[EM] Idea for Condorcet-based open list PR

[Jameson Quinn]

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.

Oops. Ignore that, I don't want to hijack the thread.

Jameson



2011/10/21 Ted Stern <araucaria.araucana at gmail.com>

> I had an idea for how Condorcet methods could be used for an open list
> Droop quota-based Proportional Representation method, as an
> alternative to STV.  I am motivated by these factors:
>
>  * I would like to avoid STV's elimination of candidates.
>
>  * I would like a PR method that requires only a single summable count
>   for each seat selection.
>
>  * I would like lower-rank preferences to be considered when choosing
>   each seat.
>
>  * I would like to give voters' higher ranks a bit more of a chance
>   than they would have with straight Approval Transferable Vote.
>
> Here's my idea:
>
> Choose a Condorcet method.  Schulze (winning votes) or any other
> robust clone-resistant method would be fine.  However, there may be
> advantages to using an approval-hybrid Condorcet method such as DMC or
> Forest Simmons' Enhanced DMC.
>
> Use some kind of ranked ballot, or rank inferred from rating.  Any
> explicit rank or rating above the bottom will be interpreted as
> Approval.
>
> Give each ballot an initial weight of 1.0.
>
> Choose a quota.  For example, I prefer the "easy" quota, (Number of
> ballots + 1) divided by (Number of Seats + 1), no truncation.  This is
> slightly different than the standard Droop quota.
>
> While seats remain unfilled,
>
>   Loop over all ballots:
>
>     Accumulate into the pairwise array, with each pairwise vote
>     scaled by ballot's weight.  Note: Calculate pairwise array only
>     for standing candidates; i.e., those who have not already won a
>     seat.
>
>     Accumulate ballot's contribution to Approval for each candidate,
>     scaled by ballot's weight.  This can be stored in the diagonal
>     entry of the pairwise matrix, if desired.
>
>     If there is only one standing candidate remaining on the ballot,
>     accumulate this ballot's weighted approval into the Locked
>     approval for this candidate.  This will be used to ensure that
>     ballots with truncated rankings are used up completely, while the
>     non-truncated ballots voting for a winner get to transfer as much
>     of their vote as possible.
>
>   Find the Winner (CW), as defined by your robust Condorcet method.
>   If that winner has total Approval that exceeds the quota, or no
>   candidates have Approval over the quota, that CW is seated.
>
>   Then let
>
>      T   = Approval Total for CW.
>      L   = Locked Approval Total for CW
>      Q   = Approval Quota
>      QML = max(Q - L, 0.0)
>      TML = max(max(T,Q) - L, eps), where eps is a small epsilon, say 1.e-9.
>
>   The fraction of CW's total score sum used up, U, from all unlocked
>   ballots voting for CW, is:
>
>      U   = QML / TML
>
>   Apply the following rescale factor to each ballot that contains an
>   explicit ranking of the CW:
>
>      F = 1.0 - U
>
>   [For efficiency, rescaling can be delayed until the ballot loop on
>   the next iteration.]
>
>   What do you do if the normal CW's Approval does not exceed the
>   quota, but there are other candidates whose score *does* exceed the
>   quota?
>
>   In that case, compute a *reduced* Condorcet Winner from among those
>   candidates whose total Approval score exceeds the quota.
>
>   This step is where an Approval-hybrid method such as DMC may be
>   useful -- if there are candidates above the quota, you can
>   terminate the search for the CW once you've descended below the
>   quota.
>
>   In the single-winner case, this would also differentiate this
>   method from standard Schulze, for example.
>
>   Note that in the other already-handled case where all candidates
>   have scores below the quota, there is no vote transfer -- each
>   ballot ranking the CW is used up completely.
>
> ---------
>
> This method is Droop Proportional: If a faction of voters approves M
> candidates, and has at least L * Q votes, then L candidates out of
> those M will be seated.
>
> This method would have significantly larger overhead than this
> Bucklin-based PR method, Graded Approval Transferable Vote, which is
> quite similar:
>
>   https://github.com/dodecatheon/graded-approval-transferable-vote
>
> Why would a Condorcet-based variant of Approval Transferable Vote be
> an improvement over a Bucklin-based method?  The main difference is
> that lower rank preferences will be considered for each seat, so cases
> where two "clone" candidates have very close first-rank totals will be
> decided more robustly.
>
> I think that in practice, the first few seats of any faction will be
> chosen similarly by either method.  The difference will come in the
> last one or two seats, especially if the remaining votes are very
> close to the quota.
>
> It might also seem like choosing the CW for each seat may be using too
> low an approval and thus use up too much of each ballot that rank's
> that seat's CW.
>
> However, in the Bucklin-based method, the "approval" total does not
> necessarily include the total votes at any explicit rank.  It depends
> on how low the threshold has descended.  So the actual approval score
> used to calculate vote transfer will probably be quite similar when
> comparing the methods, and the amount on each ballot that is used up
> will also be similar.
>
> Any thoughts?  I'll try coding up a version of this eventually ...
>
> Ted
> --
> araucaria dot araucana at gmail dot com
>
> ----
> Election-Methods mailing list - see http://electorama.com/em for list info
>
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