Truncation with Condorcet

[David Marsay]

In response to my:

Date sent:        Thu, 27 Aug 1998 16:56:41 +0000
Subject:          Re: Truncation with Condorcet

Mike Ositiff recently asked me about using 'total votes - votes 
against' or 'votes for - votes against' in tie-breaks. He cites the 
following:

Majority rule
Getting rid of the lesser-of-2-evils problem
Avoiding need for defensive strategy

and say:
>Votes-against is the measure that does the job. Again, these
>standards aren't just important to us--they're widely agreed
>upon.

I had thought that the answer was 'obviously' votes for - votes 
against. This raises three issues:
a) What did Condorcet mean?
b) Why does Mike prefer 'votes against'?
c) Which is better?

I note:

a) What did Condorcet mean?

Condorcet is not very explicit about plurality. However in his 
treatment of the probability of achieving a given plurarility (in
Essay...) he consistently deals with the probability of voting for and
against separately, even though it would have been a considerable
simplification to note that they add to 1 if abstentions are not
allowed. If we assume that abstentions are allowed then his formulae
are only correct if plurality means votes for minus votes against.

b) Why does Mike prefer 'votes against'?

It seems to me that the criteria Mike cites are at least equally 
applicable to many variations of the method you propose, including
that I cite.

c) Which is better?

Dodgson gives an alternative method in which one minimizes the number
of votes one needs to switch to bring the ballots into line with the
result. For no abstentions this is the same as Tideman/Shulze 
'beats-path' variant of Condorcet. For abstentions one can take an 
abstention on a pair-wise ranking as half-way between the two 
possible rankings (to be fair).

Alternatively, Young considered Kemeny's distance function, in which
the distance between a tied ranking and a definite ranking is half
that for a definite contradiction.

These methods are equivalent to Condorcet with plurality as 
'for - against'. I do not think a recasting of Dodgson and 
Young/Kemeny with just votes against is very natural.

Votes against suffers from truncation, as in Blake's example that 
prompted my previous message on this topic. Votes for - against does 
not.

Abstentions will tend to strengthen cycles, which intuitively seems 
undesirable. Trying to develop a simple example has given me a 
headache.


The bottom line is, what is the case for 'votes against'?
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