As near as I know the following PR method based on Range/Score style ballots is new. This method is based on maximizing a measure of "goodness" of representation to be specified later. Slates of candidates are nominated individually for consideration, because in general there are too many possible slates to consider every one of them (due to combinatorial explosion). Among the nominated slates, the one with the best measure of "goodness" of PR is elected. To reduce the abstraction, suppose that there are only 100 candidates and that only five vacancies to be filled. Suppose further, that there are ten thousand ballots (one for each of ten thousand voters). Given a subset S of five candidates, we decide how good it is as follows: Order the set S according to their Range totals, so that the highest to lowest score order is c1, c2, ...c5. This order only comes into play to determine the cyclic order of play as the candidates "choose up teams" so to speak. Ballots are assigned to each of the candidates cyclically so that the ballot most favorable to c1 goes to c1's pile, of the remaining the one most favorable to c2, goes to c2's pile, etc. like the way we used to choose teams when we were in grade school. (Eventually we'll get to how to automate judgment of favorability. Be patient) After 2000 times around the circle, each pile will contain exactly 2000 ballots. (Thanks for your patience.) For our purposes the relative favorability of ballot V for candidate C is the probability that V would elect C if it were drawn in a lottery; i.e. V's rating of C divided by the sum of all of V's ratings for the candidates in S including C. What happens when one of more of the candidates is not shown any favorability by any of the remaining ballots? The other candidates continue augmenting their piles until they reach their quotas (two thousand each in this case), and the remaining ballots are assigned by comparing them to the official public ballots of the candidates whose piles are not yet complete. (We won't worry about the details of that for now.) For each candidate C in S add up all of the ratings over all of the ballots in the pile, but not the ratings for candidates outside of S. Divide this number by the total possible, which in this case is two thousand times five or ten thousand. We now have five quotients, one for each candidate. Multiply these five numbers together and take the fifth root. This geometric mean is the "goodness" score for the slate. Among the nominated slates, elect the "best" one, i.e. the one with the highest "goodness." It is easy to show that this method satisfies proportionality requirements. And (I believe) it takes into account "out-of pile" preferences as much as possible without destroying proportionality. No time for proofs or examples right now, but first, any questions about the method? **** ---- Election-Methods mailing list - see https://electorama.com/em for list info |
I continue to be amazed that someone like Forest Simmons, who was an early writer about what later was called Asset Voting, as named by Warren Smith, and was simply a tweak on STV when invented by Charles Dodgson in the 1880s, which can produce *perfect* representation, beyond mere "proportional representation," still works on complex single ballot deterministic methods that must compromise and prevent true chosen representation in favor of some sort of theory of "optimised goodness." Asset Voting can very simply produce unanimous election of seats, where the seat represents a quota of voters who have unanimously consented, directly or, probably much more commonly in large-scale elections, through chosen "electors," I call them, who becomes a proxy for the anonymous voters, for the election of the seat. Asset allows the electors --- those who become public voters -- to cooperate and collaborate for the selection of seats. The "Electoral College," fully and accurately, represents all
the voters. But it might be very large, possibly too large to even
meet directly. But all that is needed is a way for electors to
communicate and to register their vote assignments for the
creation of seats. They could use delegable proxy to advise them
how to transfer. Asset was actually tried once, and it produced a result that most would have considered impossible. 17 voters, five candidates for a three-seat steering committee. A rather sharp division, but the final result was the election of the three seats with every vote represented directly or by proxy. Before the third seat was elected, nevertheless two seats were elected, so the steering committee, if needed, could have made any decision by unanimous vote (two agreeing) through the representation of two-thirds of the electorate. It could, in theory, have decided to elect the third seat by using the Droop quota. (the quota had not actually been specified, and the one who called the election and created the process was not totally sophisticated on Asset, which does not need to specify an election deadline, it can, instead, leave that last seat or seats open, and consider the rest of the electorate as Robert's rules considers unrecognizable ballots: they count for determining "majority" but not as a vote for or against any candidate. So, say, there is a jurisdiction with a million voters. It is decided that an optimal assembly would be 49, so the basis for a quota could be 50, and thus the quota would be 2% of the votes cast. This allows that if all electors assign their votes in exact measure to seats, 50 seats could be elected, but that outcome is improbable. (But if it happens, whoopee!), so, normally, 49 seats might be elected. Instead of electing the last seat by plurality, depending on some deadline for vote assignments, I have suggesting leaving the election open. Further, those electors could be consulted by the Assembly. This is the power of having public voters who, collectively, represent the entire electorate. Unanimous election of seats. Bayesian regret, zero. No losers.
Minimal damage to the dregs. No expensive or extensive campaigning
necessary. (I expect that the tradition would develop rapidly that
one would only vote for people one could meet face-to-face,
because how else can they truly represent? But this would be
voluntary, not coerced. Basically, anyone who registers as an
elector may participate further, the only requirement being a
willingness to vote publicly (which is already required of elected
representatives!) The U.S. Electoral College was a brilliant invention, knee-capped
by the party system. It did not, however, represent the people,
but jurisdictions, specifically states. It could have been
reformed to represent the people, but it went, instead, toward
representing the party majority in each state, in effect, becoming
a rubber stamp and creating warped results. Asset is simple, close to tradition, but actually revolutionary,
a dramatic shift from the entire concept of "elections" as
contests. The voters would be 100% represented in the College,
though voluntary choices, no need to consider "electability,"
hence no need for "voting strategy." Choose the available person,
from a large universe, you most trust. As Dodgson noticed, that
was relatively easy for ordinary voters, much easier than sensibly
ranking many candidates. That complexity is completely unnecessary
with Asset. No votes are wasted. The only tweak I see as needed involves making it difficult to coerce votes, by making it impossible to know that a specific person did *not* vote for one. This would not be needed for small NGO elections. In public elections, I'd have a set of known candidates who received substantial votes in a prior election, or something like that, and when electors register, they would assign their own vote to another candidate. If they receive less than N votes, their vote would be transferred as directed. If they receive N votes, they become an elector. Electors would not vote in the election, so, if they receive N votes, they get their own declared vote back and so they have N+1 votes to transfer. N should be the minimum size necessary to ensure that they cannot know that a specific person did not vote for them. If they do not receive N votes, their received votes are privately added to the total for the candidate they chose when registering. So N might be two, but if it is a bit higher, it could provide increased security. 3 might be completely adequate, together with it being very illegal to coerce votes. No more original content below. On 4/10/2019 5:08 PM, Forest Simmons
wrote:
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In reply to this post by Forest Simmons
One thing I forgot: as each ballot is added to a candidate's pile, it must be scaled by the favorability of the ballot to that candidate. That reduces the max total for each pile from ten thousand
[messages referred to] Today's Topics: Because of the scaling by favorability, the max possible total is only two thousand, not ten thousand.
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In reply to this post by Forest Simmons
[pesky text editor sent my previous attempt prematurely] Correction to an important oversight: as a ballot is added to a candidate's pile it must be multiplied by that ballot's favorability for the candidate. Later the normalization step that makes the max possible goodness equal to one, must take this adjustment into account. I'll indicate these corrections with inline edits of the original message below. Before I go to Lomax, let me just mention that other measures of favorability may turn out to be better, particularly those based on singular value decompositions (SVD's) like the ones Warren is working on. Now to Lomax: I fully agree that Charles Dodgson's solution (that we now call Asset Voting) is the simplest and best over all method for most multi-winner and even single winner elections. And I appreciate your insights relative to the electoral college and other historical context, as well as your practical suggestions for implementation of the method or a version of the method, since it seems that tweaks about the details are irresistible. And any other method can be combined with Asset Voting, like the Australians do by allowing the voters to vote the party line or copy candidate cards. But mostly, some of us are interested in finding the mathematical limitations of methods based on Range/Score/Cardinal Rating style ballots. Thanks for your valuable comments. [messages being replied to] Before throwing each ballot into its respective pile, scale it by its favorability (as defined below) to the candidate inot whose pile it is being thrown.
Actually, because of the nature of the favorability factor by which each ballot has already been scaled, no ballot can contribute more than one unit to the total, so the max total is the number of ballots in the pile, namely two thousand, not five thousand.
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Hi Forest, this is in reaction to your new "card dealing" PR multiwinner
voting method. > "Slates of candidates are nominated individually for consideration, because in general there are too many possible slates to consider every one of them (due to combinatorial explosion)." --My belief is, either (a) a computer can enumerate all binomial(C,W) possible W-winner subsets of the C candidates, or (b) too many for the computer. In case (b), I have arguments that virtually any multiwinner method is inherently ridiculously ultra-vulnerable to strategy. Suggesting none should be used in this regime. > "For our purposes the relative favorability of ballot V for candidate C is the probability that V would elect C if it were drawn in a lottery; i.e. V's rating of C divided by the sum of all of V's ratings for the candidates in S including C." --this measure seems suspicious if there could be many cloned candidates. If we had ballots of the form "you (the voter) have 100 points to distribute to the candidates in any way you please" it would be less suspicious. But then voters would likely be strategically motivated to give one 100 & all others 0. > What happens when one of more of the candidates is not shown any favorability by any of the remaining ballots? The other candidates continue augmenting their piles until they reach their quotas (two thousand each in this case), and the remaining ballots are assigned by comparing them to the official public ballots of the candidates whose piles are not yet complete. (We won't worry about the details of that for now.) --sounds like a freaking major "detail" to me. And if the candidates, or the voters, produce low-information ballots (such as plurality style voting) then seems likely there will just not be enough information in there to allow any intelligent way of deciding how to assign them. > For each candidate C in S add up all of the ratings over all of the ballots in the pile, but not the ratings for candidates outside of S. Divide this number by the total possible, which in this case is two thousand times five or ten thousand. Actually, because of the nature of the favorability factor by which each ballot has already been scaled, no ballot can contribute more than one unit... --I am not seeing that. Favorability is in [0,1]. Ratings are nonnegative integers. The product can be any nonnegative rational, not upperbounded by 1. > We now have five quotients, one for each candidate. Multiply these five numbers together and take the fifth root. This geometric mean is the "goodness" score for the slate. Among the nominated slates, elect the "best" one, i.e. the one with the highest "goodness." --you could have omitted taking the 5th root, and just used the product as the goodness. And also, you could have avoided multiplying things, and instead summed the logs of things, and just used that sum-of-logs as the goodness. These are equivalent restatements. As I daresay you already knew. > I believe it takes into account "out-of pile" preferences as much as possible without destroying proportionality. --that sounds like it could be the beginning of some interesting train of thought. But I don't know what it is. > any questions about the method? --I have not figured out what it is. Pseudocode might help. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step) ---- Election-Methods mailing list - see https://electorama.com/em for list info |
On 4/11/19, Warren D Smith <[hidden email]> wrote:
> Hi Forest, this is in reaction to your new "card dealing" PR multiwinner > voting method. --another reaction: in the method you described, there are a lot of exact ties. You say stuff like "choose the next card to deal as the one most favorable to candidate 3" but there will be many tied-for-most. I presume you break all such ties randomly. In your example with 10000 voters & 5 winners, I think it would be common for there to be more than 1000 such randomness-assisted events. That is a lot of randomness. So the "quality function" would actually not be a function of the ballots and winner-set alone, it also would be a function of a truckload of random coin tosses. That is problematic. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step) ---- Election-Methods mailing list - see https://electorama.com/em for list info |
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