Forest Simmons fsimmons at pcc.edu
Wed Mar 21 22:18:27 PST 2001

```Mike, the definition of Dyadic Approval has evolved slightly since the
original, so I welcome this chance to summarize to date. Plus I know that
Tom Ruen and other recent list members are interested, too.

Dyadic Approval is a generalization of both Approval and Condorcet.

It generalizes Condorcet by stair stepping (in a consistent way that
doesn't encourage empty categories) degrees of preference, as well as by
retaining the pairwise aspect of Condorcet.

It generalizes Approval by allowing the voter to specify within various
previously defined categories of Approval which of these candidates would
be approved if they were the only ones left.

I find it is easiest to start with a sample voted ballot, and then
interpret it, and explain how it is scored and put into matrix form for

Once the total matrix is in, the winner can be determined from the matrix
by SSD or any other good Condorcet method. However I am going to recommend
another method that takes advantage of the hierarchical structure of the
ballots for use when there are (generalized) Condorcet cycles (cycles of
pairwise preferences among candidates).

Here's a sample voted ballot with eight candidates:

A > B >> C > D >>> E > F >> G > H

This one is ideal ... a precisely balanced binary tree with one leaf at
the end of each twig. I'll show some more typical and extreme ones
presently, to demonstrate the range of possibilities.

We interpret this ballot as follows. The voter supports (on the coarse
level of ordinary Approval) candidates A through D, and rejects E through
H.

However, if candidates A through D were to be eliminated (by
assassination, Approval runoff, heart attacks, etc.) then among the
remaining candidates E and F would be supported, while G and H would be
disapproved. Similarly, if all of the candidates except C and D were
eliminated, then the voter would approve only C.  I think you can see the
pattern.

For scoring, we note that the coarsest distinction is made by a string of
three inequality signs: >>>, so the score of any pairwise comparison of a
pair that straddles that string receives a weight of 1. Then any pair that
straddles >> (but not >>>) receives a weight of one half.  Similarly, any
pairwise comparison between candidates on opposite sides of a single >
symbol receives a weight of one fourth.

Another voted ballot (for seven candidates this time) could look like the
following:

A >> B > C >>> D >>>> E, F >>> G

In this ballot the voter chose to not distinguish between E and F.  All
comparisons between E and F add zeros to the corresponding positions
in the scoring matrix. Furthermore, the underlying binary tree is not as
well balanced; some possible categories are missing.

The coarsest comparisons are the ones straddling the string of four
inequality signs ( >>>> ) so they are the ones that receive a weight of
one on this ballot.  By the time we get down to the comparison between B
and C, it only gets a weight of one eighth.

In particular, the comparison between the pair A and G  gets the same
weight as the comparison between D and E, so there is no temptation to
insincerely move D further up the list or E further down the list.

Two extreme possibilities are:

A > B >> C >>> D >>>> E >>>>> F >>>>>> G

and

A >>>>>> B >>>>> C >>>> D >>> E >> F > G

In the first of these two extreme cases, all comparisons relative to G
have a weight of one. Only the comparison between A and B has a weight of
1/32 . If this were a zero info election, it would mean the utilities were
skewed towards the upper end of the scale.

In the second case, all comparisons relative to A receive a weight of one,
while the comparison between F and G has a weight of 1/32 .  If this were
a zero info election, it would mean the utilities were skewed toward the
low end of the scale.

(To be continued tomorrow)

Coming up tomorrow:

(1) the binary string representation of the ballots.

(2) natural hierarchy resolution of cycles.

(4) how to use these same style ballots for Approval Runoff.

Forest

On Thu, 22 Mar 2001, MIKE OSSIPOFF wrote:

>
>
> I can't find the original complete definition of Dyadic Approval in
> the EM archives. I know it's in there, but I haven't found it. Could
> you send me the definition?
>
> As I was saying, I always want to check out new method proposals,
> because there's always the possibility that one of them will turn
> out to be best at least in some important ways.
>
> Mike
>
> _________________________________________________________________
>
>

```