[EM] Consensus and PR methods

Jameson Quinn jameson.quinn at gmail.com
Thu Mar 8 08:46:07 PST 2018

Few, but more than 2. Harmful equilibria are stable with 2 parties; never
stable for long with 3 or more.

And those who don't like any of the viable options (whether there are 2 or
3 or 10) should be able to vote in a way that reflects that, without
necessarily having their votes ignored. That means any good voting method
should allow cross-party voting and/or transfers somehow.

2018-03-07 21:44 GMT-05:00 Jack Santucci <jms346 at georgetown.edu>:

> Political scientists like their parties to be few and disciplined. This is
> said to promote accountability.
> Sent from my iPhone
> On Mar 7, 2018, at 20:11, Richard Lung <voting at ukscientists.com> wrote:
> 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>
> 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
>> Democracy Science series 3 free e-books in pdf:
>> https://plus.google.com/106191200795605365085
>> E-books in epub format:
>> https://www.smashwords.com/profile/view/democracyscience
>> ---
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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:https://plus.google.com/106191200795605365085
> E-books in epub format:https://www.smashwords.com/profile/view/democracyscience
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