[EM] Ranked Phragmen (QPQ) satisfies proportional completion

[Ross Hyman]

Proof that the ranked Phragmen method (QPQ) satisfiesproportional completion.  
The Phragmen Method
The Phragmen method is, in my opinion, a beautifullyelegant way to elect multiple winners proportionally from Approvalballots.  It was introduced to this listby Olli Salmi in 2002.  http://groups.yahoo.com/group/election-methods-list/message/10068I think it would be the simplest and most elegant wayto modify the nomination procedure for the Hugo.
Salmi generalized Phragmen to ranked ballots and themethod was developed further by Douglas Woodall in Voting Matters,where he christened it QPQ.  http://www.votingmatters.org.uk/ISSUE17/I17P1.PDFThe method is, in my opinion, a very beautiful andelegant STV-like method.  I have alsowritten about it in Voting Matters.  http://www.votingmatters.org.uk/ISSUE28/I28P2.pdf 
Proportional Completion
Proportional completion is a way to complete ballotsthat do not rank every candidate.  It wasdeveloped by Markus Schulze and used in his multiple winner proportionalrepresentation version of his beat path voting method, Schulze STV.  http://m-schulze.9mail.de/schulze2.pdf
Proportional completion completes incompletely rankedballots so as to preserve the version of the Condorcet matrix that uses matrixelements of the form VA>B/(VA>B + VB>A), that is it preserves theproportion of voters amongst those who have expressed an opinion.  Proportional completion is achieved bydividing up incomplete ballots proportionally. For example, if for all ballots that do have a ranking between A and B,60% rank A>B and 40% rank B>A then each incomplete ballot that does notprovide a ranking between A and B is divided up so that 60% of it ranks A>Band 40% ranks B>A.  
I define the Proportional Completion voting methodscriterion to be satisfied when a voting method produces the same result ifincomplete ballots are proportionally completed or not.
Terminology:V_A is the total number of ballots that contribute toelecting A.  Ballots are included inwhich Candidate A is the highest ranked hopeful (not elected and not excluded)candidate.S_A is the sum of the loads of all of the ballots thatcontribute to electing A. V_act is the total number of active ballots.  These are the ballots that rank at least onehopeful candidate. V_inact is the total number of inactive ballots.  These are the ballots that do not rank anyhopeful candidates.  They only rankelected and excluded candidates. S_inact is the sum of the loads of all of the inactiveballots.V_tot is the total number of ballots.  V_tot = V_act + V_inact.  Unlike all of the other variables listedabove, it does not change throughout the procedure.
Proof that the ranked Phragmen method (QPQ)satisfies proportional completion.  Without proportional completion, the loadto elect candidate A is s_A= (S_A +1)/V_AThis is also the load that will now be oneach ballot that contributed to electing A.  

With proportional completion, the same formula applies but V_A and S_Aare modified to:V^pc_A = V_A + V_inact *( V_A/V_act)and S^pc_A = S_A + S_inact*(V_A/V_act).
The proportionally completed load to electA is therefore s^pc_a = (S_A + S_inact * (V_A/V_act)  +1) / ( V_A + V_inact*(V_A/V_act))= (V_act/V_tot) * s_A + S_inact/V_totThe load with proportional completion isjust a scale and a shift of the load without completion.  It is the same scale and shift for eachcandidate.  A scale and shift does notchange the relative order of the loads so the same candidates would be electedin the same order. The same scale and shift applies to the load to determine ifa candidate should be excluded.  So the rankedPhragmen (QPQ) satisfies proportional completion. - Ross Hyman  
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