Discussion:
[EM] Plurality Condorcet
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r***@sbcglobal.net
2018-10-13 18:32:04 UTC
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Here is a simple plurality-based Condorcet election method.

The method consists of a series of rounds. Each round permanently eliminates
1 candidate and elects all other uneliminated candidates to the next round.
Repeat until just 1 candidate is elected to the next round. That candidate
is the winner of the election.

V_A equals the sum of ballots with candidate A the highest ranked hopeful
candidate.

1. All elected candidates from the previous round are declared hopeful
(If this is the first round, all candidates are hopeful).
2. Identify hopeful candidate A with the largest V_A. Declare candidate
A elected. Repeat until there is just one remaining hopeful candidate.
Eliminate the one remaining hopeful candidate. This ends the round.

The method is Condorcet since eliminated candidates are the losers of a
two-candidate election and Condorcet winners cannot lose an election with
two candidates.

Example

7 R D P

6 P D R

5 D P R

Round 1: R is the plurality winner and is elected to the next round. D is
next plurality winner is elected to the next round (ballots that had R as
the highest ranked hopeful candidate now have D the highest ranked hopeful
candidate). P is eliminated

Round 2 D is the plurality winner and the winner of the election.

Generalization to elect N candidates proportionally.

The method consists of a series of rounds. Each round permanently eliminates
1 candidate and elects all other uneliminated candidates to the next round.
Continue until just N candidates are elected to the next round. Elect those
N candidates.

V_A equals the sum of ballots with candidate A the highest ranked hopeful
candidate.

S_A equals the sum of seat values of all ballots with candidate A the
highest ranked hopeful candidate.

Steps for a round.

1. All elected candidates from the previous round are declared hopeful
(If this is the first round, all candidates are hopeful). All ballots are
assigned seat value s=0.
2. If there are more than N+1 hopeful candidates, identify hopeful
candidate A with the largest V_A. Declare candidate A elected. Repeat until
there are N+1 hopeful candidates.
3. Identify hopeful candidate A with largest priority V_A/(S_A+1).
Assign new seat value s= (S_A+1)/V_A to all ballots with candidate A the
highest ranked hopeful candidate. Declare candidate A elected. Repeat until
there is just one remaining hopeful candidate. Eliminate the one remaining
hopeful candidate. This ends the round.

The method won't eliminate candidates that are the winners of ever N+1
candidate election that they are in.

Elect 2.

100 A1>A2

49 B1>B2

48 C1>C2

47 D1>D2

46 E1>E2

45 F1>F2

Round 1: Step2 elects A1,A2,B1,B2,C1,C2,D1,D2,E1. Step 3 elects E2,F1. F2 is
eliminated.

Round 2: Step2 elects A1,A2,B1,B2,C1,C2,D1,D2. Step 3 elects E1,F1. E2 is
eliminated.

Round 3: Step2 elects A1,A2,B1,B2,C1,C2,D1. Step 3 elects D2,E1. F1 is
eliminated.

Round 4: Step2 elects A1,A2,B1,B2,C1,C2. Step 3 elects D1,E1,. D2 is
eliminated.

Round 5: Step2 elects A1,A2,B1,B2,C1. Step 3 elects C2,D1. E1 is eliminated.

Round 6: Step2 elects A1,A2,B1,B2. Step 3 elects C1, D1. C2 is eliminated.

Round 7: Step2 elects A1,A2,B1. Step 3 elects B2, C1. D1 is eliminated.

Round 8: Step2 elects A1,A2. Step 3 elects B1,C1. B2 is eliminated.

Round 9: Step2 elects A1. Step 3 elects A2, B1. C1 is eliminated.

Round 10: Step 3 elects A1,A2. B1 is eliminated.

A1,A2 are elected.

Elect 2

20 A B

12 B C D A

14 C D B A

15 D C B A

Round 1: Step 2 elects A. Step 3 elects B,C. D is eliminated.

Round 2. Step 3 elects C,B. A is eliminated

B,C are elected.

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