It certainly fills out the matrix of available methods, but I'm not sure when I'd actually use it. I'm more interested in hybrid geographical/proportional methods these days, as I think such methods have the best chance to prosper in the US or UK. It would be nice to even keep existing district boundaries. I think that PR-SODA is very close to being able to do this; yes, it's cheating, but the basic idea is to elect candidates and then assign each district one winner per elected party. Yeah, that means that some reps might have 1 district of constituents, while others might have 2; but otherwise it actually works surprisingly well.<div>
<br></div><div>Oops. Ignore that, I don't want to hijack the thread.</div><div><br></div><div>Jameson<br><div><br></div><div><br><br><div class="gmail_quote">2011/10/21 Ted Stern <span dir="ltr"><<a href="mailto:araucaria.araucana@gmail.com">araucaria.araucana@gmail.com</a>></span><br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">I had an idea for how Condorcet methods could be used for an open list<br>
Droop quota-based Proportional Representation method, as an<br>
alternative to STV. I am motivated by these factors:<br>
<br>
* I would like to avoid STV's elimination of candidates.<br>
<br>
* I would like a PR method that requires only a single summable count<br>
for each seat selection.<br>
<br>
* I would like lower-rank preferences to be considered when choosing<br>
each seat.<br>
<br>
* I would like to give voters' higher ranks a bit more of a chance<br>
than they would have with straight Approval Transferable Vote.<br>
<br>
Here's my idea:<br>
<br>
Choose a Condorcet method. Schulze (winning votes) or any other<br>
robust clone-resistant method would be fine. However, there may be<br>
advantages to using an approval-hybrid Condorcet method such as DMC or<br>
Forest Simmons' Enhanced DMC.<br>
<br>
Use some kind of ranked ballot, or rank inferred from rating. Any<br>
explicit rank or rating above the bottom will be interpreted as<br>
Approval.<br>
<br>
Give each ballot an initial weight of 1.0.<br>
<br>
Choose a quota. For example, I prefer the "easy" quota, (Number of<br>
ballots + 1) divided by (Number of Seats + 1), no truncation. This is<br>
slightly different than the standard Droop quota.<br>
<br>
While seats remain unfilled,<br>
<br>
Loop over all ballots:<br>
<br>
Accumulate into the pairwise array, with each pairwise vote<br>
scaled by ballot's weight. Note: Calculate pairwise array only<br>
for standing candidates; i.e., those who have not already won a<br>
seat.<br>
<br>
Accumulate ballot's contribution to Approval for each candidate,<br>
scaled by ballot's weight. This can be stored in the diagonal<br>
entry of the pairwise matrix, if desired.<br>
<br>
If there is only one standing candidate remaining on the ballot,<br>
accumulate this ballot's weighted approval into the Locked<br>
approval for this candidate. This will be used to ensure that<br>
ballots with truncated rankings are used up completely, while the<br>
non-truncated ballots voting for a winner get to transfer as much<br>
of their vote as possible.<br>
<br>
Find the Winner (CW), as defined by your robust Condorcet method.<br>
If that winner has total Approval that exceeds the quota, or no<br>
candidates have Approval over the quota, that CW is seated.<br>
<br>
Then let<br>
<br>
T = Approval Total for CW.<br>
L = Locked Approval Total for CW<br>
Q = Approval Quota<br>
QML = max(Q - L, 0.0)<br>
TML = max(max(T,Q) - L, eps), where eps is a small epsilon, say 1.e-9.<br>
<br>
The fraction of CW's total score sum used up, U, from all unlocked<br>
ballots voting for CW, is:<br>
<br>
U = QML / TML<br>
<br>
Apply the following rescale factor to each ballot that contains an<br>
explicit ranking of the CW:<br>
<br>
F = 1.0 - U<br>
<br>
[For efficiency, rescaling can be delayed until the ballot loop on<br>
the next iteration.]<br>
<br>
What do you do if the normal CW's Approval does not exceed the<br>
quota, but there are other candidates whose score *does* exceed the<br>
quota?<br>
<br>
In that case, compute a *reduced* Condorcet Winner from among those<br>
candidates whose total Approval score exceeds the quota.<br>
<br>
This step is where an Approval-hybrid method such as DMC may be<br>
useful -- if there are candidates above the quota, you can<br>
terminate the search for the CW once you've descended below the<br>
quota.<br>
<br>
In the single-winner case, this would also differentiate this<br>
method from standard Schulze, for example.<br>
<br>
Note that in the other already-handled case where all candidates<br>
have scores below the quota, there is no vote transfer -- each<br>
ballot ranking the CW is used up completely.<br>
<br>
---------<br>
<br>
This method is Droop Proportional: If a faction of voters approves M<br>
candidates, and has at least L * Q votes, then L candidates out of<br>
those M will be seated.<br>
<br>
This method would have significantly larger overhead than this<br>
Bucklin-based PR method, Graded Approval Transferable Vote, which is<br>
quite similar:<br>
<br>
<a href="https://github.com/dodecatheon/graded-approval-transferable-vote" target="_blank">https://github.com/dodecatheon/graded-approval-transferable-vote</a><br>
<br>
Why would a Condorcet-based variant of Approval Transferable Vote be<br>
an improvement over a Bucklin-based method? The main difference is<br>
that lower rank preferences will be considered for each seat, so cases<br>
where two "clone" candidates have very close first-rank totals will be<br>
decided more robustly.<br>
<br>
I think that in practice, the first few seats of any faction will be<br>
chosen similarly by either method. The difference will come in the<br>
last one or two seats, especially if the remaining votes are very<br>
close to the quota.<br>
<br>
It might also seem like choosing the CW for each seat may be using too<br>
low an approval and thus use up too much of each ballot that rank's<br>
that seat's CW.<br>
<br>
However, in the Bucklin-based method, the "approval" total does not<br>
necessarily include the total votes at any explicit rank. It depends<br>
on how low the threshold has descended. So the actual approval score<br>
used to calculate vote transfer will probably be quite similar when<br>
comparing the methods, and the amount on each ballot that is used up<br>
will also be similar.<br>
<br>
Any thoughts? I'll try coding up a version of this eventually ...<br>
<br>
Ted<br>
<font color="#888888">--<br>
araucaria dot araucana at gmail dot com<br>
<br>
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</font></blockquote></div><br></div></div>