On Tue, Jan 10, 2017 at 3:56 PM, Kristofer Munsterhjelm <km_elmet@t-online.de> wrote: > On 01/10/2017 09:13 PM, Monkey Puzzle wrote: >> If Top Two is required, I would prefer Approval voting, then include the >> Approval winner (AW1), plus the approval winner after all AW1-approving >> ballots are removed. This would be clone independent and would >> generally tend to include candidates from two different parties. It is >> basically a 2-person multiwinner election using Approval reweighted voting. > > That goes a bit too far in the other direction. Consider a profile like > this: > > 99: A B > 1: C > > It seems pretty clear that the candidates to go to the second round > should be A and B, but Approval-and-removal will pick either {A, C} or > {B, C}. Of course, the real world probably won't have this kind of > pathological election situation, but the bias is still there to a lesser > degree: it disproportionately picks "extremists" for the second seat > (i.e. candidates whose voters wouldn't vote for the first winner). This is an intriguing corner case. Let's define a goal, and see if we can find a system that meets it. Possible constraints: * An open primary that allows for two viable candidates to emerge * A general election where the largest percentage possible has expressed approval of at least one candidate in the primary Approve-and-removal seems to result in the outcome you suggest. Worse, the lack of summability makes the method uncomfortably complex. It seems that it's worth exploring the full range: #1: 51: A B 49: C #2 70: A B 30: C #3: 99: A B 1: C Scenario #1 seems to clearly call for some form of approval-and-removal to ensure proportionality. A vs C would be respectable choices for the general. Scenario #3 is an extreme case where a crackpot is able to gain the approval of a tiny slice of the electorate, and the general election seems a waste of time. A vs B seems the best choice for the general. However, what about scenario #2? Allowing A & B to move the general seems to marginalize the views of C in a way that seems like a disadvantage when considered against old-school, closed-primary FPTP. The percentage chosen in my scenario #2 was somewhat arbitrary, and was one that involved working within the two candidate general election constraint. Perhaps that number is worth playing with. Or, perhaps "number of candidates in the general election" is instead a number worth playing with. > If your complexity budget is so that you can't do anything more complex > than approval-and-remove, go with approval-and-remove because it's > better than just picking two Approval winners right out. But if you can, > the following might be better: > > A little bit more complex: First pick the Approval winner. Then randomly > remove ballots that approved of this winner until you've either removed > every ballot that approves of the winner, or 1/3 of the total number of > ballots. Then pick the Approval winner by the remaining ballots (ignore > the first winner if he's still number one). > > This is closer to Droop-proportional, but has a vote management > incentive. The following mitigates the vote management incentive, but is > more complex still: > > First pick the Approval winner W. > For each other candidate X: > Until you have removed 1/3 of the total number of ballots or every > ballot that approves W, first remove ballots that approve W but not X, > then ballots that approve both W and X. Count X's approval according to > the remaining ballots after removal, then put the ballots you removed > back in the pile so you can repeat for the next candidate. > > The candidate with the greatest thus counted approval score gets the > second seat in the runoff. (This is essentially the constraint method > with two seats and Approval.) This deserves more thought than I can give it right now. It seems worth examining further. Rob

On 1/12/2017 4:19 PM, Rob Lanphier wrote: > That goes a bit too far in the other direction. Consider a profile like > this: > > 99: A B > 1: C > > It seems pretty clear that the candidates to go to the second round > should be A and B, but Approval-and-removal will pick either {A, C} or > {B, C}. Of course, the real world probably won't have this kind of > pathological election situation, but the bias is still there to a lesser > degree: it disproportionately picks "extremists" for the second seat > (i.e. candidates whose voters wouldn't vote for the first winner). Presumably just as in normal (vote for one candidate) Top Two Runoff, there doesn't have to be a second round. If the most approved candidate A is approved on more than half the ballots then just elect A. Or you could reweight the ballots that supported A to something above zero, so that a faction with a big enough majority is able to fill both runoff spots. Chris On 1/12/2017 4:19 PM, Rob Lanphier wrote: > On Tue, Jan 10, 2017 at 3:56 PM, Kristofer Munsterhjelm > <km_elmet@t-online.de> wrote: >> On 01/10/2017 09:13 PM, Monkey Puzzle wrote: >>> If Top Two is required, I would prefer Approval voting, then include the >>> Approval winner (AW1), plus the approval winner after all AW1-approving >>> ballots are removed. This would be clone independent and would >>> generally tend to include candidates from two different parties. It is >>> basically a 2-person multiwinner election using Approval reweighted voting. >> That goes a bit too far in the other direction. Consider a profile like >> this: >> >> 99: A B >> 1: C >> >> It seems pretty clear that the candidates to go to the second round >> should be A and B, but Approval-and-removal will pick either {A, C} or >> {B, C}. Of course, the real world probably won't have this kind of >> pathological election situation, but the bias is still there to a lesser >> degree: it disproportionately picks "extremists" for the second seat >> (i.e. candidates whose voters wouldn't vote for the first winner). > This is an intriguing corner case. Let's define a goal, and see if we > can find a system that meets it. > > Possible constraints: > * An open primary that allows for two viable candidates to emerge > * A general election where the largest percentage possible has > expressed approval of at least one candidate in the primary > > Approve-and-removal seems to result in the outcome you suggest. > Worse, the lack of summability makes the method uncomfortably complex. > > It seems that it's worth exploring the full range: > #1: > 51: A B > 49: C > > #2 > 70: A B > 30: C > > #3: > 99: A B > 1: C > > Scenario #1 seems to clearly call for some form of > approval-and-removal to ensure proportionality. A vs C would be > respectable choices for the general. > Scenario #3 is an extreme case where a crackpot is able to gain the > approval of a tiny slice of the electorate, and the general election > seems a waste of time. A vs B seems the best choice for the general. > > However, what about scenario #2? Allowing A & B to move the general > seems to marginalize the views of C in a way that seems like a > disadvantage when considered against old-school, closed-primary FPTP. > > The percentage chosen in my scenario #2 was somewhat arbitrary, and > was one that involved working within the two candidate general > election constraint. Perhaps that number is worth playing with. Or, > perhaps "number of candidates in the general election" is instead a > number worth playing with. > > > >> If your complexity budget is so that you can't do anything more complex >> than approval-and-remove, go with approval-and-remove because it's >> better than just picking two Approval winners right out. But if you can, >> the following might be better: >> >> A little bit more complex: First pick the Approval winner. Then randomly >> remove ballots that approved of this winner until you've either removed >> every ballot that approves of the winner, or 1/3 of the total number of >> ballots. Then pick the Approval winner by the remaining ballots (ignore >> the first winner if he's still number one). >> >> This is closer to Droop-proportional, but has a vote management >> incentive. The following mitigates the vote management incentive, but is >> more complex still: >> >> First pick the Approval winner W. >> For each other candidate X: >> Until you have removed 1/3 of the total number of ballots or every >> ballot that approves W, first remove ballots that approve W but not X, >> then ballots that approve both W and X. Count X's approval according to >> the remaining ballots after removal, then put the ballots you removed >> back in the pile so you can repeat for the next candidate. >> >> The candidate with the greatest thus counted approval score gets the >> second seat in the runoff. (This is essentially the constraint method >> with two seats and Approval.) > This deserves more thought than I can give it right now. It seems > worth examining further. > > Rob > ---- > Election-Methods mailing list - see http://electorama.com/em for list info > > > ----- > No virus found in this message. > Checked by AVG - www.avg.com > Version: 2016.0.7996 / Virus Database: 4749/13750 - Release Date: 01/11/17 > >

On 1/10/2017 6:03 AM, Erik Moeller wrote: Practical questions for this group: - Do you think this system would benefit from iterative improvements? - If so, what would those changes look like? Is going back to party-based primary elections with 1-2-3 ballots and pairwise counting an iterative improvement? If so, that's what I would recommend. An open primary (of the type used in CA) is vulnerable to the following strategy. If there are more Democratic voters, then Republicans can offer just two candidates and quietly fund "spoiler" Democratic candidates. That means there would be 4 Democratic candidates and 2 Republican candidates. Because single-mark ballots are used, and vulnerable to vote splitting, the likely result would be that both runoff candidates will be the two Republicans, and then a Republican is sure to win, even though that's exactly the opposite of what a fair election would yield (if there are more Democratic voters). I think the most important point is that there would be no "need" (or advantage) to open primaries if primaries used fair ballots and fair counting. In other words, this open-primary "reform" is only seen as worthwhile because primary elections use primitive single-mark ballots that are so easily taken advantage of. As a reminder, the whole point of primary elections is to avoid vote-splitting in general elections because general elections use single-mark ballots. Why do so many election "reformers" suggest changes that continue to use single-mark ballots? Single-mark ballots are the source of the unfairness, and REAL reform HAS to use something better than single-mark ballots. Richard Fobes On 1/10/2017 6:03 AM, Erik Moeller wrote: > Since a 2010 ballot initiative [1], California uses a nonpartisan > blanket primary for non-presidential elections. This means that > candidates from all parties and independent ones run against each > other in a primary round, and the top two advance to the general > election. > > Multiple candidates from the _same_ party can run in the primary, > meaning that you may have vote splitting effects. Frequently, the top > two general election candidates are from the same party. This is by > design, though it remains to be seen how voters would respond to e.g., > two gubernatorial choices from the same party. You can see some > example results here: > > https://en.wikipedia.org/wiki/California_State_Assembly_election,_2016 > > An interesting effect you can see here is the large number of > candidates who make it to the general election on a fraction of > write-in votes. Some of these districts would likely have been > uncontested in prior elections. For example, Angela Rupert from > District 46 got 131 (!) write-in votes, and managed to get 43.9% / > 60,658 votes in the general. Quite a jump! > > There's a large turnout gap between primary and general, and the > electorate is different. [2] As a consequence, small parties have > generally feared a loss of visibility, and opposed such measures in > the states where they have been proposed. The change certainly hasn't > helped them in California so far -- only the Libertarians made a small > showing in the 2016 elections, without any seats. > > Practical questions for this group: > - Do you think this system would benefit from iterative improvements? > - If so, what would those changes look like? > > It almost goes without saying that I think proportional representation > voting is the most desirable long term change. But if there are > smaller changes that would improve this system in the near term, CA > may be a viable target for reform, by law or by ballot initiative. > > Personally, I think the vote splitting and spoiler effects in the > primary stage are problematic, and having candidates from the same > party in the general is undesirable. The use of fractions of write-in > votes also seems hackish and suggests ballot access problems in the > primary stage. I can imagine multiple ways to address this, but I'd be > curious what this group thinks. > > Warmly, > Erik > > [1] https://en.wikipedia.org/wiki/California_Proposition_14_(2010) > [2] http://www.ppic.org/content/pubs/report/R_514EMR.pdf > ---- > Election-Methods mailing list - see http://electorama.com/em for list info

On 01/12/2017 06:49 AM, Rob Lanphier wrote: > On Tue, Jan 10, 2017 at 3:56 PM, Kristofer Munsterhjelm > <km_elmet@t-online.de> wrote: >> That goes a bit too far in the other direction. Consider a profile like >> this: >> >> 99: A B >> 1: C >> >> It seems pretty clear that the candidates to go to the second round >> should be A and B, but Approval-and-removal will pick either {A, C} or >> {B, C}. Of course, the real world probably won't have this kind of >> pathological election situation, but the bias is still there to a lesser >> degree: it disproportionately picks "extremists" for the second seat >> (i.e. candidates whose voters wouldn't vote for the first winner). > > This is an intriguing corner case. Let's define a goal, and see if we > can find a system that meets it. > > Possible constraints: > * An open primary that allows for two viable candidates to emerge > * A general election where the largest percentage possible has > expressed approval of at least one candidate in the primary > > Approve-and-removal seems to result in the outcome you suggest. > Worse, the lack of summability makes the method uncomfortably complex. > > It seems that it's worth exploring the full range: > #1: > 51: A B > 49: C > > #2 > 70: A B > 30: C > > #3: > 99: A B > 1: C > > Scenario #1 seems to clearly call for some form of > approval-and-removal to ensure proportionality. A vs C would be > respectable choices for the general. > Scenario #3 is an extreme case where a crackpot is able to gain the > approval of a tiny slice of the electorate, and the general election > seems a waste of time. A vs B seems the best choice for the general. > > However, what about scenario #2? Allowing A & B to move the general > seems to marginalize the views of C in a way that seems like a > disadvantage when considered against old-school, closed-primary FPTP. > > The percentage chosen in my scenario #2 was somewhat arbitrary, and > was one that involved working within the two candidate general > election constraint. Perhaps that number is worth playing with. Or, > perhaps "number of candidates in the general election" is instead a > number worth playing with. Suppose we use Droop quota logic. In each of these, the Droop quota is 33 + 1/3, say 33 for simplicity's sake. So in scenario 1, the surplus is 18; in scenario 2, it is 37, and in scenario 3, it's 66. Thus the outcome would be {A, C} in #1 and {A, B} in #2 and #3. >> If your complexity budget is so that you can't do anything more complex >> than approval-and-remove, go with approval-and-remove because it's >> better than just picking two Approval winners right out. But if you can, >> the following might be better: >> >> A little bit more complex: First pick the Approval winner. Then randomly >> remove ballots that approved of this winner until you've either removed >> every ballot that approves of the winner, or 1/3 of the total number of >> ballots. Then pick the Approval winner by the remaining ballots (ignore >> the first winner if he's still number one). >> >> This is closer to Droop-proportional, but has a vote management >> incentive. The following mitigates the vote management incentive, but is >> more complex still: >> >> First pick the Approval winner W. >> For each other candidate X: >> Until you have removed 1/3 of the total number of ballots or every >> ballot that approves W, first remove ballots that approve W but not X, >> then ballots that approve both W and X. Count X's approval according to >> the remaining ballots after removal, then put the ballots you removed >> back in the pile so you can repeat for the next candidate. >> >> The candidate with the greatest thus counted approval score gets the >> second seat in the runoff. (This is essentially the constraint method >> with two seats and Approval.) > > This deserves more thought than I can give it right now. It seems > worth examining further. I haven't had time to look at it thoroughly, but it seems to be summable, which ordinary approval-and-removal isn't (as you said). To calculate X's adjusted Approval score, you only need: - The number of voters who approved both W and X - The number of voters who approved W but not X - The number of voters in total (for the 1/3 calc) If you have the number of voters who approved W but not X, and you have the unadjusted Approval counts for every candidate, you can get the number of voters who approved both W and X. So it seems all you need is: - The number of voters in total: O(1) - For each candidate X: the number of voters approving X: O(n) in total - For each pair of candidates W, X: the number of voters who approved W but not X: O(n^2) in total which gives summability with order n^2. So we have an interesting contrast: One method is simple to explain but not summable. The other one is more complex, but is summable. (It's vaguely Condorcetian if you read "approves W but not X" as W>X.) The 1/3 threshold would probably mean it fails IIB. It should be possible to fix this (at the cost of more complexity still), with the general idea being to eliminate candidates with the fewest Approvals until the winner has at least 1/3 of the remaining votes.

     - The number of voters who approved W but not X

     - The number of voters in total (for the 1/3 calc)

Kristofer's latest comment is basically a specialized rehash of a topic we were discussing on this list last October, on a thread I started about looking for some sort of Approval/Concorcet hybrid + top two runoff. What he is illustrating is that it is possible to use a two-seat multiwinner election technique to do this, and that with Approval it is possible to do this summably. Kristofer is also using a variant of Droop-Bucklin reweighted voting that uses constraints to discourage free-riding. In the case of a top-two runoff, I don't think that free-riding defection would be as much of a problem, because the quota would be large enough that defection could lead to losing both the Approval winner and the second-seat position. Other comments on the O(N^2) arrays for summable calculation inserted in blue below. On Thu, Jan 12, 2017 at 6:30 AM, Kristofer Munsterhjelm < km_elmet@t-online.de> wrote: > On 01/12/2017 06:49 AM, Rob Lanphier wrote: > >> On Tue, Jan 10, 2017 at 3:56 PM, Kristofer Munsterhjelm >> <km_elmet@t-online.de> wrote: >> >>> That goes a bit too far in the other direction. Consider a profile like >>> this: >>> >>> 99: A B >>> 1: C >>> >>> It seems pretty clear that the candidates to go to the second round >>> should be A and B, but Approval-and-removal will pick either {A, C} or >>> {B, C}. Of course, the real world probably won't have this kind of >>> pathological election situation, but the bias is still there to a lesser >>> degree: it disproportionately picks "extremists" for the second seat >>> (i.e. candidates whose voters wouldn't vote for the first winner). >>> >> >> This is an intriguing corner case. Let's define a goal, and see if we >> can find a system that meets it. >> >> Possible constraints: >> * An open primary that allows for two viable candidates to emerge >> * A general election where the largest percentage possible has >> expressed approval of at least one candidate in the primary >> >> Approve-and-removal seems to result in the outcome you suggest. >> Worse, the lack of summability makes the method uncomfortably complex. >> >> It seems that it's worth exploring the full range: >> #1: >> 51: A B >> 49: C >> >> #2 >> 70: A B >> 30: C >> >> #3: >> 99: A B >> 1: C >> >> Scenario #1 seems to clearly call for some form of >> approval-and-removal to ensure proportionality. A vs C would be >> respectable choices for the general. >> Scenario #3 is an extreme case where a crackpot is able to gain the >> approval of a tiny slice of the electorate, and the general election >> seems a waste of time. A vs B seems the best choice for the general. >> >> However, what about scenario #2? Allowing A & B to move the general >> seems to marginalize the views of C in a way that seems like a >> disadvantage when considered against old-school, closed-primary FPTP. >> >> The percentage chosen in my scenario #2 was somewhat arbitrary, and >> was one that involved working within the two candidate general >> election constraint. Perhaps that number is worth playing with. Or, >> perhaps "number of candidates in the general election" is instead a >> number worth playing with. >> > > Suppose we use Droop quota logic. In each of these, the Droop quota is 33 > + 1/3, say 33 for simplicity's sake. So in scenario 1, the surplus is 18; > in scenario 2, it is 37, and in scenario 3, it's 66. Thus the outcome would > be {A, C} in #1 and {A, B} in #2 and #3. > > If your complexity budget is so that you can't do anything more complex >>> than approval-and-remove, go with approval-and-remove because it's >>> better than just picking two Approval winners right out. But if you can, >>> the following might be better: >>> >>> A little bit more complex: First pick the Approval winner. Then randomly >>> remove ballots that approved of this winner until you've either removed >>> every ballot that approves of the winner, or 1/3 of the total number of >>> ballots. Then pick the Approval winner by the remaining ballots (ignore >>> the first winner if he's still number one). >>> >>> This is closer to Droop-proportional, but has a vote management >>> incentive. The following mitigates the vote management incentive, but is >>> more complex still: >>> >>> First pick the Approval winner W. >>> For each other candidate X: >>> Until you have removed 1/3 of the total number of ballots or every >>> ballot that approves W, first remove ballots that approve W but not X, >>> then ballots that approve both W and X. Count X's approval according to >>> the remaining ballots after removal, then put the ballots you removed >>> back in the pile so you can repeat for the next candidate. >>> >>> The candidate with the greatest thus counted approval score gets the >>> second seat in the runoff. (This is essentially the constraint method >>> with two seats and Approval.) >>> >> >> This deserves more thought than I can give it right now. It seems >> worth examining further. >> > > I haven't had time to look at it thoroughly, but it seems to be summable, > which ordinary approval-and-removal isn't (as you said). > > To calculate X's adjusted Approval score, you only need: > - The number of voters who approved both W and X > For this first item, you can store a candidates's approval and approval of other candidates on the same ballot in the same NxN array. The approval winner W is then the candidate with the largest diagonal entry: W = candidate i for which A_{i,i} > A_{j,j} for all j != i. > - The number of voters who approved W but not X > I think this second array may not be necessary. If W is candidate i, and X is candidate j, and the first array above has entry A_{i,i} for W's approval and A_{i,j} for voters who approve both W and X, then the number of voters who approve W but do not approve X would be A_{i,i} - A_{i,j}. > - The number of voters in total (for the 1/3 calc) > > If you have the number of voters who approved W but not X, and you have > the unadjusted Approval counts for every candidate, you can get the number > of voters who approved both W and X. So it seems all you need is: > > - The number of voters in total: O(1) > - For each candidate X: the number of voters approving X: O(n) in total > - For each pair of candidates W, X: the number of voters who approved W > but not X: O(n^2) in total > > which gives summability with order n^2. > > So we have an interesting contrast: One method is simple to explain but > not summable. The other one is more complex, but is summable. > > (It's vaguely Condorcetian if you read "approves W but not X" as W>X.) > > The 1/3 threshold would probably mean it fails IIB. It should be possible > to fix this (at the cost of more complexity still), with the general idea > being to eliminate candidates with the fewest Approvals until the winner > has at least 1/3 of the remaining votes. > There are some variants one could consider instead of Kristofer's somewhat complex constraint method. If you do the simpler Bucklin (or Approval) reweighted voting, the ballot reweighting for ballots that approve W is F = 1 - min[ (Q / A_w), 1 ] where Q = the quota, and A_w = the approval winner. In other words, if the approval for W is less than the quota, all W-approving ballots are reweighted to zero. The number of ballots to be reweighted can be calculated using the O(N^2) arrays Kristofer has already described. One perpetual problem with reweighting methods like this is that truncated ballots, those on which there are no remaining candidates after W is seated, will be "eliminated" regardless, so the reweighting factor on non-truncated ballots is too large. It is possible to correct this with a modification to the reweighting factor that removes the bullet-vote ballots first. If you accumulate another O(N) vector B, where B_i = the number of ballots that bullet vote for candidate i, then you can reweight only the non-bullet-voting ballots using F = 1 - min[ (Q - B_w) / max[ (A_w - B_w), epsilon ], 1 ] Aside from reweighting factor adjustment, there are various choices one can make for the quota. If you choose the Droop quota [Q = Total/3], a large party that gets more than 66.6% of the Approval vote can dominate both positions. If you use the Hare quota [Q = Total/2], party dominance is only possible with 75+% of the Approval vote. In the interest of promoting a diverse choice for the runoff, I think using the Hare quota would be preferable here. Droop vs. Hare has a much larger effect in multiwinner elections with 4 or fewer winners. Ted (AKA Monkey Puzzle AKA dodecatheon)

     - The number of voters who approved W but not X

     - The number of voters in total (for the 1/3 calc)

I should clarify that the thread I'm referring to started on 2016-11-10, and the topic was "Top-two Approval Pairwise Runoff (TTAPR)". Strategic concerns in the thread included pushover, chicken dilemma (defection), clone resistance, favorite betrayal, and if preferences are included, Condorcet compliance. In the California or Louisiana style top-two primary, we have an electorate that generally doesn't participate in the primary elections in high numbers, but is confused by large numbers of candidates in the general election. They want some way of winnowing down the field. So primary preferences are not a reliable metric for the general election result. Michael Ossipoff made the following comment in that thread: It's *[Ted's note: Approval top-two runoff] *been rejected in EM discussion > because, if you decide to approve Favorite, in addition to Compromise: > Favorite can knock Compromise out of the runoff, when Compromise, but not > Favorite can win in the runoff. In the case of the Approval two-seat methods both Kristofer and I are suggesting, however, it is unlikely that either Favorite or Compromise would either win one of the seats or the runoff election unless they form a sufficiently large coalition together to compete against the opposing seat. Frango ut patefaciam -- I break so that I may reveal On Thu, Jan 12, 2017 at 1:14 PM, Monkey Puzzle <araucaria.araucana@gmail.com > wrote: > Kristofer's latest comment is basically a specialized rehash of a topic we > were discussing on this list last October, on a thread I started about > looking for some sort of Approval/Concorcet hybrid + top two runoff. > > What he is illustrating is that it is possible to use a two-seat > multiwinner election technique to do this, and that with Approval it is > possible to do this summably. > > Kristofer is also using a variant of Droop-Bucklin reweighted voting that > uses constraints to discourage free-riding. > > In the case of a top-two runoff, I don't think that free-riding defection > would be as much of a problem, because the quota would be large enough that > defection could lead to losing both the Approval winner and the second-seat > position. > > Other comments on the O(N^2) arrays for summable calculation inserted in > blue below. > > On Thu, Jan 12, 2017 at 6:30 AM, Kristofer Munsterhjelm < > km_elmet@t-online.de> wrote: > >> On 01/12/2017 06:49 AM, Rob Lanphier wrote: >> >>> On Tue, Jan 10, 2017 at 3:56 PM, Kristofer Munsterhjelm >>> <km_elmet@t-online.de> wrote: >>> >>>> That goes a bit too far in the other direction. Consider a profile like >>>> this: >>>> >>>> 99: A B >>>> 1: C >>>> >>>> It seems pretty clear that the candidates to go to the second round >>>> should be A and B, but Approval-and-removal will pick either {A, C} or >>>> {B, C}. Of course, the real world probably won't have this kind of >>>> pathological election situation, but the bias is still there to a lesser >>>> degree: it disproportionately picks "extremists" for the second seat >>>> (i.e. candidates whose voters wouldn't vote for the first winner). >>>> >>> >>> This is an intriguing corner case. Let's define a goal, and see if we >>> can find a system that meets it. >>> >>> Possible constraints: >>> * An open primary that allows for two viable candidates to emerge >>> * A general election where the largest percentage possible has >>> expressed approval of at least one candidate in the primary >>> >>> Approve-and-removal seems to result in the outcome you suggest. >>> Worse, the lack of summability makes the method uncomfortably complex. >>> >>> It seems that it's worth exploring the full range: >>> #1: >>> 51: A B >>> 49: C >>> >>> #2 >>> 70: A B >>> 30: C >>> >>> #3: >>> 99: A B >>> 1: C >>> >>> Scenario #1 seems to clearly call for some form of >>> approval-and-removal to ensure proportionality. A vs C would be >>> respectable choices for the general. >>> Scenario #3 is an extreme case where a crackpot is able to gain the >>> approval of a tiny slice of the electorate, and the general election >>> seems a waste of time. A vs B seems the best choice for the general. >>> >>> However, what about scenario #2? Allowing A & B to move the general >>> seems to marginalize the views of C in a way that seems like a >>> disadvantage when considered against old-school, closed-primary FPTP. >>> >>> The percentage chosen in my scenario #2 was somewhat arbitrary, and >>> was one that involved working within the two candidate general >>> election constraint. Perhaps that number is worth playing with. Or, >>> perhaps "number of candidates in the general election" is instead a >>> number worth playing with. >>> >> >> Suppose we use Droop quota logic. In each of these, the Droop quota is 33 >> + 1/3, say 33 for simplicity's sake. So in scenario 1, the surplus is 18; >> in scenario 2, it is 37, and in scenario 3, it's 66. Thus the outcome would >> be {A, C} in #1 and {A, B} in #2 and #3. >> >> If your complexity budget is so that you can't do anything more complex >>>> than approval-and-remove, go with approval-and-remove because it's >>>> better than just picking two Approval winners right out. But if you can, >>>> the following might be better: >>>> >>>> A little bit more complex: First pick the Approval winner. Then randomly >>>> remove ballots that approved of this winner until you've either removed >>>> every ballot that approves of the winner, or 1/3 of the total number of >>>> ballots. Then pick the Approval winner by the remaining ballots (ignore >>>> the first winner if he's still number one). >>>> >>>> This is closer to Droop-proportional, but has a vote management >>>> incentive. The following mitigates the vote management incentive, but is >>>> more complex still: >>>> >>>> First pick the Approval winner W. >>>> For each other candidate X: >>>> Until you have removed 1/3 of the total number of ballots or every >>>> ballot that approves W, first remove ballots that approve W but not X, >>>> then ballots that approve both W and X. Count X's approval according to >>>> the remaining ballots after removal, then put the ballots you removed >>>> back in the pile so you can repeat for the next candidate. >>>> >>>> The candidate with the greatest thus counted approval score gets the >>>> second seat in the runoff. (This is essentially the constraint method >>>> with two seats and Approval.) >>>> >>> >>> This deserves more thought than I can give it right now. It seems >>> worth examining further. >>> >> >> I haven't had time to look at it thoroughly, but it seems to be summable, >> which ordinary approval-and-removal isn't (as you said). >> >> To calculate X's adjusted Approval score, you only need: >> - The number of voters who approved both W and X >> > > For this first item, you can store a candidates's approval and approval of > other candidates on the same ballot in the same NxN array. > > The approval winner W is then the candidate with the largest diagonal > entry: W = candidate i for which A_{i,i} > A_{j,j} for all j != i. > > >> - The number of voters who approved W but not X >> > > I think this second array may not be necessary. If W is candidate i, and > X is candidate j, and the first array above has entry A_{i,i} for W's > approval and A_{i,j} for voters who approve both W and X, then the number > of voters who approve W but do not approve X would be A_{i,i} - A_{i,j}. > > >> - The number of voters in total (for the 1/3 calc) >> >> If you have the number of voters who approved W but not X, and you have >> the unadjusted Approval counts for every candidate, you can get the number >> of voters who approved both W and X. So it seems all you need is: >> >> - The number of voters in total: O(1) >> - For each candidate X: the number of voters approving X: O(n) in total >> - For each pair of candidates W, X: the number of voters who approved W >> but not X: O(n^2) in total >> >> which gives summability with order n^2. >> >> So we have an interesting contrast: One method is simple to explain but >> not summable. The other one is more complex, but is summable. >> >> (It's vaguely Condorcetian if you read "approves W but not X" as W>X.) >> >> The 1/3 threshold would probably mean it fails IIB. It should be possible >> to fix this (at the cost of more complexity still), with the general idea >> being to eliminate candidates with the fewest Approvals until the winner >> has at least 1/3 of the remaining votes. >> > > There are some variants one could consider instead of Kristofer's somewhat > complex constraint method. > > If you do the simpler Bucklin (or Approval) reweighted voting, the ballot > reweighting for ballots that approve W is > > F = 1 - min[ (Q / A_w), 1 ] > > where Q = the quota, and A_w = the approval winner. In other words, if > the approval for W is less than the quota, all W-approving ballots are > reweighted to zero. The number of ballots to be reweighted can be > calculated using the O(N^2) arrays Kristofer has already described. > > One perpetual problem with reweighting methods like this is that truncated > ballots, those on which there are no remaining candidates after W is > seated, will be "eliminated" regardless, so the reweighting factor on > non-truncated ballots is too large. It is possible to correct this with a > modification to the reweighting factor that removes the bullet-vote ballots > first. > > If you accumulate another O(N) vector B, where B_i = the number of ballots > that bullet vote for candidate i, then you can reweight only the > non-bullet-voting ballots using > > F = 1 - min[ (Q - B_w) / max[ (A_w - B_w), epsilon ], 1 ] > > Aside from reweighting factor adjustment, there are various choices one > can make for the quota. If you choose the Droop quota [Q = Total/3], a > large party that gets more than 66.6% of the Approval vote can dominate > both positions. If you use the Hare quota [Q = Total/2], party dominance > is only possible with 75+% of the Approval vote. > > In the interest of promoting a diverse choice for the runoff, I think > using the Hare quota would be preferable here. Droop vs. Hare has a much > larger effect in multiwinner elections with 4 or fewer winners. > > Ted (AKA Monkey Puzzle AKA dodecatheon) >