[EM] two bit ratings

[Forest Simmons]

On Thu, 20 Sep 2001, Richard Moore wrote:

> Forest Simmons wrote:
> 
> > Clarification:
> > 
> > It satisfies a version of the Condorcet Criterion relativized to the
> > ballot type:
> > 
> > If there is a candidate A such that for any other candidate B, it is the
> > case that candidate A is preferred over candidate B on more ballots than
> > candidate B is preferred over candidate A, then candidate A is the
> > method's designated winner. 
> > 
> > This is the closest we can get to the Condorcet Criterion for any method
> > that doesn't require complete rankings.
> 
> 
> It is interesting to note that Approval meets this "Relaxed 
> Condorcet Criterion". But I have to temper my enthusiasm by 
> noting that, so does Plurality.
> 
> Richard
> 

I was hoping you wouldn't notice that about Plurality :-)

And I hope Condorcet advocates who allow truncated ballots aren't
disappointed when they find out Plurality satisfies the same version of
the Condorcet Criterion that their favorite method does :-) 

Of course, Condorcet methods that allow truncated ballots are still
extremely likely to pick the true Condorcet Winner if there is one, while
Plurality has an excellent chance of missing the true CW, especially when
voters are inclined to vote sincerely or have insufficient information
with which to strategize effectively.

The more expressive the ballot type the more demanding this Relative
Condorcet Criterion is.

So, although standard Approval (based on the one bit ballot) satisfies
this Criterion, Five Slot Approval does not :-(

Similarly, if voters were allowed to express all of there preferences, but
only their first ranked preference was used in deciding the winner, then
that version of plurality would not satisfy the Relative Condorcet
Criterion.

The more expressive the ballot type, the more likely the Relative
Condorcet Winner will also be the true CW.

And as Rob LeGrand has often pointed out, voter strategy tends to
reinforce this effect. So even in Plurality when mildly sophisticated
voters have a modicum of information to strategize with, there is a good
chance that if there is a true CW, that candidate will get a majority, and
thereby become the Relative CW as well.

In the case at hand (two bit rating ballots) satisfaction of the Relative
Condorcet Criterion is a modest, but nice, feature (in my totally unbiased
opinion).

Forest