[EM] Dyadic Rated Pairs

[Forest Simmons]

On Fri, 21 Sep 2001, Roy wrote:

> Forest Simmons wrote:
> ...
> > It seems natural to me to reckon the weak relations
> > (A > B and C > D) at half the rate of the stronger relations
> > (the four that straddle the double inequality).
> ...
> > The method generalizes in an obvious way to any number of 
> > candidates.
> 
> Not entirely obviously: is >>> twice as strong as >>, or merely 1.5
> times as strong?

For dyadic ballots, in which a well balance ballot has twice as many >>'s
as >>>'s, it seems that the most natural weight factor would be two rather
than 1.5.

In other systems where only one relation of each strength is allowed, it
would be more natural to make the strength directly proportional to the
number of >'s in the string.

I admit that I prefer methods like Universal Approval that don't assign
numerical weights to the various strengths.

A way to modify Dyadic Rated Pairs so as not to assign weights is as
follows:

Just use the order of pair strengths on each ballot to get a group ranking
of the order of pair strengths via Ranked Pairs of Pairs. This would
involve a Super Pairwise Matrix with N^4 entries, where N is the number of
candidates.


> 
> I'm going to bring up again (just for the record) that:
> 1. Assigning numbers to ranked preferences introduces "data" that
> isn't necessarily an accurate interpretation of the voters'
> expressions.
> 2. CR type ballots can be used for this, with the voters supplying the
> numbers.
> 
> I also think that your method, when multiple ballots are summed, still
> depends on a winning margin. It's just a weighted margin, isn't it?
> 

Not really a margin. Take the following example:

45 A > B >> C
35 C > A >> B
20 B > C >> A

The pair strengths are ...

S({B,C})= 2*45+2*35+20 = 180
S({C,A})= 2*45+35+2*20 = 165
S({A,B})= 45+2*35+2*20 = 155

[No margins, because no subtraction]

So according to Dyadic Rated Pairs the over all ranking is

B > C > A

Despite the fact that A is the Ranked Pairs winner, the SSD winner, the
Approval Winner, the Universal Approval winner, the Dyadic Approval
winnner, the ACMA winner, and even the IRV winner.

This example is enough to get me to abandon Dyadic Rated Pairs.

True, the {B,C} contest is the one the voters seem to be most passionate
about, but that might just be because the media has whipped up an
artificial frenzy between Gore and Bush, when the race of substance
is between Nader and the corporate candidates.

But still, I have gained some valuable insights from considering this
method.

For one, Ranked Pairs would do better to use margins of squared vote
totals, rather than margins of votes. 

Here's why: suppose that two pairs have the same margin of winning minus
losing votes.  Wouldn't it make sense to go with the pair having the
greatest total number of voters expressing a preference? 

Note that (winning - losing)*(winning + losing) = winning^2 - losing^2 .

On the other hand, if two pairs have the same number of voters expressing
preferences, then naturally, the greatest margin would have to rule.

Using the above product takes both of these considerations into account
without destroying the anti symmetry of the margin matrix.

The product may be thought of as (Intensity of preference)*(Margin of
preference).

Forest