[EM] IRV inconsistency

[Blake Cretney]

On Mon, 14 May 2001 15:45:33 -0700 (PDT)
Forest Simmons <fsimmons at pcc.edu> wrote:

> Here's an inconsistency of IRV that I wish somebody had told me
about
> before I submitted my article to the Green Voice.
> 
> It is possible for a candidate to "win" every precinct without
winning the
> election.
> 
> In other words, if the "winner" of each precinct is calculated by
applying
> the rules of IRV to the ballots from each individual precinct before
> applying IRV to the entire collection of ballots, it can happen that
> candidate B wins in every precinct while some other candidate wins
the
> election.
> 
> This weirdness cannot happen in Approval, for example.

I'm always suspicious of the Weirdness Avoidance Standard.  My
impression is that a lot about public choice doesn't fit with our
intuitions.  So, our intuitions aren't always reliable.

Since Consistency violation can occur in Condorcet methods too,
including Ranked Pairs, I defend it on my web site.  Here's the
defense.  I'm going to suggest a similar, although purely hypothetical
situation, and show that the intuition against consistency violation
can lead to error.

Imagine that you are asking a magical oracle about which is the best
candidate. The oracle only accepts questions comparing two candidates.
 The oracle is sometimes wrong, but always gives an accurate
prediction of its likelihood of being correct.  This is meant to be
vaguely similar to the situation of relying on majority decisions to
get a best guess for best winner.

The oracle claims
A>B has a 80% chance of being correct
B>C has a 70% chance of being correct
C>A has a 60% chance of being correct 

Clearly the oracle is wrong about one of these answers.  Since the
oracle is least certain about C>A, it makes sense to guess that this
is where the mistake is made.  We can then declare A the likely best
candidate.  

Now, someone else goes to the oracle to ask about the same candidates.
 The oracle gives the following information 

A>C 51%
A>B 100%
C>B 100%

On the basis of this information alone, A appears to be the best
candidate. 

Let me point out, that the two sessions with the oracle are not
inconsistent.  They may be unlikely, but they both could happen.  

Now, by consistency, we would expect that A should also win if the
information from the two interviews are combined.  The combined
information gives these responses: 

A>B 80% chance
B>C 70% chance
C>A 60% chance
A>B 100% chance
C>B 100% chance 

Since we know the 100% statements are true, the B>C can be dropped as
false, and the A>B 80% can be dropped as redundant.   This gives,

A>B 100% chance
C>B 100% chance
A>C 51% chance
C>A 60% chance 

So, clearly, since C>A is more likely than A>C, if follows that C is
most likely the best candidate.   This of course violates consistency,
but in this case, appears reasonable.

---
Blake Cretney