[EM] Condorcet divisor method proportional representation

[Ross Hyman]

A Condorcet divisor method proportional
representation procedure is presented that is a variant of Nicolaus Tideman’s Comparison
of Pairs of Outcomes by Single Transferable Vote (CPO-STV) and Shultz STV but
requires the determination of fewer candidate set comparisons than either.  The method will produce the same result as a party
list election that uses the same divisor method provided that each voter votes
their party’s list.  The procedure is a Condorcet variant of the procedure presented in the February 2011 issue of Voting Matters.


 

For an N-seat
election, one primary election electing N
candidates must be performed for each set of N + 1 candidates.  For
example, for a two-seat election involving candidates A,B,C and D, primary
elections for the candidate sets ABC, ABD, ACD and BCD are held.

 

For each of these primary elections, the
winning set and its priority over loosing sets is determined by the following
procedure (the method is presented for the d’Hondt divisor method but is easily
generalized to other divisor methods.): 


Step 1. Every candidate in the N+1 primary sub-election candidate set
is hopeful and every candidate not in that set is excluded.  The seat value of every ballot is set to zero. 

Step 2. The priority, PC,
for each hopeful candidate C that is the topmost hopeful candidate on at least
one ballot is determined from PC
= VC/(SC+1) where VC
is the total number of ballots where C is the topmost hopeful candidate and SC is the sum of the seat
values of ballots where C is the topmost hopeful candidate.  The candidate with the highest priority is
elected. If the total number of elected candidates is N, the count is ended and
the N elected candidates are declared
the winning candidate set of the primary with its priority over losing sets equal
to the priority of the Nth elected candidate.  Otherwise, if candidate C is elected, the
seat value for each ballot that contributed to electing C is increased to (SC+1)/VC. Repeat Step 2 until N candidates are elected. 

 

Each loser set from a primary contains the
candidate from the primary candidate set that is not in the winning set plus N-1 additional candidates from the winning
set. For a two-seat election in which AB is the winning set of the primary
candidate set ABC, AC and BC are the loser sets.  

 

Only the priority of the winning set for
each primary is calculated.  The method
determines the priorities of fewer relations than Shultz STV but still elects
the Condorcet winner candidate set if there is one since the Condorcet winner candidate
set cannot be a losing set.  

 

Once every primary election has been held, winning
set > losing set relations are then elected from highest priority to lowest.
However, if electing a relation would violate transitivity then that relation is
excluded instead of elected.  In
practice, only loosing sets that are the winning set of at least one primary
election need be considered.  When every
relation has been elected or excluded, the highest ranked candidate set is
declared the elected candidate set.  An
example with a Condorcet cycle is the two-seat election presented in Election
1.

 

Election 1

7 A B C D

6 B C D A

5 C D A B

4 D A B C

 

Primary ABC

11 ABC

6 BCA

5 CAB

AB > AC and AB > BC. Priority: 8.5

 

Primary ABD

7 ABD

6 BDA

9 DAB

AD > AB and AD > BD.  Priority: 8

 

Primary ACD

7 ACD

11 CDA

4 DAC

CD > AC and CD > AD. Priority: 7.5

 

Primary BCD

13 B C D

5 C D B

4 D B C

BC > BD and BC > CD. Priority: 9

 

The winning sets are AB, AD, CD and
BC.  Since a candidate set must be a
winning set in at least one primary to win the election, only relations
involving winning sets need be considered. 
The relevant candidate relations are

 

BC > CD. Priority: 9

AB > BC. Priority: 8.5

AD > AB. 
Priority: 8

CD > AD. Priority: 7.5

AB > CD. 
Priority: 6.5

 

Transitivity can be preserved by electing
relations in priority order that preserve transitivity and excluding those that
do not.  When the three highest priority
relations are elected, they produce the transitive candidate set order AD >
AB > BC > CD.  The next highest
priority relation CD > AD is excluded since the higher priority relations have
determined that AD > CD.  According to this procedure, candidates A and
D are elected.
-Ross Hyman


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