I think this whole "data compression" idea by Forest Simmons for multiwinner elections is brilliantly simple. The obvious first worry about Forest Simmons' idea is that averaging too-large sets of ballots as step 1, destroys all hope for proportional representation voting. E.g. imagine we averaged ALL the ballots as step 1, then we'd only have one amalgamated ballot, and from it we clearly would be unable to elect a PR parliament. But Simmons is being smarter about it, he is reducing everything down to C amalgamated ballots, if C is the number of candidates -- not one. I.e. he amalgamates all the ballots that score X highest, for each candidate X. And that actually seems acceptable, in the sense that with 100% "racist" voters in a situation with "colored" voters and candidates, you would not lose any information in this way! Thus, you could still do proportional representation in "racist" situations, which by some reckonings is all you need -- i.e. if the definition of PR is "yields color-proportionality in racist situations." However, there would be a problem if we sort of had "2-level racism." E.g. suppose every voter had both a "color" and a "secondary color." Ditto for candidates. Voters give candidates score 7 if agree on color, and bonus score 3 if agree on secondary color. In that kind of situation, Simmons' "data compression" method would lose information and presumably lose the ability to deliver (the more clever sort of) PR that it ought to. Also, Toby Pereira and (at his urging) me too, like to think about elections in which there are 2 kinds of candidates -- colored and uncolored -- and voters give same-color candidates score 10, other-color candidates score 0, and uncolored candidates get a score that depends only on the candidate not on the voter. But it seems to me Simmons' data compression technique already is lossless for these elections. And it seems to me we can devise other "data compression methods" which do not lose information in the Color+SecondaryColor scenarios. For example we could amalgamate all ballots which both score X highest, and score Y highest among candidates getting a lower score than X; put all those ballots into bin(X,Y), then amalgamate all ballots within any one bin. This results in at most (C-1)*C ballots after compression -- and at most (C-1)*C^2 approval-style ballots after both the compression and a Pereira transform -- in a C-candidate election. The other brilliant thing about Simmonsesque data compression is, this permits multiwinner PR elections to be "counted in precincts." -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

I like your idea about C*(C-1) factions determined by 1st and 2nd choices. I'm sure you've already thought of the obvious way of dealing with top equal and second equal ratiings, but for the record, here it is: Suppose the respective scores on a certain ballot B for the respective candidates are 100%, 100%, 80%, 80%, 80%, 70%, 60%, ... Each of the six bins in the set {bin(X, Y)| {X,Y} in the Cartesian product of {c1, c2} and {c3, c4, c5}} should get a copy of B with weight 1/6 . This weight will be used in the respective weighted averages of the ballots in the respective bins. Also for the record, here is a suggestion for how to turn ordinal (i.e.ranked preference style) ballots into cardinal (i.e. range style) ballots, so that Pereira's transform, and so forth,can be used iin that context: (1) Find the respective random ballot probabilities p1, p2, ... for the respective candidates c1, c2, ... (2) For each candidate c_i ranked above bottom on ballot B, assign the score r_i = one minus the sum of the p_k such that c_k is ranked strictly above c_i on ballot B. Remarks: Since there is no candidate ranked above a top ranked candidate, rule (1) gives every top ranked candidate a score of one. Unfortunately step (1) cannot be done at the precinct level, unless we are willing to make do with the precinct estimates of the random ballot probabilities. "Random ballot probabilities" can be interpreted in various ways. For example we could use the "implicit approval" random ballot model for the respective probabilities. I like this version better because it would give candidate C a higher rating in the following scenario: 45 A1=A2=A3>C 55 B1=B2=B3>C The respective standard random ballot probabilities (.15, .15, .15, .55/3, .55/3, .55/3, 0) would yield score vectors ... 45 (1, 1, 1, 0, 0, 0, .55) 55 (0, 0, 0, 1, 1, 1, .45) Random implicit approval ballot would give C a probability of 1/4 if ties were to be resolved by coin tosses, or p_C=100% if ties were to be resolved by further ballot draws. In the latter case (which I prefer) the probability vector would be (0, 0, 0, 0, 0, 0, 1), and the resulting score vectors would be 45 (1, 1, 1, 0, 0, 0, 1) 55 (0, 0, 0, 1, 1, 1, 1) In the former case (of coin toss tie breaking) the probability vector would be (.11, .11, .11, .14, .14, .14, .25), and the score vectors would be 45 (1, 1, 1, 0, 0, 0, .67) 55 (0, 0, 0, 1, 1, 1, .58) In either case, C is the range winner. On Tue, Dec 15, 2015 at 9:50 AM, Warren D Smith <warren.wds@gmail.com> wrote: > I think this whole "data compression" idea by Forest Simmons for > multiwinner elections is brilliantly simple. > > The obvious first worry about Forest Simmons' idea is that averaging > too-large sets of ballots > as step 1, destroys all hope for proportional representation voting. > > E.g. imagine we averaged ALL the ballots as step 1, then we'd only > have one amalgamated ballot, and from it we clearly would be unable to > elect a PR parliament. > > But Simmons is being smarter about it, he is reducing everything down > to C amalgamated ballots, if C is the number of candidates -- not one. > I.e. he amalgamates all > the ballots that score X highest, for each candidate X. > > And that actually seems acceptable, in the sense that with 100% > "racist" voters in > a situation with "colored" voters and candidates, > you would not lose any information in this way! > Thus, you could still do proportional representation in "racist" > situations, which > by some reckonings is all you need -- i.e. if the definition of PR is > "yields > color-proportionality in racist situations." > > However, there would be a problem if we sort of had "2-level racism." > E.g. suppose every voter had both a "color" and a "secondary color." > Ditto for candidates. Voters give candidates score 7 if agree on > color, and bonus score > 3 if agree on secondary color. > > In that kind of situation, Simmons' "data compression" method would lose > information and presumably lose the ability to deliver (the more > clever sort of) PR > that it ought to. > > Also, Toby Pereira and (at his urging) me too, like to think about > elections in which there are 2 kinds of candidates -- colored and > uncolored -- and voters > give same-color candidates score 10, other-color candidates score 0, > and uncolored candidates get a score that depends only on the candidate not > on the voter. But it seems to me Simmons' data compression technique > already is lossless for these elections. > > And it seems to me we can devise other "data compression methods" > which do not lose information in the Color+SecondaryColor scenarios. > For example we could amalgamate all ballots which both score X highest, > and score Y highest among candidates getting a lower score than X; > put all those ballots into bin(X,Y), then amalgamate all ballots > within any one bin. > This results in at most (C-1)*C ballots after compression -- and at most > (C-1)*C^2 approval-style ballots after both the compression > and a Pereira transform -- in a C-candidate election. > > The other brilliant thing about Simmonsesque data compression is, > this permits multiwinner PR elections to be "counted in precincts." > > -- > Warren D. Smith > http://RangeVoting.org <-- add your endorsement (by clicking > "endorse" as 1st step) >

There is a flaw (perhaps merely aesthetic?) in the Simmons "data compression" idea: discontinuity. That is, suppose I am a voter. I rate candidates X and Y high. I gradually lower my X-rating while gradually raising my Y-rating. Kaboom! -- Suddenly, when they cross, my ballot flies into the Y-bin instead of the X-bin. This yields a discontinuous change in the "quality of the winner set" function (if our voting method were based on such a function). However, if this is considered a flaw, it is one that is easily repaired. Split each ballot into C weighted copies in a C-candidate election (instead of the simpler Simmons course of usually making all weights 0 except for one that is 1). The weight for copy J is K / (E + (maxscore - score[J])^2) where the constant K>0 is chosen to make all the C different weights sum to 1, and the constant E>0 is arbitrary and Simmons' original approach arises in the limit E --> 0+. Or other formulas also may be used, of course. You may prefer one which becomes 0 when score[J]=0. Anyhow, when anything of this ilk is done the quality function becomes smooth and continuous everywhere, no discontinuities. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)