<html><body><div style="color:#000; background-color:#fff; font-family:times new roman, new york, times, serif;font-size:12pt"><div class="MsoNormal"><span style="background-color: transparent;">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.</span></div><div class="MsoNormal" style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; background-color: transparent; font-style: normal;"><span style="background-color: transparent;"><br></span></div><div class="MsoNormal" style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; background-color: transparent; font-style: normal;"><span style="background-color: transparent;">Proportional Socially Ranked Condorcet Election Method for Electing n Winners</span><br></div><div class="MsoNormal" style="color: rgb(0, 0, 0);
font-size: 16px; font-family: 'times new roman', 'new york', times, serif; background-color: transparent; font-style: normal;"><o:p></o:p></div><div class="MsoNormal"><br></div><div class="MsoNormal" style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; background-color: transparent; font-style: normal;">Requirements<br>
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.). <o:p></o:p></div><div class="MsoNormal"><br></div><div class="MsoNormal" style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; background-color: transparent; font-style: normal;">Proportionality Property<br>
The method satisfies Droop proportionality if the method used to choose n
winners from n+1 candidates satisfies Droop proportionality.<o:p></o:p></div><div class="MsoNormal"><br></div><div class="MsoNormal" style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; background-color: transparent; font-style: normal;">Socially Ranked Condorcet Property<br>
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. <o:p></o:p></div><div class="MsoNormal"><br></div><div class="MsoNormal" style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; background-color: transparent; font-style: normal;">Definitions<br>
Each candidate is either hopeful, elected, or excluded. <br>
Each n+1 candidate election is either active or inactive. <br>
An electable candidate is a hopeful candidate that is a winner of every active
n+1 candidate election in which it runs.<br>
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.<br>
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.<o:p></o:p></div><div class="MsoNormal"><br></div><div class="MsoNormal" style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; background-color: transparent; font-style: normal;">Method<br>
Initially, all candidates are hopeful and all n+1 candidate elections are
active. <br>
Elect all electable candidates. If there
are no electable candidates, exclude the lowest ranked hopeful candidate. Repeat until there are n elected candidates.<o:p></o:p></div><div class="MsoNormal"><br></div><div class="MsoNormal" style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; background-color: transparent; font-style: normal;">Example 1, 2 seats<o:p></o:p></div><div class="MsoNormal">45 ABCD<o:p></o:p></div><div class="MsoNormal">40 ADCB<o:p></o:p></div><div class="MsoNormal">15 ACBD<o:p></o:p></div><div class="MsoNormal">Condorcet Social ranking A>C>B>D<o:p></o:p></div><div class="MsoNormal">Active elections (Candidates: Winners, Looser)<o:p></o:p></div><div class="MsoNormal">ABC: AC, B<o:p></o:p></div><div class="MsoNormal">ABD: AB, D<o:p></o:p></div><div class="MsoNormal">ACD: AC, D<o:p></o:p></div><div class="MsoNormal">BCD: BD, C<o:p></o:p></div><div class="MsoNormal">Candidate A, the only electable candidate,
is elected. The election BCD:BD, C is made inactive. The
remaining active elections are<o:p></o:p></div><div class="MsoNormal">ABC: AC,B<o:p></o:p></div><div class="MsoNormal">ABD: AB,D<o:p></o:p></div><div class="MsoNormal">ACD: AC,D<o:p></o:p></div><div class="MsoNormal">Now C is electable and is elected. The winners are A and C. <o:p></o:p></div><div class="MsoNormal"><br></div><div class="MsoNormal" style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; background-color: transparent; font-style: normal;">Example 2 Woodall’s Torpedo, 2 seats<o:p></o:p></div><div class="MsoNormal">11 AC<o:p></o:p></div><div class="MsoNormal">9 ADEF<o:p></o:p></div><div class="MsoNormal">10 BC<o:p></o:p></div><div class="MsoNormal">9 BDEF<o:p></o:p></div><div class="MsoNormal">10 CA<o:p></o:p></div><div class="MsoNormal">10 CB<o:p></o:p></div><div class="MsoNormal">10 EFDA<o:p></o:p></div><div class="MsoNormal">11 FDEB<o:p></o:p></div><div
class="MsoNormal">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)<o:p></o:p></div><div class="MsoNormal">Active Elections<o:p></o:p></div><div class="MsoNormal">ABC: AB, C<o:p></o:p></div><div class="MsoNormal">ABD: AB, D<o:p></o:p></div><div class="MsoNormal">ABE: AB, E<o:p></o:p></div><div class="MsoNormal">ABF: AB, F<o:p></o:p></div><div class="MsoNormal">ACD: CD, A<o:p></o:p></div><div class="MsoNormal">ACE: CE, A<o:p></o:p></div><div class="MsoNormal">ACF: CF, A<o:p></o:p></div><div class="MsoNormal">ADE: DE, A<o:p></o:p></div><div class="MsoNormal">ADF: DF, A<o:p></o:p></div><div class="MsoNormal">AEF: EF, A<o:p></o:p></div><div class="MsoNormal">BCD: CD, B<o:p></o:p></div><div class="MsoNormal">BCE: CE, B<o:p></o:p></div><div class="MsoNormal">BCF: CF, B<o:p></o:p></div><div class="MsoNormal">BDE: DE, B<o:p></o:p></div><div class="MsoNormal">BDF: DF, B<o:p></o:p></div><div class="MsoNormal">BEF: EF, B<o:p></o:p></div><div class="MsoNormal">CDE: CD, E<o:p></o:p></div><div
class="MsoNormal">CDF: CF, D<o:p></o:p></div><div class="MsoNormal">CEF: CE, F<o:p></o:p></div><div class="MsoNormal">DEF: DE, F<o:p></o:p></div><div class="MsoNormal">There are no electable candidates. The lowest socially ranked hopeful candidate is
B. It is excluded. The active elections are:<o:p></o:p></div><div class="MsoNormal">ACD: CD, A<o:p></o:p></div><div class="MsoNormal">ACE: CE, A<o:p></o:p></div><div class="MsoNormal">ACF: CF, A<o:p></o:p></div><div class="MsoNormal">ADE: DE, A<o:p></o:p></div><div class="MsoNormal">ADF: DF, A<o:p></o:p></div><div class="MsoNormal">AEF: EF, A<o:p></o:p></div><div class="MsoNormal">CDE: CD, E<o:p></o:p></div><div class="MsoNormal">CDF: CF, D<o:p></o:p></div><div class="MsoNormal">CEF: CE, F<o:p></o:p></div><div class="MsoNormal">DEF: DE, F<o:p></o:p></div><div class="MsoNormal">C is now electable and is elected. The active elections are now:<o:p></o:p></div><div class="MsoNormal">ACD: CD, A<o:p></o:p></div><div class="MsoNormal">ACE: CE, A<o:p></o:p></div><div class="MsoNormal">ACF: CF, A<o:p></o:p></div><div class="MsoNormal">CDE: CD, E<o:p></o:p></div><div class="MsoNormal">CDF: CF, D<o:p></o:p></div><div class="MsoNormal">CEF: CE,
F<o:p></o:p></div><div class="MsoNormal">There are no other electable candidates. The lowest ranked hopeful candidate, F, is
excluded. The active elections are:<o:p></o:p></div><div class="MsoNormal">ACD: CD, A<o:p></o:p></div><div class="MsoNormal">ACE: CE, A<o:p></o:p></div><div class="MsoNormal">CDE: CD, E<o:p></o:p></div><div>
</div><div class="MsoNormal">The electable candidate, D, is elected. The winners are C and D.</div> </div></body></html>