[EM] Consensus and PR methods
Richard Lung
voting at ukscientists.com
Wed Mar 7 17:11:39 PST 2018
If my old memory serves me tolerably well, isn't this paper something
like an article entitled The Best of Both Worlds, where the authors did
a survey of a tendency for European electoral systems, over the decades,
to have decreased their average magnitude. I forget the details, just
about everything actually. But it may have gone something like: the
constiuencies shrank and the thresholds got higher.
It was an informative statistical survey.
But I think it went awry on what academics are fond of calling
"normative" considerations. Or on the stricture of David Hume, that what
is, is not necessarily right.
I would have put to the authors, as a critic. That was this trend, they
so diligently exposed, but the moving to a "sweet spot" for political
incumbents, with precious little to do with democracy and effective
elections for the voters?
from
Richard Lung.
On 07/03/2018 19:19, Jack Santucci wrote:
> Consensus in academia? Maybe that cigarettes cause cancer. Maybe.
>
> I jest.
>
> This paper may be helpful:
> http://personal.lse.ac.uk/hix/Working_Papers/Carey-Hix-AJPS2011.pdf
>
> On Wed, Mar 7, 2018 at 2:14 PM, Richard Lung <voting at ukscientists.com
> <mailto:voting at ukscientists.com>> wrote:
>
> So, the academic world has no consensus or standard model of
> election method?
>
>
> On 03/03/2018 19:57, Kristofer Munsterhjelm wrote:
>
> Say we have a consensus method M that works by choosing the
> council C that minimizes the maximum penalty p(C, v) for the
> voter that maximizes this penalty. That is, the method finds C
> according to
>
> C = arg min max p(c, v)
> c v
>
> where ties are broken in a leximax fashion (i.e. considering
> next to max, then next to next to max and so on). Furthermore
> let the penalty "nonnegative" in the sense that any voter with
> a real preference has at least as great a penalty as a voter
> with no preference (the zero voter, as it were).
>
> Now let the modified consensus method M' be one that has the
> same optimization objective, but the method is permitted to
> remove a Droop quota of votes to help minimize the penalty.
>
> So M says "what council displeases the most displeased voter
> the least?", while M' says "what council displeases the most
> displeased voter the least, if we can discard a Droop quota of
> voters from consideration?"
>
> Then, are there any properties for p that makes M' satisfy
> Droop proportionality? Can we in general turn consensus
> methods of this form into PR methods by adding a "you can
> discard a Droop quota" rule?
>
> If we can, then we easily get a multiwinner version of
> Bucklin/MJ by doing the following:
>
> Let g(c, v) be the grade voter v gives to the least preferred
> candidate in c.
>
> Let the consensus method M be
>
> C = arg max min g(c, v)
> c v
>
> Let M' permit the method to remove a Droop quota, i.e. if |V|
> is the number of voters, and V is the set of voters itself:
>
> C' = arg max c:
> max x subset of V so that |x| = |V|/(seats+1):
> min v in V \ x:
> g(c, v)
>
> For a single-winner election, M' is (up to tiebreaker) just
> MJ, because for each potential winner c, the removal step will
> remove the voters who grade c the worst, and the Droop quota
> for single-winner is a majority. Then the voter grading the c
> the worst after half of the voters have been removed is just
> the median voter.
>
>
>
> Some thoughts about two-winner remove-voter minimax Approval
> follow. Reasoning about what voter removal actually does can
> get kinda hairy, thus I may very well be wrong:
>
> In minimax Approval, p(c, v) is the Hamming distance between c
> and voter v's ballot, i.e. the number of candidates in c but
> not approved by v plus the number of candidates approved by v
> not in c.
>
> Say we have an analogous Droop criterion for Approval: if more
> than k Droop quotas approve of a set of j candidates (and
> nobody else), then at least min(k, j) of these must be elected.
>
> For two winners, there are these possibilities:
> 1. no Droop constraints
> 2. k = 2, j >= 2
> 3. k = 2, j = 1
> 4. k = 1, j >= 1
> 5. k = 1, j = 1
>
> 1. is no problem, because we can elect anyone we want without
> running afoul of the Approval DPC.
>
> 2. Since there can only be three Droop quotas in total, when
> we're considering A = {C_1, C_2} with C_1 and C_2 in the set
> of j candidates (call it J), we can eliminate all but the
> J-voters and the maximum penalty is j-2.
> In contrast, for some B = {C_x, C_y} not a subset of j, the
> best it can do is eliminate a Droop quota of the J-voters. In
> the best case (for B), everybody but the J-voters approve of B
> alone. But there still remains a Droop quota (plus one voter)
> of the J-voters, and each of them gives penalty j. So A is
> preferred to B.
> If B = {C_1, C_x}, then even if everybody but the J-voters
> approve of B alone, the J-voters give penalty j-1. So A is
> still preferred to B.
>
> 3. Same as in 2, but let A = {C_1, C_x}, J = {C_1}. With A, we
> eliminate so that only the J-voters are left, and then max
> penalty is 1 (for C_x). Furthermore, every remaining voter
> gives penalty 1. Let B = {C_x, C_y}. In the best case for B, a
> Droop quota of J-voters are eliminated and we have a Droop
> quota plus one left. These all give penalty 2, which is worse
> than penalty 1. So A is preferred to B.
>
> 5. Let A = {C_1, C_x} and B = {C_x, C_y}. In the best case for
> B here, two Droop quotas minus a voter approve only of B, and
> the remaining Droop quota plus one voter approves of J =
> {C_1}. Eliminating all but one of the J-voters gives a max
> penalty of 3 from that one J-voter: one point for not having
> C_1, and two points for having C_x and C_y. A eliminates one
> of the two B-approving Droop quotas and gets a penalty of 1
> from every remaining voter, which is better.
> Note that I assume that C_x is approved by the B-voters. If
> that were not the case, then {C_x, C_y} would already be
> beaten by some {C_z, C_y} where C_z is. Note also that I don't
> consider the case where the B-voters also approve of a whole
> load of other candidates, with the idea of raising the penalty
> under A. The problem is that because only two candidates can
> be elected, this would also raise their penalty under B.
>
> 4. Let A = {C_1, C_x} and B = {C_x, C_y}. The best case for B
> has worst penalty j+2, since after a Droop quota of J-voters
> have been eliminated, there remains a single voter who only
> approves of J. After eliminating some of the B-voters, A gets
> penalty j from the J-voters (j-1 for the members of J not part
> of {C_1, C_x} and one more for C_x which is not approved by
> them), and one penalty point from the B-voters.
> Here it'd seem that adding loads of candidates to the B-voters
> would make things hard. Can it be salvaged?
>
> Suppose there are J-voters and C-voters. B is a subset of C.
> When considering outcome B, before excluding a Droop quota,
> every J-voter gives a penalty of j+2 and every C-voter gives a
> penalty of c-2 where c=|C|.
> Under outcome A, before excluding, every J-voter gives j, and
> every B-voter gives c (-1 for having C_x, +1 for having C_1).
> If j+2 > c, then we're in the domain above, and no problem.
> If c > j+2, then the excluded candidates under both A and B
> are C-voters.
> So under B we have a Droop quota of C-voters with penalty c-2,
> and a Droop quota plus one of J-voters at j+2.
> Under A we have a Droop quota of C-voters with penalty c, and
> a Droop quota plus one of J-voters at j.
>
> So unless I made a mistake, Hamming distance is not good
> enough. But I might have made a mistake, because it seems
> strange that even in ordinary minimax Approval, a coalition
> can increase its power by approving a lot of clones. E.g.
> suppose in ordinary minimax Approval that there are two
> coalitions of almost equal size:
>
> n+1: A B
> n: C1 C2 C3 ... Cq
>
> {A, B} gets worst penalty q+2 (there are n of these and n+1
> zeroes)
> {A, C1} gets worst penalty q (n voters like C1 but not A)
> {C1, C2} gets worst penalty q-2 (n voters give this penalty,
> and then n+1 give penalty 4).
>
> ... does that mean an arbitrarily small minority can force its
> own council to win by just approving enough clones that they
> set the worst penalty in every outcome? That feels rather wrong.
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>
> --
> Richard Lung.
> http://www.voting.ukscientists.com
> <http://www.voting.ukscientists.com>
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>
>
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>
>
> --
> Jack Santucci, Ph.D.
> Independent scholar
> http://www.jacksantucci.com
--
Richard Lung.
http://www.voting.ukscientists.com
Democracy Science series 3 free e-books in pdf:
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