[EM] Arrow's Theorem - The Return (again)

[John B. Hodges]

>Date: Sat, 2 Aug 2003 13:17:36 -0700 (PDT)
>Subject: Re: [EM] Arrow's Theorem - The Return (again)
>From: "Alex Small" <asmall at physics.ucsb.edu>
>
>(snip much)
>
>Here's a good argument for why Approval flunks IIAC:
>
>Say that the electorate is as follows:
>
>30 (A>B)>C
>30 (C)>A>B
>40 (B)>A>C
>
>The parentheses indicate which candidates a particular group of voters
>approved.  In this case, B wins with 70 votes.
>
>Now give all of the voters a ballot again, but don't include candidate C
>on it.  If the people who approved both A and B the first time around are
>rational they will only approve A this time.  If the people who only
>approved C the first time around are rational they will only approve A
>this time.  A now wins 60-40.
>
>A mathematician at Northwestern published a paper on this premise a few
>years back.  He used more sophisticated assumptions about voter behavior,
>but the final conclusion was that Approval Voting flunks IIAC.  I wasn't
>impressed when I ran across it.  If we can use a single counter-example to
>prove that a statement is false, then a long proof is unnecessary.  But I
>suspect he's from the Saari school of voting theory (Saari used to teach
>at Northwestern), and really wanted to drive home that Approval Voting
>isn't perfect.  (I'm sure everybody here, Donald especially, was
>completely shocked to learn that Approval Voting isn't perfect.  ;)
>
>Alex

JBH here- I was reading Dummett recently, and his comment on the 
relevance of the Independence from Irrelevant Alternatives Criterion 
was that it implied that there was no "straightforward" voting 
system, i.e. that there was no voting system that never gave 
incentives for insincere reporting of the voter's preferences.

Alex's example above shows the possibility of a "spoiler effect", 
i.e. in a two-person race between A and B, A wins, but adding C to 
the race gives the victory to B. Those who vote for C give the 
election to their last choice, so if they understood the situation in 
advance, they would abstain from voting for their first choice. That 
is significant; up to now I thought that Monotonicity implied the 
absence of a spoiler effect. If avoiding the spoiler effect depends 
on a voting method satisfying IIAC, there will be very, very few 
methods that qualify.  (Does anybody KNOW of any?)

Then there is the question of how MUCH of a problem this is; 
different methods may all share the same set of flaws, but to 
different degrees. Plurality suffers from the spoiler effect very 
reliably, even if the third party is small. IRV suffers from it 
differently; it is center parties that spoil, not wing parties, and 
this happens only when the wing party grows larger than its 
neighboring centrist party. If Approval also suffers from a spoiler 
efect, we have to get some estimate of the relevant circumstances.
-- 
----------------------------------
John B. Hodges, jbhodges@  @usit.net
Do Justice, Love Mercy, and Be Irreverent.