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Arrow's Theorem and Overstatements

By       Message Paul Cohen       (Page 1 of 1 pages)     Permalink    (# of views)   5 comments, 2 series

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Perhaps it is just hard-wired into the minds of the human species to make it a practice of overstating our case or perhaps it is something we just learn as a strategy for impressing or persuading others. Whatever the reason, we all are guilty of it from time to time and we all should be wary of it in evaluating what others say. But too often we are apt to ignore that cautionary advice.

An article I read recently claimed that by virtue of Arrow's theorem we know that all voting methods are defective. Now while I have little doubt that every possible voting method would be defective in the opinion of someone, that is not exactly what Arrow's theorem says. Arrow's theorem actually shows that ranked voting systems are defective in a particular, well-defined way. A ranked voting system (in the sense of Arrow's theorem) is one in which each voter is asked to vote by specifying a ranking of all of the candidates in order of preference. But that is a very special kind of voting system and Arrow's theorem says nothing whatever about many other voting systems.

A suspicious mind might wonder whether this overly broad conclusion is an accident or whether it is purposely drawn by ranked voting advocates so as to suggest that we simply must be satisfied with a demonstrably defective voting system. Such a person would actually prefer that rather dismal conclusion to the alternative and more natural realization that we might best turn to something other than a ranked voting system. Fortunately there are many good alternatives to ranked voting systems.

There is an understandable attraction to ranked voting that derives from a mistaken impression that by ranking the candidates a voter's opinions can fully, accurately and completely expressed. This fallacy was addressed in the earliest of these articles concerning ranked voting. The fact that the ranking may overstate the feelings of a voter (who is forced to invent preferences in ranking where none exist) seems just to be ignored; we are just so accustomed to overstatement and tend to dismiss it as being normal and harmless but it does disguise that the voter is in some instances indifferent, something that needs to be reflected in any complete and accurate measure of voter preference. But even that is not the end of the story.

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Actual implementations of ranked voting generally relax its definition to only require a voter to specify a ranking of just a few of the candidates; this is no longer a ranked voting system. So even in the relaxed real-world version of ranked voting, the conclusion of Arrow's theorem possibly might not be justified. Nonetheless, Arrow's theorem does suggest that if we want a voting system without serious defects, we would best be advised to look elsewhere than to ranked voting systems. But despite this evidence from Arrow's theorem, there remains strong support for the particular form of ranked voting called Instant-Runoff Voting (IRV), so the topic needs further attention. Is the real-world version of IRV defective? An example could show it to be.

Here we have seen three instances of overstatement -- in the interpretation of Arrow's theorem as applying to all voting systems (not just ranked ones), in the actual overstatement of preferences by the voters, and finally in applying Arrow's theorem to voting systems that allow voters not to rank all of the candidates. In everyday parlance we may be accustomed to overstatement, but here we are applying a mathematical theorem where precise adherence to details in definitions and assumptions is critically important in this arena. In mathematics, it is often the case that seemingly minor changes to a definition or to assumptions can lead to very different conclusions.

The subject of this series of articles is balanced voting, and since none of the balanced systems are ranked this is the first time that it has seemed appropriate to even mention Arrow's theorem. Still, due to the great popularity of IRV, that particular ranked system has been discussed in several of the articles. One reason I wrote this article was to have an opportunity to make available a brief survey of these articles for someone who might be primarily interested in IRV.

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In an early article and again in a more recent one, we addressed difficulties with the very rational behind the invention of IRV. And in yet another article we provided a concrete example to clearly illustrate a serious defect in an IRV election. These articles should make it clear that IRV falls well short of being the perfect answer to better elections.

Finally turning to a slightly different topic, I should take this opportunity to make it clear that Instant-Runoff Balanced Voting (IRBV) is a balanced voting system and not a ranked system. At first glance, IRBV may seem like a ranked voting system, but like other balanced systems, it is not actually a ranked system and so Arrow's theorem does not say anything about IRBV. However, IRBV is modeled on IRV and it shares much the same appeal as IRV though it lacks many of its deficiencies. However, as with IRV, our tradition of counting the votes in a distributed manner rather than at a single central place is very problematic if not completely impossible. More insight into the relationship between IRBV and IRV is developed in one of the later articles.

 

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A concerned citizen and former mathematician/engineer now retired and living in rural Maine.

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Series: "Ranked Voting"

A First Experience with Ranked Voting (Article) (# of views) 06/13/2018
Instant Runoff - Balanced once Again (Article) (# of views) 04/06/2017
Ranked Voting (Article) (# of views) 03/07/2017
View All 9 Articles in "Ranked Voting"
Total Views for the Series: 6693   

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