[EM] Dyadic Approval and Bubble Sorted Approval (fwd)

[Forest Simmons]

Here's most of a message I sent to Ted Stern recently, but I'm not sure if 
his new email server allowed it past the filter.

I like the idea of Grade ballots and the use of Cardinal Ratings for seeding 
the bubble sort.

I'm not sure how much temptation there would be to distort the ratings, though. 
Would ratings tend to gravitate to the extremes?

I would be more inclined to use Dyadic Approval ballots to refine the initial 
order, rather than Cardinal Ratings.

A dyadic approval ballot expresses a heirarchy of approval like

    A>B=C>>D=E>F>>>G>>>>H>I>>J>K>>>L>M>>N

The strength of the pairwise comparison is determined entirely by the strength 
of the strongest inequality that the two candidates straddle, so there is not 
as much temptation to crowd the candidates towards the extremes.

So G beats H with the same strength that A beats N, as in approval.

And C beats D with the same strength that A beats F.

And K beats L with the same strength that H beats N.

You can seed the bubble sort with approval (aka the highest strength 
relation on each ballot), replacing all but the strongest relations with 
equality.

The first bubble sort is then done by resurrecting the second strongest 
relations.

Etc, until the last bubble sort uses all of the order relations.

In this particular method the stronger relations are not given more value than 
the weaker ones when both are in play, but they do come into play sooner.

Of course there are other ways to use dyadic ballots, many of which were 
suggested three or four years ago on the EM list.

Dyadic ballots can be constructed from CR ballots if the CR ballots have a 
resolution that is a whole power of 2, like 8, 16, 32, etc.

The upper half of the range is separated from the lower half by the strongest 
relation.  Then each half is similarly divided by the next strongest relation, 
etc.

Forest