Discussion:
[EM] A stronger proportionality criterion than Droop and a weaker no harm criterion than LNH
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r***@sbcglobal.net
2018-10-06 17:39:46 UTC
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I define a proportionality criterion for ranked ballot methods that is
stronger than Droop proportionality and I show that the Phragme'n method
obeying later-no-harm/help (LNH) does not satisfy it. I conjecture that the
criterion is incompatible with LNH, but I do not prove it. I then propose a
weakening of LNH that I believe is compatible with the stronger
proportionality criterion. I provide two variants of Phragme'n that obey
this weakened LNH criterion and I provide an example ballot set for which
the variants obey the stronger proportionality criterion. But I do not
supply a general proof.



Definition of D'Hondt Open List Proportional Representation:

If each voter is a partisan who only votes for candidates of their own party
but in any order they like, then a method satisfying this property elects
the same number of candidates from each party as would be elected from
D'Hondt List PR.



Claim: ranked Phragme'n obeying Later No Help/Harm does not obey D'Hondt
Open List Proportional Representation.

Proof:

Example Elect 2.

100 A1>A2

49 B1>B2

48 C1>C2

47 D1>D2

46 E1>E2

45 F1>F2

D'Hondt List PR elects A1 with priority 100 and A2 with priority 100/2 =50.

For ranked Phragme'n obeying Later No Help/Harm, one first calculates the
quota as (100+49+48+47+46+45)/3 = 111.67. No candidate's priority is bigger
than the quota so all candidates with zero priority and the candidate with
the lowest nonzero priority are excluded including A2. QED.



Claim: Any method obeying Later No Harm/Help cannot obey D'Hondt Open List
Proportional Representation for elections for more than 1 winner. Not
proven.



Definition of N-Later no Harm/Help (N-LNH).

Consider any set of N candidates. That set of candidates cannot be helped or
harmed if the order of candidates ranked lower than all those N candidates
on a ballot are changed or any of those candidates are deleted. Example of
2-LNH: Consider a ballot A>B>C>D>E. The candidate set (A,B) cannot be
helped or harmed if the order of C,D,E is changed or any of those candidates
deleted. The candidate sets (A,C) and (B,C) cannot be helped or harmed by
interchanging D and E or by deleting D and/or E. The candidate sets (A,D) ,
(B,D) , and (C,D) cannot be helped or harmed by deleting E. 1-LNH is the
same as conventional LNH and N-LNH is a weaker version of 1-LNH.



Definition of First Variant Phragme'n method for electing N candidates.

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 uneliminated. Elect those N candidates.




Definitions

Truncated ballot: A truncated ballot is a copy of an original ballot
truncated so that all candidates ranked lower than the Nth highest ranked
elected candidate from the previous round have been removed. If this is the
first round then a truncated ballot retains only the first N ranked
candidates.

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

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



Steps for a round.

1.Create the set of truncated ballots for this round from the original
ballots. All elected candidates from the previous round are declared hopeful
(If this is the first round, all candidates are hopeful). All truncated
ballots are assigned seat value s=0.

2.Identify hopeful Candidate A with largest V_A/(S_A+1) on the truncated
ballots for this round. Assign new seat value s= (S_A+1)/V_A to all
truncated ballots with candidate A the highest ranked hopeful candidate.
Declare candidate A elected to the next round.

3.Repeat step 2 until only one hopeful candidate remains. Eliminate that
candidate. This ends the round.



Definition of Second Variant Phragme'n method for electing N candidates.

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 uneliminated. Elect those N candidates.




Definitions

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

S_A equals the sum of seat values of all truncated 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.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 to the next round.
Repeat Step 2 until N-1 candidates have been elected to the next round.

3.Identify hopeful candidate A with smallest priority V_A/(S_A+1). Declare
candidate A permanently eliminated. Elect all remaining hopeful candidates
to the next round. This ends the round.



Proof that both Phragme'n variants obeys N-LNH:

For each round, no more than the N highest ranked hopeful candidates on each
ballot are looked at. No lower ranked candidates can influence the result
of a round.



Proof that both N-LNH Phragme'n variants satisfy D'Hondt Open List
Proportional Representation for the example given above.

Example

Elect 2.

100 A1>A2

49 B1>B2

48 C1>C2

47 D1>D2

46 E1>E2

45 F1>F2

Both variants elect A1 at the beginning of each round after which candidate
A2 has a priority of 50, higher than any other candidate, so it is never
eliminated.



Claim that N-LNH obeying Phragme'n variants satisfy D'Hondt Open List
Proportional Representation for any ballot set.

Unproved.



The N-LNH variants do not require a determination of quota. But both
variants can be sped up with a quota check. For example, the second
variant's step 3 can be changed to:

3.Identify hopeful candidate A with largest priority V_A/(S_A+1). If this
priority is larger than the quota V_Active/(S_Active+2) (where V_Active and
S_Active are for all ballots ranking at least one hopeful candidate) then
declare candidate A and all N-1 candidates elected to the next round the
winners of the election and end the count. Otherwise, Identify hopeful
candidate A with smallest non-zero priority V_A/(S_A+1). Declare candidate
A, along with all candidates with zero priority, permanently eliminated.
Elect all remaining hopeful candidates to the next round. This ends the
round.



I believe (without proof) that the second variant (but not the first) also
has the property that if each voter is a partisan who only votes for
candidates of their own party, and the method elects N_A candidates from
party A, Then an election for N_A seats using just the ballots for Party A
will elect the same set of candidates from party A.
r***@sbcglobal.net
2018-10-06 17:48:26 UTC
Permalink
Oops. The second variant does not require truncated ballots but I forgot to
remove that from the definitions of V_A and S_A for the second variant.
They should read

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.

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