[EM] condorcet loser elimination PR

James Green-Armytage jarmyta at antioch-college.edu
Mon Jul 21 15:27:04 PDT 2003

```Dear Chris, and other election methods fans,

This is going back a little bit, but you suggested a form of Single
Transferable Vote PR where instead of eliminating the candidate with the
fewest top-choice votes, you eliminate the Condorcet loser, or some
approximation of a Condorcet loser based on a Condorcet-completion
principle.

(First of all, it's interesting to me that you suggested a completion
mechanism to fill the absence of a genuine Condorcet loser; I've never
heard of that. I imagine, though, that it should be a simple matter of
putting a regular Condorcet completion principle into reverse somehow...)
Anyway, I'm sorry to say that I don't think your suggestion would work
very well as far as creating a proportional result. I did some
pen-and-paper experiments with it, and I came up with a few examples that
illustrate why this is the case. I will share a couple of those examples
with you here.
The first example uses a Droop quota (votes÷(seats+1)), and the second
example uses a Hare quota(votes÷seats). I happen to prefer the Hare quota,
because it is more fair to smaller parties than Droop, but I will leave
that as a subject for a later posting.
To keep the examples simple, I have given several candidates exactly one
quota. That shouldn't affect the test of your method though, because the
distinguishing feature of your method deals with eliminations, not
surpluses. To make the examples more realistic, though, one could also
imagine that these examples begin from the stage right after the surpluses
have been transferred.

Example #1
4 seats
Quota: 20 (Droop)

20: A, B, C, D, E
10: B, A, C, D, E
10: B, C, A, D, E
10: C, B, D, A, E
10: C, D, B, E, A
9: D, C, E, B, A
9: D, E, C, B, A
19: E, D, C, B, A
3: F

The outcome using regular STV is ABCE. That is also the outcome using
CPO-STV, or sequential STV. I hope that you can agree that this seems like
the most fair outcome. The outcome using the method you suggest, however,
is ABCD.
The imaginary political reality behind this example is a simple spectrum
from A to B, where voters vote for candidates nearest to their first
choices on the spectrum before they vote for candidates who are further
away from their first choices. F is kind of an outlier in his own world.
The F voters don't give a damn about any of the other candidates.
Anyway, the difference between the two outcomes is whether D or E gets
elected. Plain STV, CPO-STV, and sequential STV favor E over D because
s/he quite simply has more votes. D, however, is closer to the center, and
so is favored by the method you suggest. That is, if votes for candidates
are arranged along a spectrum, then Condorcet does favor the center of the
spectrum. That's correct for single-winner elections, but not correct in
proportional representation.
Hmm, maybe the Hare example will make it more clear.

Example #2
5 seats
Quota: 20 (Hare)

20: A, B, C, D, E, F
10: B, A, C, D, E, F
10: B, C, A, D, E, F
10: C, B, D, A, E, F
10: C, D, B, E, A, F
10: D, C, E, B, F, A
10: D, E, C, F, B, A
1: E, F, D, C, B, A
19: F, E, D, C, B, A

This example is more extreme, so it should make my point a lot clearer.
The outcome using plain STV, CPO-STV, and sequential STV is ABCDF. The
outcome using the method you suggest, however, is ABCDE. That seems more
drastically unfair than the last example, right? So, in answer to your
wondering, yes, I do think that it is unfair to have that last seat pulled
towards the center by voters who have already invested their votes in
getting other candidates up to quota.

I do think that you have quite an interesting point, though, about the
strategy incentives in STV for placing a marginal candidate above a more
popular sincere favorite. I'm not quite sure whether anything can be done
about that, while still having truly proportional results.
The Meek version of STV is designed to solve a related, though different,
strategy incentive. That is, using Newland Britain, imagine two voters.
One votes for 1. a candidate who will surely be eliminated, 2. his popular
sincere favorite, then 3. a marginal candidate. Another votes just for 1.
his popular sincere favorite, and 2. a marginal candidate.
The first voter's vote counts in full for the marginal candidate, but the
second voter's vote only counts in fraction.
Meek solves this by bringing in later votes from eliminated candidates to
retained by that candidate so that it balances out. I think that Meek is
an improvement on Newland Britain (assuming that you are able to calculate
the results via computer).
But, again, Meek answers a somewhat different concern. The point you
bring up is quite legitimate, and as far as I can tell still applies using
Meek and Meek CPO-STV. I would be interested to know if there is any
system that avoids that strategy incentive... I would also be surprised.

The second idea you bring up is definitely not proportional, I think.
That is, the idea of PR by selecting the Condorcet winner, deleting a
fraction of the ballots that contributed to him/her, and selecting the new
Condorcet winner, until you have filled all the seats. That's it, right?
For example, let's assume a political spectrum once again. The first
Condorcet winner is a centrist, right? And somehow you eliminate the
voters who contributed most to that candidate (I don't know how you would
do that either, but maybe you can assume that we are taking a chunk out of
the centrist vote.). Who will be the next Condorcet winner? There might be
fewer centrist votes now, but the votes on either side still serve to feed
central candidates to win against the sides. My guess is that if you are
electing 5 seats, and the votes are organized into a spectrum with people
ranking all or most of the candidates, this method will more or less elect
the 5 most centrist candidates. Which means that it isn't proportional.

Anyway, I wish that it was that easy to combine the merits of Condorcet
and STV, but apparently it isn't. God knows that CPO-STV is forbiddingly
complicated, but that may be the only way to really go about doing it for
real.
There are some other methods that come fairly close. Sequential STV is
one, especially if you use CPO-STV as the tiebreaker rather then the
method its inventors proposed.
Another one is this:
1. Do an STV count.
2. Find which votes and fractions of votes are responsible for electing
each successful candidate.
3. Make each of those pools of votes and vote fractions into a series of
separate electorates, one for each seat.
4. Conduct a separate Condorcet election in each of these electorates to
determine the final winner for each seat.
This one is not bad. It probably doesn't have any disadvantages from
regular STV other than complexity, and it should increase the chances of a
CPO outcome, hence increasing precision, allowing for more candidates to
run on similar platforms, etc. Surely it takes less computing power than
CPO-STV. The only problem is that the electorate are already defined by
the initial winners, and so the results are biased towards them.
Anyway, this is why I provisionally support CPO-STV. It's not because I'm
a complexity freak; it's just that combining STV and Condorcet seems to be
trickier than one might assume.

Well, that's my two cents. I hope that you find it useful. Let me know if
you have any comments, questions, or disagreements.

all my best,
James

[end of original message]

>
Chris Benham wrote:
>I  have two main ideas on this subject. It seems to me that  STV  is
>impeccable in principle, EXCEPT for  the horrible, arbitary feature it
>has in common with  IRV of  eliminating the candidate who happens to
>have the lowest total at the time it is convenvenient to eliminate a
>candidate. So why not eliminate the Condorcet loser instead?
> Here is a specific proposal:  Ranked ballots, equal preferences and
>truncation allowed. Quota is (1/n+1)+ 0.0000001. Equal first preferences
>are split into fractions of a vote. All candidates with a quota of first
>preferences are elected, and overflow lower preferences are distributed
>. The part of the overflow consisting of "exhausted" ballots can just be
>treated as equal preference for the unmarked candidates and distributed
>fractionally  (This seems correct in principle and saves mucking around
>with having to change the quota.)  After the overflow any candidates who
>now have a quota are elected.  If there are any seats now left
>vacant,then counting all the votes at full value, of  the remaining
>candidates eliminate the Condorcet loser (using I'm-not-sure-what
>completion method).  Distribute this candidate's preferences and elect
>any candidate who now has a quota. If any seats are still vacant,
>repeat, etc.
>My second idea is this:  Elect the CW (completed however), and then
>depending on how many seats there are to be filled, fractionally mark
>down some of the ballots according to their contribution to electing the
>winner, and taking into account the "wasted" vote. Repeat  until the
>desired number of candidates are elected. The details of  exactly which
>ballots to mark down by exactly how much  I haven't yet thought about,
>but I shouldn't think it is a huge problem.

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