, or anyone looking for a fun mathematical excursion. Sadly, the paperback (which I bought in graduate school for $25) is no longer in print and the price has jumped quite a bit. [1] http://www.amazon.com/Basic-Geometry-Voting-Donald-Saari/dp/3540600647 http://www.amazon.com/Basic-Geometry-Voting-Donald-Saari/dp/...
- tokenadult 13y ago
  Thanks for the tip on the book. It looks like the same mathematician author later wrote a more popular, less technical book on voting, Chaotic Elections! A Mathematician Looks at Voting, that may be of interest to several people following the thread here. http://www.amazon.com/Chaotic-Elections-Mathematician-Looks-Voting/dp/0821828479/ http://www.amazon.com/Chaotic-Elections-Mathematician-Looks-...
dllthomas 13y ago
Range voting still has its issues - every voting methodology has some issues - but range voting pretty well dominates plurality voting IMO.
- kybernetikos 13y ago
  This is definitely a point worth making. Arrows theorem is an extremely interesting result, but I'm fed up of hearing it brought out as an excuse for not changing from a plurality system. Yes, every social choice mechanism has issues, but some have an awful lot more issues than others.
  - baddox 13y ago
    The point of bringing up Arrow's theorem is to debunk people who claim that certain voting systems are "more democratic" or "better" in some objective sense. What they should be saying is that they prefer a certain voting system because they value certain attributes of voting systems over others.
    - YokoZar 13y ago
      There are reasonable and unreasonable definitions of "fair" and "better". Some voting systems have to have very very unreasonable (to a normal human) definitions of "fair" to be considered superior by any sort of consistent logic.
    - kybernetikos 13y ago
      Well if that's the point, I definitely disagree with it. Some voting systems genuinely are more democratic or better than others in an objective sense. For example, a voting system where one person determined by their status in society has their vote count and everyone elses is discarded is objectively less democractic than a voting system where a ballot is selected at random from all voters and is taken to determine the result. Neither of these systems is as democratic as a system where everyone votes for their favourite and the one with the most votes wins, and that in turn is not as democratic as a condorcet-loser system, where it's not possible for a candidate that the majority rank last to win. Just because all of the options have problems, it doesn't mean that there aren't options that are completely dominated by others.
tunesmith 13y ago
I don't often see this opinion, but I think the Impossibility Theorem is over-rated. It doesn't state you can't build a "perfect voting system". It states that you can't build a voting system that will always conform to the different criteria that Arrow chose for the theorem. But it doesn't attempt to prove or justify that those four criteria are absolutely required for a "perfect voting system". "Independence Of Irrelevant Alternatives" is thought to be particularly problematic. There definitely are thought experiments that demonstrate that IIA is not as desirable a criteria (to rule out a voting system) as you would want. If a voting ballot runs afoul of IIA, it can point more to a confused voting population than a flawed voting method. For instance, the Condorcet Criterion runs afoul of IIA. The Condorcet Criterion states that if a candidate would beat every other candidate one-on-one, that candidate would be the winner. Apparently it is possible to introduce additional candidates (that don't change the relative rankings between pre-existing candidates) such that a Smith Set is instead produced. However, it is not possible to introduce additional candidates such that a different Condorcet Winner is produced. But let's think about what that says. A Smith Set is not a problem in itself. It means an indecisive or conflicted voting population. It is interesting that additional candidates can expose a conflicted voting population, a confusion that actually already was there; just covered up by the lack of candidates. In other words, if IIA changes a result from a Condorcet Winner to a Smith Set, it is an indication that the population was not given enough choice in the original ballot. In that sense, IIA is actually useful, and not a flaw. It does not point out a flaw with the Condorcet voting mechanism; it points out a flaw with the limited choice of candidates (and time to research/understand those candidates) for that vote. This is part of the problem with voting theory, is that they make unwarranted theoretical assumptions, such that a vote will always offer sufficient choice to a voting population. At any rate, I believe IIA is a flawed criterion, and therefore the Impossibility Theorem, while true, is less useful than it is celebrated to be.
- YokoZar 13y ago
  Arrow's Theorem has to be one of the most over-celebrated yet uninteresting bits of math out there. I myself even thought it was a cool way of challenging alternative voting systems when I first started studying them. But, as you say, Arrow's Theorem is simply too broad with it's definition of "fair". It's entirely reasonable for the voters to have a list of candidates they hate and a list of 3 candidates they would prefer, and for there to be a rock-paper-scissors situation among those top three candidates. It's entirely possible to have a voting system that only chooses from among the rock-paper-scissors options, and indeed every Condorcet system will make such a choice. Those options are the Smith set. The tragedy is that Arrow's Theorem is often used to justify systems (like plurality or range voting) that often won't choose from among those top three, and might even choose the Condorcet loser! I'm not sure what a good definition of "fair" is, but I'd settle for not picking the candidate that would lose to literally every other candidate in a one-on-one election.

_dps 13y ago

I will add to your list of references an excellent book, "Basic Geometry of Voting" by D. Saari [1]. It covers all major alternative voting systems in an accessible mathematical framework that clearly highlights their tradeoffs (necessitated by Arrow's theorem) in simple diagrams. I strongly recommend this book for anyone interested in multi-participant decision systems, or anyone looking for a fun mathematical excursion. Sadly, the paperback (which I bought in graduate school for $25) is no longer in print and the price has jumped quite a bit. [1] http://www.amazon.com/Basic-Geometry-Voting-Donald-Saari/dp/3540600647 http://www.amazon.com/Basic-Geometry-Voting-Donald-Saari/dp/...

tokenadult 13y ago
Thanks for the tip on the book. It looks like the same mathematician author later wrote a more popular, less technical book on voting, Chaotic Elections! A Mathematician Looks at Voting, that may be of interest to several people following the thread here. http://www.amazon.com/Chaotic-Elections-Mathematician-Looks-Voting/dp/0821828479/ http://www.amazon.com/Chaotic-Elections-Mathematician-Looks-...

dllthomas 13y ago

Range voting still has its issues - every voting methodology has some issues - but range voting pretty well dominates plurality voting IMO.

kybernetikos 13y ago
This is definitely a point worth making. Arrows theorem is an extremely interesting result, but I'm fed up of hearing it brought out as an excuse for not changing from a plurality system. Yes, every social choice mechanism has issues, but some have an awful lot more issues than others.
- baddox 13y ago
  The point of bringing up Arrow's theorem is to debunk people who claim that certain voting systems are "more democratic" or "better" in some objective sense. What they should be saying is that they prefer a certain voting system because they value certain attributes of voting systems over others.
  - YokoZar 13y ago
    There are reasonable and unreasonable definitions of "fair" and "better". Some voting systems have to have very very unreasonable (to a normal human) definitions of "fair" to be considered superior by any sort of consistent logic.
  - kybernetikos 13y ago
    Well if that's the point, I definitely disagree with it. Some voting systems genuinely are more democratic or better than others in an objective sense. For example, a voting system where one person determined by their status in society has their vote count and everyone elses is discarded is objectively less democractic than a voting system where a ballot is selected at random from all voters and is taken to determine the result. Neither of these systems is as democratic as a system where everyone votes for their favourite and the one with the most votes wins, and that in turn is not as democratic as a condorcet-loser system, where it's not possible for a candidate that the majority rank last to win. Just because all of the options have problems, it doesn't mean that there aren't options that are completely dominated by others.

tunesmith 13y ago

I don't often see this opinion, but I think the Impossibility Theorem is over-rated. It doesn't state you can't build a "perfect voting system". It states that you can't build a voting system that will always conform to the different criteria that Arrow chose for the theorem. But it doesn't attempt to prove or justify that those four criteria are absolutely required for a "perfect voting system". "Independence Of Irrelevant Alternatives" is thought to be particularly problematic. There definitely are thought experiments that demonstrate that IIA is not as desirable a criteria (to rule out a voting system) as you would want. If a voting ballot runs afoul of IIA, it can point more to a confused voting population than a flawed voting method. For instance, the Condorcet Criterion runs afoul of IIA. The Condorcet Criterion states that if a candidate would beat every other candidate one-on-one, that candidate would be the winner. Apparently it is possible to introduce additional candidates (that don't change the relative rankings between pre-existing candidates) such that a Smith Set is instead produced. However, it is not possible to introduce additional candidates such that a different Condorcet Winner is produced. But let's think about what that says. A Smith Set is not a problem in itself. It means an indecisive or conflicted voting population. It is interesting that additional candidates can expose a conflicted voting population, a confusion that actually already was there; just covered up by the lack of candidates. In other words, if IIA changes a result from a Condorcet Winner to a Smith Set, it is an indication that the population was not given enough choice in the original ballot. In that sense, IIA is actually useful, and not a flaw. It does not point out a flaw with the Condorcet voting mechanism; it points out a flaw with the limited choice of candidates (and time to research/understand those candidates) for that vote. This is part of the problem with voting theory, is that they make unwarranted theoretical assumptions, such that a vote will always offer sufficient choice to a voting population. At any rate, I believe IIA is a flawed criterion, and therefore the Impossibility Theorem, while true, is less useful than it is celebrated to be.

YokoZar 13y ago
Arrow's Theorem has to be one of the most over-celebrated yet uninteresting bits of math out there. I myself even thought it was a cool way of challenging alternative voting systems when I first started studying them. But, as you say, Arrow's Theorem is simply too broad with it's definition of "fair". It's entirely reasonable for the voters to have a list of candidates they hate and a list of 3 candidates they would prefer, and for there to be a rock-paper-scissors situation among those top three candidates. It's entirely possible to have a voting system that only chooses from among the rock-paper-scissors options, and indeed every Condorcet system will make such a choice. Those options are the Smith set. The tragedy is that Arrow's Theorem is often used to justify systems (like plurality or range voting) that often won't choose from among those top three, and might even choose the Condorcet loser! I'm not sure what a good definition of "fair" is, but I'd settle for not picking the candidate that would lose to literally every other candidate in a one-on-one election.