[EM] Dyadic ballots (was "...encouraging truncation")

[Forest Simmons]

On Fri, 14 Mar 2003, [iso-8859-1] Kevin Venzke wrote:

> I suspect that this system (applying a Condorcet
> method to the ballots) is identical to Borda with a
> fixed number of ranks.  I'd bet that voters would use
> Approval strategy (give only 15's and 0's).
>
> Given a Borda ballot of A>B>C>D>E, the points given
> are A 4, B 3, C 2, D 1, E 0.  If you make a Condorcet
> matrix reflecting these points, you'll get the same
> winner:
>    A   B   C   D   E
> A  .   1   2   3   4
> B  0   .   1   2   3
> C  0   0   .   1   2
> D  0   0   0   .   1
> E  0   0   0   0   .
>
> Measuring the degree of preference essentially means
> letting this voter vote four times for the A>E
> proposition.  The value of mere relative ranking is
> diluted.  I'm not sure this can be overcome.

Your reaction is typical for the first exposure to methods based on dyadic
ballots.  The main problem with the Borda scoring approach is that it
doesn't have fixed boundaries for the relative preference strengths.

In the dyadic ballot A>B>>C>D the strength of the AD preference is exactly
the same as the strength of the BC preference because the max strength
boundary straddled is of the same type for both. In your Borda example of
A>B>C>D, the AD preference would be three times as strong as the BC
preference.

If A were Favorite and B were Compromise, there would be no advantage on
the dyadic ballot to rank B over A in the contest between B and C or
between B and D.

On the Borda ballot B would gain advantage by switching places with A,
i.e. by burying Favorite.

In the recent method I suggested no entry on any pairwise matrix computed
from any ballot is anything other than a zero or a one.

Think of it as pairwise matrices constructed from CR ballots of
successively greater resolution starting with resolution two (i.e.
approval).

When it comes to pairwise matrices, the main advantage of using CR ballots
over ranked ballots is that equality is allowed at the top and middle as
well as the bottom.

Now to be more specific about the method.

Initialize the order with the pairwise matrix from the resolution two
ballots.

Locally Kemenize (i.e. bubble sort) with the pairwise matrix from the
resolution four ballots.

Then locally Kemenize again with the pairwise matrix from the resolution
eight ballots.

Finally, locally Kemenize again with the pairwise matrix from the full
resolution ballots.

The method satisfies the Condorcet Criterion as well as any other method
(such as MAM) that allows equal rankings and has ballots of resolution
limited to sixteen. In any case the winner is a member of the Smith set.

The dyadic ballots are merely a way of combining the resolution 2, 4, 8,
and 16 ballots into one. Equivalently you could start with an high
resolution CR ballot with all ratings represented as binary point
expansions between .00000... and .11111... .

To get the various low resolution ballots, truncate (rather than round)
the ballot ratings to various numbers of binary places.

A rating of .1101001011101 could be truncated to .1101 for a resolution
16 ballot, or  .1101001 for a resolution 128 ballot, etc.

Forest