h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

I was wondering how well the Instant Runoff Voting (IRV) system answers the issues raised by Arrow's theorem. For those who are unfamiliar with the concept, I have a good link-to-link below.

I have a great link on this page (http://www.bbc.co.uk/h2g2/guide/A520705) to a site that shows step by step how the Muppets elected their new CEO using the IRV system. Farcical, yes, but incredibly educational as well.


Mikey

Hi Mikey - did you expect to see me here?

IRV fails to meat the independance from (irrelevant) alternatives criterion (IIAC), which is, in practice, why most election methods have this problem - Condorcet, etc, all fail the same.

Personally, I don't see that IIAC is terribly important - the idea has been put forward that losing candidates (or policies, whatever) are not irrelevant because they provide more information. For example, a vote of A>B is not as informative as a vote of A>X>Y>Z>B. In combination with many other votes, of course.

One slightly weaker variant of IIAC, which is actually achievable, is Local Independance from Irrelevant Alternative Criterion. To understand this, you first need to understand the Smith Set. So:

The Smith Set is the smallest possible set of candidates such that all candidates inside the set pairwise beat all candidates outside the set. For example, if we have votes as follows:

33% A>B>C
33% B>C>A
33% C>A>B

Then A beats B, B beats C, and C beats A. So here the Smith Set is {A,B,C}. If one candidate beats all the other candidates, then the Smith Set will be just that one candidate.

Now, LIIAC says that if a candidate is removed from the election, and that candidate is NOT in the Smith Set, then that candidate must have no effect on the outcome. Good Condorcet methods pass this, and IRV does not.

Oh, IRV also fails the second condition - that of 'monotonicity'. So it manages to fail two of arrow's criteria, rather than just the minimum one. Fortunately, that doesn't happen very often - but it does happen.