[EM] Proportional Socially Ranked Condorcet Election Method

[Ross Hyman]

I think all the bugs are out and the claimed properties are satisfied.  I think this is the most natural way to combine Condorcet with STV.

Proportional Socially Ranked Condorcet Election Method for Electing n Winners


Requirements
The method requires a social ranking of the candidates (such as Schulz, Tideman, etc.) and a proportional method
for choosing n winners from any set of n+1 candidates (STS, QPQ, etc.).  

Proportionality Property
The method satisfies Droop proportionality if the method used to choose n
winners from n+1 candidates satisfies Droop proportionality.

Socially Ranked Condorcet Property
The n winners will either all win in any n+1 candidate election in which all of
them are running or the candidate that does not win will have higher social
rank than the candidate that replaced it.  Corollary:  If the n winners are
the n lowest ranked candidates running in an n+1 candidate election they are
all guaranteed to win. 

Definitions
Each candidate is either hopeful, elected, or excluded.  
Each n+1 candidate election is either active or inactive.  
An electable candidate is a hopeful candidate that is a winner of every active
n+1 candidate election in which it runs.
Electing a candidate means to change its status from hopeful to elected and to change
the status of all active n+1 candidate elections in which the elected candidate
does not run to inactive.
Excluding a candidate means to change its status from hopeful to excluded and
to change the status of all active n+1 candidate elections in which the
excluded candidate runs to inactive.

Method
Initially, all candidates are hopeful and all n+1 candidate elections are
active. 
Elect all electable candidates.  If there
are no electable candidates, exclude the lowest ranked hopeful candidate.  Repeat until there are n elected candidates.

Example 1, 2 seats
45 ABCD
40 ADCB
15 ACBD
Condorcet Social ranking A>C>B>D
Active elections (Candidates: Winners, Looser)
ABC: AC, B
ABD: AB, D
ACD: AC, D
BCD:  BD, C
Candidate A, the only electable candidate, is elected.  The election BCD:BD, C is made inactive. The
remaining active elections are
ABC: AC,B
ABD: AB,D
ACD: AC,D
Now C is electable and is elected.  The winners are A and C.  

Example 2 Woodall’s Torpedo, 2 seats
11 AC
9 ADEF
10 BC
9 BDEF
10 CA
10 CB
10 EFDA
11 FDEB
Condorcet Social ranking A>C>D>E>F>B
(actually the placement of A is ambiguous but it does not matter where it is
placed.  The method returns the same two
winners regardless)
Active Elections
ABC: AB, C
ABD: AB, D
ABE: AB, E
ABF: AB, F
ACD: CD, A
ACE: CE, A
ACF: CF, A
ADE: DE, A
ADF: DF, A
AEF: EF, A
BCD: CD, B
BCE: CE, B
BCF: CF, B
BDE: DE, B
BDF: DF, B
BEF: EF, B
CDE: CD, E
CDF: CF, D
CEF: CE, F
DEF: DE, F
There are no electable candidates.  The lowest socially ranked hopeful candidate is
B.  It is excluded.  The active elections are:
ACD: CD, A
ACE: CE, A
ACF: CF, A
ADE: DE, A
ADF: DF, A
AEF: EF, A
CDE: CD, E
CDF: CF, D
CEF: CE, F
DEF: DE, F
C is now electable and is elected.  The active elections are now:
ACD: CD, A
ACE: CE, A
ACF: CF, A
CDE: CD, E
CDF: CF, D
CEF: CE, F
There are no other electable candidates.  The lowest ranked hopeful candidate, F, is
excluded.  The active elections are:
ACD: CD, A
ACE: CE, A
CDE: CD, E
The electable candidate, D, is elected.  The winners are C and D.
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