Re: [EM] Anyone got a good analysis on limitations of approval andrange voting?
Dear folks, there is another assumption in Arrow's theorem which people almost always forget: Determinism. Methods which use some amount of chance can easily meet all his other criteria, the most trivial example of this being again Random Ballot (i.e. pick a ballot uniformly at random and copy its ranking as the group's ranking). Some people think this violates the no-dictator requirement, but it doesn't since a dictator would be a person determined *beforehand*. Yours, Jobst Raph Frank schrieb: The theorem states (from wiki) that there is no method which has the following properties: * If every voter prefers X over Y, then the group prefers X over Y. * If every voter prefers X over Y, then adding Z to the slate won't change the group's preference of X over Y. * There is no dictator. All 3 of those conditions are met for range. The only problem is that adding Z could cause renormalisation changes in how people vote. A voter who votes A: 100 B: 0 might change vote to: A: 100 B: 50 Z: 0 after Z is added. Thus changing the difference between A and B for that ballot. Ranked systems allow full ranking. Adding another candidate just requires that you insert the candidate into the rank order. With range this might not be possible. If the candidate has a rating outside the max and min, a voter may have to rescale their prior preferences. If the assumption is that voters are just allowed add a rating for Z and not change any of their other ratings, then it meets the 3 conditions and thus is a counter example to Arrow's theorem. Election-Methods mailing list - see http://electorama.com/em for list info Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] What does proportional representation MEAN? And list of known PR methods (know any more?)
On Tue, Nov 17, 2009 at 5:02 PM, Warren Smith warren@gmail.com wrote: Kristofer Munsterhjelm asked me what proportional representation (PR) means. At this time it is probably unwise to make a too-precise definition since every PR voting method seems to obey a different proportionality theorem. I say you should just assess each theorem on a case by case basis to see if you like it. But a somewhat imprecise definition is: I would say that any voting method which elects W winers from N candidates (arbitrary 0WN) with the property that under an assumption of 'standard racist' voter behavior, it always elects the same proportions of different-'color' candidates as the voters (provided enough candidates of each color run) up to some reasonable error bound is PR. However * what is the 'standard racist' voter behavior? * what are the 'error bounds'? (Once they get poor enough, they would no longer be acceptable, but I propose no precise threshhold) I am not sure referring to racism is a good plan :). Something like Any group representing more than N/M of the voters, where M is the number of seats to be filled, must be able to guarantee that N of their candidates are elected (assuming they run enough candidates). I think all PR methods meet this (as it is the Hare quota). Also, a reasonable definition for ranked methods is: http://en.wikipedia.org/wiki/Proportionality_for_Solid_Coalitions Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Is there a PR voting method obeying participation criterion?
This seems to be an open question at present. But it might be pretty easy to prove or disprove. A multiwinner voting method obeys participation if an extra voter, by voting honestly, cannot make the election result worse (in her view) than if she had not voted. (If you remove a winner and replace it with somebody that voter considers better, the winner set got better. Any change in the winner set not arising from a finite sequence of such improvement steps, is a worsening in the view of some voter.) It is fair if symmetric under permuting the candidates and voters. Conjecture: there does not exist a fair multiwinner proportional representation voting method obeying participation. -- Warren D. Smith http://RangeVoting.org -- add your endorsement (by clicking endorse as 1st step) and math.temple.edu/~wds/homepage/works.html Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] What does proportional representation MEAN? And list of known PR methods (know any more?)
Warren Smith wrote: Kristofer Munsterhjelm asked me what proportional representation (PR) means. At this time it is probably unwise to make a too-precise definition since every PR voting method seems to obey a different proportionality theorem. I say you should just assess each theorem on a case by case basis to see if you like it. But a somewhat imprecise definition is: I would say that any voting method which elects W winers from N candidates (arbitrary 0WN) with the property that under an assumption of 'standard racist' voter behavior, it always elects the same proportions of different-'color' candidates as the voters (provided enough candidates of each color run) up to some reasonable error bound is PR. However * what is the 'standard racist' voter behavior? * what are the 'error bounds'? (Once they get poor enough, they would no longer be acceptable, but I propose no precise threshhold) These differ from theorem to theorem. And for Asset Voting standard racism assumptions also are needed about the candidate-behavior. I think that the measure of proportionality should be on sets, not just candidates, because I'd like the method to be better than open party list. Because of that, I like the Droop proportionality criterion, and hope something analogous to it can be constructed for Webster, because it seems that when people *do* vote party list style, Webster beats Droop in the proportionality department (as you yourself have shown on your apportionment pages). A proportionality based on sets would also permit voters to vote for some semi-popular candidate first and a less-known independent second, and have the vote support both. Even that does have limits, though, because it would not guarantee that a vote of independent first, then semi-popular would support semi-popular if independent didn't make it, a property which I'd also like. If I can have a pony, metaphorically speaking, the method should capture people's preferences in orders of sets as well. E.g. if there are n% libertarian socialists, then the method should pick n% that are both, not just ensure n% libertarian, n% socialist. This might not be possible, and might reduce to a set covering problem even if technically possible. In any case, this is all informal. HERE'S MY LIST OF KNOWN PR VOTING METHODS: Webster, and certainly all divisor methods for party-list (it is one) already are known to obey such criteria. (The very definition of divisor method is a PR theorem.) This should include my new notion of generalized divisor methods where both multiplicative and/or additive parameters are involved. Hamilton-Vinton is one. See http://rangevoting.org/Apportion.html http://www.RangeVoting.org/NewAppo.html http://www.RangeVoting.org/BishopSim.html M.L. Balinski H. Peyton Young: Fair Representation: Meeting the Ideal of One Person, One Vote (2nd edition), Brookings Institution Press 2001 Asset voting also obeys a PR theorem. http://rangevoting.org/Asset.html paper #77 at http://www.math.temple.edu/~wds/homepage/works.html RRV also (RRV is kind of based on stealing the divisor-method idea, inside). paper #78 at http://www.math.temple.edu/~wds/homepage/works.html http://rangevoting.org/RRV.html Hare/Droop STV also. Nicolaus Tideman: The Single transferable Vote, J. Economic Perspectives 9,1 (1995) 27-38. And LPV(kappa) (logarithmic penalty voting) also. Invented by F.Simmons. Described in paper #91 at http://www.math.temple.edu/~wds/homepage/works.html Also certain PR methods which are precinct countable invented by Forest Simmons, see puzzle#15 at http://rangevoting.org/PuzzlePage.html . Finally, there was also a simple one invented by a student at University of Michigan named Tim Hull. See http://lists.electorama.com/htdig.cgi/election-methods-electorama.com/2007-April/020194.html http://lists.electorama.com/htdig.cgi/election-methods-electorama.com/2007-April/020195.html That's my list. Is anybody aware of any other PR methods? Non-divisor party list PR (closed and open, though neither is proportional within each list). Party list PR in general might be a good place to consider how much PR is PR. For instance, is Jefferson party list PR? Imperiali? Using party list gets rid of the question of proportionality of *what*, because there can only be proportionality of two things, and these two don't interfere: of party lists, and of candidates within those lists (if open list PR). I have also constructed some proportional and semiproportional methods. One simple method is multiwinner Bucklin, which goes like this: do ordinary Bucklin until some candidate has the support of at least a Droop quota. Elect him and remove from all ballots, reweighting the ballots who contributed to his victory, then restart. I also tried to make a Droop proportional summable version of Bucklin, but it failed because of an ambiguity problem I called shadowing. See
Re: [EM] strategy-free Condorcet method after all!
Here's a way to incorporate this idea for large groups: Ballots are ordinal with approval cutoffs. After the ballots are counted, list the candidates in order of approval. Use just enough randomly chosen ballots to determine the Lull winner with 90% confidence: let L(0) be the candidate with least approval. Then for i = 0, 1, 2, ... move L(i) up the list until some candidate L(i+1) beats L(i) majority pairwise (in the random sample). If the majority is so close that the required confidence is not attained, then increase the sample size, etc. Then with the entire ballots set, apply Jobst's Reverse Lull method: Start with candidate A at the top of the approval list. If a majority of the ballots rank A above the Lull winner (i.e. the presumed winner if A is not elected) then elect A. Otherwise, go down the list one candidate to candidate B. Let L be the top Lull winner with approval less than B. If a majority of ballots rank B above L, then elect B, else continue down the list in the same way. In each case the comparison is of a candidate C with the L(i) with the most approval less than C's approval. If the decisions are all made in the same direction as in the sample, then the Reverse Lull winner is the same as the Lull winner, but occasionally (about ten percent of the time) there will be a surprise. If a voter knew that her ballot was going to be used in the forward Lull sample, she would be tempted to vote strategically. But in a large election, most voters would not be in the sample, so there would be little point in them voting strategically. If sincerity had any positive utility at all, it would be enough to result in sincere rankings (in a large enough election). Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] What does proportional representation MEAN? And list of known PR methods (know any more?)
At 01:03 PM 11/17/2009, Raph Frank wrote: I am not sure referring to racism is a good plan :). The word racist was used. A more politically correct term would be factionally affiliated or factionally dedicated. Something like Any group representing more than N/M of the voters, where M is the number of seats to be filled, must be able to guarantee that N of their candidates are elected (assuming they run enough candidates). I think all PR methods meet this (as it is the Hare quota). Because Asset Voting is a system which can create an electoral college that elects the assembly, I prefer the Hare quota, but, of course, this leaves the question of dregs: after all the seats are assigned from candidates who are elected and the easy vote transfers are done, there may be votes which belong to candidates not elected and they are having difficulty finding a suitable compromise. If any votes are left unused, of course, there is then a vacant seat, and, less likely, more than one vacant seat. Is it necessary to fill this seat? The importance of it is not necessarily clear. The Election Science Foundation is currently used Asset Voting, experimentally, to select a steering committee of three members. It's pretty unlikely that more than one seat would remain unallocated if the Hare quota is used, but one empty seat is reasonably possible. That represents what could be approaching one-third of members who aren't represented. Now, if direct voting on issues is allowed -- if this is to become a board of a corporation, there could be some legal problems with both direct voting and the possibility of unequal voting powers, but decision-making is a separate issue from voting and voting method, per se -- and if the members of the unrepresented group still have a means to bring some matter to the attention of the short committee, it's not particularly harmful. Decisions could require, I believe, a majority of the board, which with two members requires unanimity. That, again, might not be harmful. The organization is proposing to have a separate presiding officer who would be elected by the committee, and who could vote to break ties. So, again, it would take unanimity to elect this officer, so the officer is likely to represent a compromise who would satisfy at least two-thirds of the members. Not bad. But it would be better if the remaining voters are represented. So what happens if we use the Droop quota? We end up with a single-winner election for that last seat, with the voters being those who are unrepresented plus excess votes from those elected. This member of the board might represent a fairly small number of voters, but gets equal voting power (normally). This is the reason I prefer the Hare quota, it's exact, every member elected does represent a specific group of voters, the same as the quota, with the same percentage of voting power in the committee. And then the elected board can make whatever special rules it finds appropriate to secure representation for the remaining voters. It can even, as I'd implement Asset, pick an entirely new candidate, who was not on the ballot. Ideally, for that last seat, it would maximize representation, encouraging a broad compromise among the remaining voters, and it could be flexible. The quota could be adjusted down for the last seat, but that lowering could also be restricted so that disparity of representation doesn't exceed a limit. There is little harm if a small faction has a slightly higher voting power; by definition in our problem here, it is a small faction that can only exercise power through coalition, and if the disparity is limited, it would, in a practical sense, not have true excess power, because actual voting is limited. The goal in an elected committee or assembly, as I see it, is to fairly represent as much of the electorate as possible. When the number of seats is small, that becomes quite important, how that last seat is chosen. With a hundred seats, it wouldn't be so important but, on the other hand, an assembly with a hundred seats is pretty large, it really needs to function through a (sub)committee system, direct meeting for most business becomes too cumbersome. Normally, supermajority of a deliberative body can change or suspend the rules. So ... rules could provide that, say, two-thirds of the assembly (full size, so this is an absolute supermajority based on the full assembly assuming all seats had been elected) can determine how the last seat is chosen, within certain simple parameters designed to prevent serious abuse. Note, however, that if two-thirds of any body with power wants to abuse, it can. Period. An absolute majority can, not merely a supermajority, that's the Nuclear Option in the U.S. Senate. And the nuclear option has never been exercised, to my knowledge, because of the recognized damage to the traditions of the Senate. It was bad enough that the Senate
Re: [EM] Is there a PR voting method obeying participation criterion?
At 01:08 PM 11/17/2009, Warren Smith wrote: This seems to be an open question at present. But it might be pretty easy to prove or disprove. A multiwinner voting method obeys participation if an extra voter, by voting honestly, cannot make the election result worse (in her view) than if she had not voted. Might be a small point, but voted should be defined. Under Robert's Rules, if a voter writes something on a ballot, the voter has voted, and the vote is counted in the basis for majority, and analogous for PR would be that the voter has possibly increased the quota. But we can also look at what happens if the voter votes for an irrelevant candidate. If we are going to be able to properly analyze the systems in a fair way, I think we have to assume that the voter votes for someone who is at least eligible, and that if it's Asset, the candidate actually is available to recast the vote and fairly functions as an effective representative of the voter in further process. No voting method can protect a voter from being dissatisfied with the candidate they voted for! Asset, then, could only change the outcome negatively for the voter by causing some effect due to increasing the quota. How could that happen? From the voter voting, the quota increased by a fraction. For accuracy of vote transfers later on, I recommend that exact quotas be used. In the first round, the fractional vote is irrelevant, but it would be considered when determining excess votes available for transfer. In any case, an increase in quota could cause a failure to immediately elect, or could prevent a later election. But the candidate holding this voters' vote could overcome this, still effectively casting the voter's vote to improve the outcome, should an initial election that would improve the outcome fail by one vote, being a fractional vote short. I think Asset, properly implemented, satisfies a reasonable interpretation of participation. There is no harm caused by the voter's participation that cannot be remedied by a proper recasting of the voter's vote. Ah! The voter's vote can affect more than one election. But if fractional vote transfers can be made (which I recommend) then the voter's proxy can fix the problem by spreading that vote among the affected candidates. If fractional vote transfers can't be made, then, sure, there is a technical failure which is basically roundoff error. That's silly, an example of voting criteria gone mad, separated from practical reality. It is fair if symmetric under permuting the candidates and voters. Conjecture: there does not exist a fair multiwinner proportional representation voting method obeying participation. I don't know how to apply fair. Can you give an example of a system which is not fair by this definition? That would help. Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Is there a PR voting method obeying participation criterion?
Abd ul-Rahman Lomax wrote: At 01:08 PM 11/17/2009, Warren Smith wrote: It is fair if symmetric under permuting the candidates and voters. Conjecture: there does not exist a fair multiwinner proportional representation voting method obeying participation. I don't know how to apply fair. Can you give an example of a system which is not fair by this definition? That would help. Choose the first n candidates who registered (not invariant under permutation of the candidates). Turn the first n voters who voted into dictators (not invariant under permutation of the voters). Election-Methods mailing list - see http://electorama.com/em for list info
