[EM] please read quick question about Condorcet

[Forest Simmons]

On Thu, 18 Mar 2004, James Green-Armytage wrote:

>
> Hi folks,
> 	I was just wondering what sort of Condorcet tally methods have been
> proposed to process ballots that are not only ranked, but where some
> preferences are indicated as being stronger than others. For example, a
> vote may look like this:
>
> Kucinich > Nader >> Dean >> Kerry > Edwards >> Clark >>> Lieberman >>>>
> Bush
>
> 	For example, I have heard of one where, if there is a majority rule
> cycle, the lowest-priority preferences on each ballot are changed to
> equalities. I'm wondering if anything else has been proposed along these
> lines.

Dyadic Approval ballots are in this class, and there have been a variety
of proposals for their use.

Here's one that applies whenever there is a rule that there can be only
one instance of strongest preference  (i.e. the Lieberman>>>>Bush
preference in your example) on each ballot:

The strongest preference is considered the approval cutoff.  Start by
listing the candidates in order of approval. Then demote the approval
cutoff one level, and "Locally Kemenize" the list using the preferences at
that level. Continue kemenizing using weaker and weaker preference levels,
until it has been done at the lowest level on all ballots.

What does "Kemenize" mean?

It means to adjust the order of the candidates by swapping candidates that
are out of order according to the current pairwise matrix (i.e. the
pairwise matrix based on the current preference levels).

There are various priority orders for doing this, and different priorities
for swaps can lead to different winners when there is no CW.

Here are two Kemenizations that I like:

(1) If any adjacent candidates are out of order, swap the adjacent pair
with the greatest discrepancy.  Repeat until there are no more adjacent
pairs out of order according to the current pairwise matrix.

The greatness of the discrepancy can be measured by either margins or
winning votes, so there are two versions of this method of kemenization.

Note that the method cannot cycle, since once a swap is made, it cannot be
undone.

(2) Similar to (1) except not limited to adjacent pairs.  To avoid cycling
we institute the rule that in each step the greatest discrepancy that can
be undone without introducing another discrepancy as bad or worse than the
one being undone is the one to be undone.

Since the undoable discrepancy sizes decrease monotonically the process
must come to an end.


Whether we use method (1) or (2) the lists will be "solid" from top to
bottom when we get through, i.e. there will be no adjacent pair of
candidates out of order. [A list satisfying this solid condition is said
to be locally kemeny optimal, because one kemeny move from this list will
increase the kemeny sum of distances to the ballot orders. Hence the name
"local kemenization."]


To carry out either of these methods it is enough to have the approval
totals and a pairwise matrix for each of the other level.

So if there are seven levels on the most discerning ballot, we would have
to sum six pairwise matrices.

Note that we count down from the top, so that all of the top levels are
approval levels even if on some ballots they are not indicated as strongly
as on others.

Forest
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