[EM] Re: Bucklin-Condorcet PR (also Bucklin PR)

[Chris Benham]

  Previously, on Friday, August 8, 2003  I posted a suggested 
ranked-ballot PR method that combines Generalized Bucklin and Condorcet. 
It wrongly included: "Equal preferences are divided into equal fractions 
(which sum to 1)". I now think it is fine if equal preference for A and 
B are counted Approval-style as a whole vote for each.
With this mistake removed, this is my proposal again:
Ranked ballots, equal preferences ok. Count the first preference votes. 
.If any candidates have  a Droop quota they are elected, and then reduce 
the values of the ballots which have elected members by an amount which 
sums to a Droop quota.
If more than one place remains unfilled,  proceed to to the second 
round. Add the second preference votes to the first preferences (based 
on the value of the ballots after the any reductions that were  made the 
previous round). If this gives any candidate a Droop quota, then elect 
the candidate with the highest tally. If  there is a tie, then elect the 
tied candidate who had  the bigger tally at the last round, if still 
tied then the round before that if  there was one, otherwise the 
Condorcet winner of the tied candidates based on the ballots after the 
most recent devaluations. Reduce the value of the ballots that  elected 
this winner by an amount that sums to a Droop quota. If  there is still 
more than one place unfilled and if after the latest devaluing of 
ballots any candidates have a Droop quota, then elect the one with the 
highest tally (same tie-breaking proceedure) and so on.
If there is more than one place to be filled, then add the third 
preference votes to the tallies of first and second preferences and if 
that gives any candidate a Droop quota, then as before the candidate 
with the highest tally is elected and so on.
If  proceeding in this way leads to the situation where there is one and 
only one more place to be filled, then based on the ballots after all 
the devaluations elect the Condorcet Winner.
A more simple, pure GB-style method would be to fill the last seat by 
Generalized Bucklin (GB) as well. In the following examples I will give 
both results.
This example is lifted from a June 2002  UK Electoral Reform Society 
 article "Sequential STV - a new version":

2 seats, 5028 votes, 5 candidates.
1248: A>B>C>D>E
1236: B>C>D>A>E
1224: C>D>B>A>E
1212: D>B>C>A>E
   36:  E>A>B>D>C
   36:  E>C>D>B>A
   36:  E>D>C>B>A
Droop quota =1676.
First round:   A: 1248    B:1236   C:1224    D:1212    E: 108
No candidate has a Droop quota, so the second  preference-level votes 
are added, to give
Second round:  A: 1284   B: 3696   C: 2496   D: 2472  E: 108
If  any candidate has a quota, elect the one with the highest total. 
Here 3  candidates have a quota, B has the highest total so is elected. 
All those ballots which helped elect B, i.e. those that give first or 
second preference to B, are now all reduced in  value by a Droop quota, 
so that they now sum 3696 - 1676  =  2020. So for example with the 
election of  B and the subsequent devaluations, the 1248  ABCDE ballots 
become 1248/3696 x 2020 = 682   ACDE ballots. So after these 
devaluations the votes
now are :
682:   A>C>D>E
1900 :C>D>A>E
662:   D>C>A>E
36:     E>A>D>C
36:     E>C>D>A
36:     E>D>C>A
 On these ballots  C still has a quota and so is elected, making the 
result BC  (agreeing with both plain and Sequential STV).
The second example  is taken from the same source, and is the same as 
the first except that all those voters who ranked E last now rank E 
second. The first round is the same.
First round:   A: 1248    B: 1236   C:1224   D: 1212    E:108
No Droop quota, so add the second preference-level votes:
Second round: A:1248   B:1236   C:1260   D:1248     E: 5028
E is first or second on all the ballots! So E is elected, and there is 
now no point in devaluing ALL the ballots. On the original ballots, 
disregarding E, B is the CW  and also the Bucklin winner so both the PR 
methods elect  E and  B.
Sequential STV also elects EB, but plain STV elects BC as before.

Chris Benham.



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