I have tweaked things slightly in order to make things slightly simpler and to make sure that when Range style ballots are used the method reduces to standard Range in the one winner case. The general method takes any proportional lottery based on score/range (including approval) style ballots and converts it into a PR election method. Recall that a lottery method is a system of assigning probabilities to candidates. A lottery method is "proportional" if it assigns probabilities in proportion to the respective faction sizes when all faction members vote with single-minded and exclusive loyalty in favor of their favorite. Let's suppose that we are in the context of an election where w>0 candidates are to be elected, and that there is at least one subset of candidates W of size w, such that when our lottery method is applied to W, all members of W are assigned positive probabilities. If there is only one such subset, then that is the subset selected by our method. Otherwise, for each subset W of candidates of the requisite size w, we do the following steps: (1) Partition the ballots into sets S and S' that do and don't, respectively, give at least one candidate of W a positive rating. (2) Let p be the probability that a randomly drawn ballot would be a member of the set S. Let q = 1 - p. [Note this is a change; the definitions of p and q have been switched, for esthetic reasons.] (3) Let p1, p2, ... be the respective probabilities assigned to the members of W by our lottery method, when that method is restricted to the ballot set S. (4) Let a1, a2, ... be the respective averages of the candidate ballot scores over the respective ballots that rate the respective candidates positively. [In case of approval, all of these averages will be ones] Having completed these four steps for each candidate set W of size w, elect the set W that maximizes the value of the expression min(a1*p1, a2*p2, ...)*(p), Now you I will show you the reason for the tweaks: suppose that w=1. Then the value of p1 is 1, and the value of a1 is the average rating of W's only candidate over the ballots in the set S, The average rating of W's member over S' is zero, so the average score over the union of S and S' is the weighted average a1*p1*p + 0*q, which simplifies to a1*p1*p . Therefore in the case w=1, the standard range winner wins! We have mentioned several of the various random ballot lotteries. More variations are possible. Ordinal ballots can be used via the "Implicit approval cutoff," or with the help of an explicit one. If the lottery is random nallot the only step needing an approval measure is step (4) above. There are other possible ways to adapt to ordinal ballots (ranked preference ballots). There are other proportional lotteries besides the random ballot ones. One is the so called Ultimate Lottery, a restricted version of which is called the Nash Lottery. The Nash Lottery Method picks the lottery that maximizes the product of the ballot expectations based on the lottery. It turns out that the crucial factor that makes this lottery proportional is the "homogeneity" of the ballot expectations in the probabilities. So if we widen the admissible ballots to include any homogeneous functions of the probabilities (along with the natural requirement that such functions not be decreasing in any of their arguments), the we get the Ultimate Lottery Method. Jobst Heitzig has come up with many lotteries with special attention to those with low entropy, which makes those lotteries useful in single winner elections. Why would we want to use a lottery in single winner elections? It turns out that the element of chance, when skillfully incorporated, can (more or less) remove incentives for insincere voting. The best known example of this is the "benchmark" standard random ballot lottery. In that method there is no incentive for insincere voting, but the resulting lotteries tend to be high entropy lotteries, hence not good for single winner elections. One of Jobst's best methods has two stages. The first stage generates a set of approval ballots from "thresh-hold" information supplied by the voters on their ballots. I won't go into the details of this stage, but the voters give tentative approvals that allow the method to automatically build approval ballots that would back-fire on defectors. The second stage calculates the total approvals for the respective candidates and assigns each ballot B to the candidate with the greatest total approval of any candidate approved on ballot B. The lottery probabilities are proportional to the number of ballots assigned to each candidate. For our multi-winner purposes we can replace the first stage with direct use of approval ballots or by conversion of range ballots into approval ballots via Toby's idea or by other ideas, including the "sincere approval strategy" technique, for example. Where else can we find suitable proportional lotteries? Andy Jennings found that he could take any sequential PR method and convert it into a proportional lottery by cloning all of the candidates as many times as needed, and allowing a huge number of winners. Then the candidate lottery probabilities are proportional to the number of clones they have in the winning circle. In fact it turned out that Andy;s first application of that technique generated the Nash Lottery by use of sequential PAV adapted to range ballots. More recently, Andy and I have seen that there are many ways to convert practlcally any single winner method into a sequential PR method. Putting all of this together we have the following diagram: single winner -> sequential PR -> Lottery -> multi-winner PR So in a standard way we can convert any single winner method into a multi-winner PR method that retains some of the flavor of the single winner method. Why not just stop at the sequential PR stage? That's always a possibility, but sequential methods tend to generate an hierarchy among the winners in order of election. It is sometimes better philosophically to have a method that compares (many if not all) possible winning sets (including the winning sets generated by various sequential PR methods) against each other. In this regard, Jameson has just invented (and posted the EM list) a method that compares methods on the basis of sum of squared envy. That's it for now ... On Thu, Dec 3, 2015 at 2:55 PM, Forest Simmons <fsimmons@pcc.edu> wrote: > Now I just have time to point out that in place of the random ballot > lottery we could use any proportional lottery, such as PAV or The Ultimate > Lottery. > > More on next time ... > > On Thu, Dec 3, 2015 at 2:42 PM, Forest Simmons <fsimmons@pcc.edu> wrote: > >> Continuing as promised... >> >> Now let's allow range ballots with greater resolution than mere approval >> ballots. >> >> For each w-tuple of candidates we ... >> >> (1) first divide the ballots into two sets, the set S of ballots that >> have positive support for at least one candidate of our w-tuple, and the >> complementary set S' of ballots that rate every member of our w-tuple at >> zero. >> >> (2) Then we imagine an experiment of drawing a ballot B at random from >> the entire set of ballots. Let p0 be the probability that B is a member of >> S', and let p1, p2, ... pw be the respective probabilities for the choices >> of the respective members of our w_tuple given that B is a member of S. >> [Multiplication by (1-p0) would give the respective unconditioned >> probabilities.] >> >> (3) Let a1, a2, ... be the respective candidate score averages for >> ballots in S. >> >> (4) elect the w-tuple with the greatest value of the expression >> >> min(a1*p1, a2*p2, ...) - p0. >> >> There are at least three good ways of defining the details of the random >> ballot experiment. >> >> (1) The value p3, say, is the probability that candidate 3 would be the >> highest rated candidate (from our w-tuple) on a randomly drawn ballot from >> the set S. If several candidates are tied in this respect, divide up the >> probability (from Ballot B) equally among them.. >> >> (2) The positive scores on every ballot B are normalized with respect to >> the members of our w-tuple. These normalized scores are averaged over S to >> get the respective "random ballot" probabilities. >> >> (3) Use a Toby Pereira transformation to convert each range style ballot >> into one hundred approval ballots. Then use a random approval ballot >> lottery on that set. I'll explain this in my next post ... >> >> On Thu, Dec 3, 2015 at 1:44 PM, Forest Simmons <fsimmons@pcc.edu> wrote: >> >>> Continuing as promised .. >>> >>> Suppose that there are to be w winners, and the ballots are still >>> approval style: >>> >>> For each w_tuple of candidates, let p1, p2, ... be the respective >>> probabilities of selection of the respective candidate by random ballot >>> (restricted to the w-tuple), and let p0 be the probability that a random >>> ballot would not approve any of the candidates in the w-tuple. >>> >>> Elect the w-tuple with the largest value of min(p1, p2, ...) - p0. >>> >>> To Be Continued ... >>> >>> >>> On Thu, Dec 3, 2015 at 1:30 PM, Forest Simmons <fsimmons@pcc.edu> wrote: >>> >>>> >>>> >>>> This query has lead me to some interesting ideas: >>>> >>>> Approval Ballots. Two to elect: >>>> >>>> For each pair of candidates {X1, X2}, let p1 and p2 be the respective >>>> probabilities that X1 or X2 would be selected by a random approval ballot >>>> drawing (restricted to our pair of candidates), and let p0 be the >>>> probability that a random ballot would approve neither X1 nor X2. Elect >>>> the pair with the greatest value of min(p1, p2) - p0. >>>> >>>> This actually gives two methods, since there are two natural ways of >>>> selecting a candidate by random approval ballots. >>>> >>>> The first way is to select ballots at random until the approval set for >>>> one of them has non-empty intersection with the set from which we are to >>>> select a winner. The names of the candidates are drawn randomly from a >>>> hat. The first name drawn of a candidate in the intersection set is the >>>> name of the winner. >>>> >>>> The second way starts out as above, but once the first non-empty >>>> intersection set is determined, additional ballots are drawn as needed to >>>> narrow down the intersection to one candidate, the winner. >>>> >>>> More later ... >>>> >>>> From: Forest Simmons <fsimmons@pcc.edu> >>>>> To: EM <election-methods@lists.electorama.com>, Andy Jennings >>>>> <elections@jenningsstory.com> >>>>> >>>>> >>>>> How about thie following ideas? >>>>> >>>>> Elect the pair that covers the most voters (i.e. that leaves the fewest >>>>> voters with nobody that they approved elected). In case of ties, among >>>>> tied pairs elect the one whose weaker member has the most approval. >>>>> >>>>> Or this variant: If no pair covers more than 70 percent of the voters, >>>>> elect the pair that covers the greatest number of voters. Otherwise >>>>> consider all pairs that cover at least 70 percent of the voters to be >>>>> tied. Then among tied pairs, elect the one whose weaker member has the >>>>> greatest approval. >>>>> >>>>> >>>>> From: Andy Jennings <elections@jenningsstory.com> >>>>> > To: Election Methods <election-methods@electorama.com> >>>>> > Subject: [EM] Approval ballots. Two to elect. Best method? >>>>> > >>>>> > SPAV? >>>>> > 1. Candidate with most approvals wins. >>>>> > 2. That candidate's voters have their voting weight halved (or >>>>> multiplied >>>>> > by 1/3). >>>>> > 3. Remaining candidate with most points wins. >>>>> > >>>>> > STV-like? >>>>> > 1. Choose quota Q = one-third (or one half) of voters. >>>>> > 2. Candidate with most approvals wins. (T = # of approvals) >>>>> > 3. That candidate's voters have their voting weight multiplied by >>>>> > max(1-(Q/T), 0) >>>>> > 4. Remaining candidate with most points wins. >>>>> > >>>>> > Monroe-like? >>>>> > 1. For each pair of candidates, find the voter-assignment which >>>>> maximizes >>>>> > the number of voters assigned to a candidate they approved, such >>>>> that no >>>>> > more than half the voters are assigned to one candidate. >>>>> > 2. Elect the pair which satisfies the most voters. >>>>> > >>>>> > Others? Toby, what are your favorite PR methods at the moment? Can >>>>> you >>>>> > give a short explanation of how Phragmen/Ebert would work with only >>>>> two to >>>>> > elect? >>>>> > >>>>> > >>>>> > >>>>> > Specifically, I'm worried that in practically every approval-ballot >>>>> PR >>>>> > method, if there is a candidate you really like, but are sure that >>>>> she can >>>>> > get elected without your vote, you gain an advantage by not approving >>>>> > them. Is there any method that minimizes that incentive? >>>>> > >>>>> > ~ Andy >>>>> > >>>>> >>>> >>> >> >

On 12/05/2015 01:09 AM, Forest Simmons wrote: > I have tweaked things slightly in order to make things slightly simpler > and to make sure that when Range style ballots are used the method > reduces to standard Range in the one winner case. > > The general method takes any proportional lottery based on score/range > (including approval) style ballots and converts it into a PR election > method. > > Recall that a lottery method is a system of assigning probabilities to > candidates. A lottery method is "proportional" if it assigns > probabilities in proportion to the respective faction sizes when all > faction members vote with single-minded and exclusive loyalty in favor > of their favorite. Proportionality is a good start, but I'd think one would also need proportionality among sets so that the voters aligned with multiple candidates don't have to coordinate which candidate to vote on. One possible DSV way of defining this would be: consider an approval election where each voter shows exclusive loyalty to some subset of the candidates. Then select the candidates so that the chance of the voter getting someone from his subset is maximized, where every voter is considered of equal importance. But I imagine this could be difficult for lottery methods since there are many different assignments that meet the property above. For instance: 10: A B (faction A) 10: C B (faction C) where both the lottery {0% A, 100% B, 0%C} and the lottery {50% A, 0% B, 50% C} gives each faction the same chance of getting one of the candidates of the favored subset. I guess that's part of why say, the Droop proportionality criterion, also involves the number of seats, not just the number of candidates. Does your method meet some form of set proportionality? If so, which? > Where else can we find suitable proportional lotteries? > > Andy Jennings found that he could take any sequential PR method and > convert it into a proportional lottery by cloning all of the candidates > as many times as needed, and allowing a huge number of winners. Then > the candidate lottery probabilities are proportional to the number of > clones they have in the winning circle. > > In fact it turned out that Andy;s first application of that technique > generated the Nash Lottery by use of sequential PAV adapted to range > ballots. > > More recently, Andy and I have seen that there are many ways to convert > practlcally any single winner method into a sequential PR method. > > Putting all of this together we have the following diagram: > > single winner -> sequential PR -> Lottery -> multi-winner PR Andy's way of turning any sequential PR method into a proportional lottery sounds quite a bit like how you could turn multiwinner PR into a party list method. Say you have a multiwinner method X and want to turn it into a party list method. Let the voters rate or rank each party as if they were candidates; then, if the number of seats is s, clone each candidate (representing a party) s times. Finally, run an s-seat election using the multiwinner method in question. Andy's method seems to be what you'd get if you let s go to infinity and let the method be sequential (I'm guessing to ensure convergence). In turn, that suggests that the diagram above could be expanded into: single winner -> sequential PR -> lottery -> party list PR, multi-winner PR To go from a lottery to party list PR, let each party's support be its probability of being drawn. Then use Webster (or some other party list method of choice) to quantize the probabilities into seat numbers. This won't be a perfect party list PR method because some parties could end up having no seats yet, by having nonzero probability, exclude other parties (e.g. LCR in a one-seat situation). Another option is to just repeatedly draw from the lottery (with replacement) until all s seats are filled, but that has greater variance than using a party list method for the quantization. In a similar way, you could also go from lottery to a weighted multiwinner method. Select the s candidates with greatest probability, elect them, and let their weights in the assembly be proportional to their probabilities. But I suspect (as I've been trying to get to in my clone posts) that there's an inherent cloning problem to this kind of system, and the strategic equilibrium becomes Droop quota PR. That is, unless there's some way of preventing a large faction from masquerading as multiple smaller factions and filling up more of the s slots. It might be better to hold the sum of weights constant and let s vary than holding s constant and letting the sum of weights vary.

The idea of convertin lottery methods into PR methods has been around for a long time. The obvious idea that has been tried over and over in some form or another is to run the lottery on the entire set of ballots with the entire set of candidates, and then elect the set of w candidates with the greatest winning probabilities. Since that doesn't work very well, either we have resorted to allowing the winners to carry weights with them into the assembly, or we have gone back to sequential methods with droop quotas, etc. I think a much better idea (that seems to have been entirely over-looked) is to not run the lottery on the entire set of candidates, but to run it on all of the subsets of size w,, and choose the most satisfactory of these subsets. There are three things that wouold make a subset satisfactory: (1) The candidates in the set should not have too much difference in the probabilities assigned by the lottery to their subset. In other words the least probability should be as large as possible. In other words, the entropy of the probability distribution should be as large as possible. (2) The candidates should have as high average ratings among their supporters as possible. (3) There should be few if any ballots in the set that truncate the entire subset. My suggestions are an attempt to incorporate all of these ideals into a way of comparing subsets of the appropriate size. Here's a simple example based only on ranked preference ballots that illustrates the main idea: Suppose that we want to compare two subsets W and W' of the requisite size w. Let b_i be the number of ballots that rank candidate c_i of the set W above every other candidate of W. These numbers are the sizes of the Voronoi (or Dirichlet) regions for the respective candidates of W Similarly, let b'_i be the number of ballots that rank candidate c'_i of the set W' above every other candidate of W'. Let m = min(b_i) and m'= min (b'_i). If m > m', then W is better than W' in the sense of (1) above. Since top ratings from the ballots are used preferentially, (2) is given some support. And the bigger the smallest Voronoi Region, the more likely the Voronoi Regions will come close to covering the voters, fulfilling condition (3). If you can picture the Voronoi regions in your mind, then you can understand everything I do about this and similar methods based on random ballot lotteries. Obviously the method is computationally hard (because of the number of subsets of size w). But given a few million such subsets found by other means, it can quickly compute the m's and put them in order. On Sat, Dec 5, 2015 at 2:31 AM, Kristofer Munsterhjelm <km_elmet@t-online.de > wrote: > On 12/05/2015 01:09 AM, Forest Simmons wrote: > > I have tweaked things slightly in order to make things slightly simpler > > and to make sure that when Range style ballots are used the method > > reduces to standard Range in the one winner case. > > > > The general method takes any proportional lottery based on score/range > > (including approval) style ballots and converts it into a PR election > > method. > > > > Recall that a lottery method is a system of assigning probabilities to > > candidates. A lottery method is "proportional" if it assigns > > probabilities in proportion to the respective faction sizes when all > > faction members vote with single-minded and exclusive loyalty in favor > > of their favorite. > > Proportionality is a good start, but I'd think one would also need > proportionality among sets so that the voters aligned with multiple > candidates don't have to coordinate which candidate to vote on. > > One possible DSV way of defining this would be: consider an approval > election where each voter shows exclusive loyalty to some subset of the > candidates. Then select the candidates so that the chance of the voter > getting someone from his subset is maximized, where every voter is > considered of equal importance. > > But I imagine this could be difficult for lottery methods since there > are many different assignments that meet the property above. For instance: > > 10: A B (faction A) > 10: C B (faction C) > > where both the lottery {0% A, 100% B, 0%C} and the lottery {50% A, 0% B, > 50% C} gives each faction the same chance of getting one of the > candidates of the favored subset. > > I guess that's part of why say, the Droop proportionality criterion, > also involves the number of seats, not just the number of candidates. > > Does your method meet some form of set proportionality? If so, which? > > > Where else can we find suitable proportional lotteries? > > > > Andy Jennings found that he could take any sequential PR method and > > convert it into a proportional lottery by cloning all of the candidates > > as many times as needed, and allowing a huge number of winners. Then > > the candidate lottery probabilities are proportional to the number of > > clones they have in the winning circle. > > > > In fact it turned out that Andy;s first application of that technique > > generated the Nash Lottery by use of sequential PAV adapted to range > > ballots. > > > > More recently, Andy and I have seen that there are many ways to convert > > practlcally any single winner method into a sequential PR method. > > > > Putting all of this together we have the following diagram: > > > > single winner -> sequential PR -> Lottery -> multi-winner PR > > Andy's way of turning any sequential PR method into a proportional > lottery sounds quite a bit like how you could turn multiwinner PR into a > party list method. > > Say you have a multiwinner method X and want to turn it into a party > list method. Let the voters rate or rank each party as if they were > candidates; then, if the number of seats is s, clone each candidate > (representing a party) s times. Finally, run an s-seat election using > the multiwinner method in question. > > Andy's method seems to be what you'd get if you let s go to infinity and > let the method be sequential (I'm guessing to ensure convergence). In > turn, that suggests that the diagram above could be expanded into: > > single winner -> sequential PR -> lottery -> party list PR, multi-winner PR > > To go from a lottery to party list PR, let each party's support be its > probability of being drawn. Then use Webster (or some other party list > method of choice) to quantize the probabilities into seat numbers. This > won't be a perfect party list PR method because some parties could end > up having no seats yet, by having nonzero probability, exclude other > parties (e.g. LCR in a one-seat situation). > > Another option is to just repeatedly draw from the lottery (with > replacement) until all s seats are filled, but that has greater variance > than using a party list method for the quantization. > > In a similar way, you could also go from lottery to a weighted > multiwinner method. Select the s candidates with greatest probability, > elect them, and let their weights in the assembly be proportional to > their probabilities. But I suspect (as I've been trying to get to in my > clone posts) that there's an inherent cloning problem to this kind of > system, and the strategic equilibrium becomes Droop quota PR. That is, > unless there's some way of preventing a large faction from masquerading > as multiple smaller factions and filling up more of the s slots. It > might be better to hold the sum of weights constant and let s vary than > holding s constant and letting the sum of weights vary. >