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2018-10-06 17:39:46 UTC

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.

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.