[EM] Consistency, Truncation, etc. (was CR ballots, etc.)

[Forest Simmons]

On Mon, 24 Sep 2001, Jobst Heitzig wrote:

> On 24 Sep 2001, Buddha Buck wrote:
> 
> > Hmm...  I'd love to see an example of this, since I fail to see how it
> > could happen.  
> > Therefore, the Condorcet Criterion is not
> > inconsistant.
> 
> Sorry about that: you are of course completely right! Condorcet methods
> can only be inconsistent in cases where some of the groups lacks a
> Condorcet winner.

In other words, the Condorcet Criterion is not inconsistent, but passing
the Condorcet Criterion guarantees failing the Consistency Criterion.

For a simple proof of this, see Markus Schulze's posting at

http://www.mail-archive.com/election-methods-list@eskimo.com/msg06140.html

Markus' proof doesn't touch Demorep's ACMA, because ACMA doesn't satisfy
the Condorcet Criterion (as Martin Harper slyly pointed out in message
06154).  But here's an example that shows ACMA fails the criterion: 

Precinct 1
40 A > B >> C
35 B > C >> A
25 C > A >> B

Precinct 2
51 B > A >> C
49 A > C >> B

All candidates pass the 50% approval quota (the first A in ACMA) in the
first precinct as well as the entire electorate, but not in the second
precinct by itself. (If precinct 2 were the whole electorate, C would be
eliminated.)

When restricted to precinct 1, the race has no Condorcet winner (the C in
ACMA) but B has Most Approval (the MA in ACMA). So B wins in precinct 1. 
B also wins in precinct 2 as the majority winner, therefore B is the ACMA
winner in both precincts. 

In the actual election, however, candidate A is both the Approval winner
and the Condorcet winner, as well as the ACMA winner. 

So B is the ACMA winner in both precincts, but not the over-all ACMA
winner.  ACMA does not satisfy the Consistency Criterion.

However, like Ranked Pairs, SSD, etc. when ACMA fails the Condorcet
Criterion it is because of the lack of decisive winners in the
sub-elections. (The Approval tie breaker had to be invoked in the first
precinct, and margin of victory was small in the other precinct.) 

On the other hand, IRV can give a decisive win to a candidate in all of
the precincts without that candidate winning the election. (However,
nobody can find this out because of the way ballots are counted in IRV.)

<snip>

> 
> > So are you suggesting that, for instance, in an election between A, B,
> > C and D, the following truncated ballot:
> > 
> > A>B>C
> 
> (What does truncated ballot mean? The question is how we should interpret
> this incomplete preference information...)
> 
> > but not for:
> > 
> > A>D
> > D>A
> > B>D
> > D>B
> > C>D
> > D>C
> > 
> > i.e., it would count neither for nor against D?
> 
> Yes! I completely think so. Since the voter did not express preferences
> concerning D, he might just be completely uninformed about D and might
> just want to delegate the decision about D to the other voters.
> 

Suppose that there are an hundred voters in an election with three
candidates A, B, and C, and that the first two of these are well
known, well liked, and about equally matched.

Suppose that C is known to only about 10 voters, and that these voters all
like him much better than either A or B. This dark horse C could be a
write-in candidate. 

The votes come out like this:

46 A > B > NOTB
44 B > A > NOTB
10 C > NOTB > A=B

(NOTB is the approval cutoff)

If the truncations of the first two factions are considered as
delegating the decision about C to the 10 voters that know something
about C, rather than ranking C at the bottom of the first two factions,
then C becomes the Condorcet winner, which seems a rather undesirable
outcome to me.

Demorep would take care of this by eliminating C from the contest in the
first step of ACMA.

Forest