[EM] Hello everybody!

[Forest Simmons]

Welcome to the list. I checked out your website.  I like what you are
doing.  Some of us have suggested interactive ballots, but have been too
lazy or lacking in time or skills to do that, let alone provide a whole
service like your project envisions.

We have been exploring the use of dyadic ballots, which are too
complicated for the layman to vote without interactive help.

There are a lot of great uses of dyadic ballots, but most of the readers
of this list consider them to be pipe dreams for the distant future
because the ballots require not only a preference order among the
candidates, but also a (partial) order in the strengths of preference.

If a voter can vote an approval ballot, then with interactive help she can
vote a dyadic ballot.

First distinguish the Approved from the Unapproved. That determines
whether the first (leftmost) rating digit is a zero or a one.

Then among the approved candidates which would be approved if all of the
unapproved candidates were eliminated? These are the candidates of type AA
whereas the rest of the approved candidates are of type AU. 

Similarly, among the unapproved can be divided into two groups, the UA's
and the UU's.

This process continues until the voter can make no further distinctions.

Representing A's and U's with ones and zeros, respectively, we have (for
example)

000 > 001 >> 010 > 011 >>> 100 > 101 >> 110 > 111

(The relative gap sizes make sense numerically only if you think of the
numerals as base three or higher.) 

Some slots might be empty of candidates, while others might have multiple
candidates.

The job of the questioner is to find the biggest gap in preference of the
voter within the subset of candidates being considered at that stage.

Another approach is to get a preference order first, as you do in your
website, and then adjust gap strengths until the voter can no longer say
that one preference is stronger than another.

The gap strengths are not necessarily quantified in methods based on
dyadic ballots, although in some cases they are.  Usually it is the order
of strength that counts the most, if not exclusively.

For example in Approval Runoff based on dyadic ballots, at each stage of
the runoff, on any given ballot the approval cutoff is the strongest
relation still in operation, i.e. still being straddled by two or more
candidates.

For a method that does quantify the gap strengths to get an estimate of a
group consensus as to their relative strength (i.e. a group ranking of the
gap strengths), see my recent posting "Dyadic Rated Pairs" which takes
advantage of this group ranking of gap strengths to mimic Ranked Pairs
in such a way as to retain all the good properties of Ranked Pairs, while
satisfying the Favorite Betrayal Criterion to boot.

[The method could be relativized via pairwise comparisons of the pair
strengths, so that quantification would be unnecessary, but that would
involve carrying along approximately n^4 quantities in the precinct
summaries.]

There is still a lot of exploration to be done in this fertile field.

I hope that what I just dashed off wasn't too confusing :-)

Best Wishes,

Forest