Kristofer Munsterhjelm | 19 Oct 14:08 2014

Party lists and candidate multiwinner elections

Say we have two settings: one is an ordinary multiwinner election with, 
say, 10 seats. The other is a party list PR election to a very large 
assembly (say 500 seats), but where the number of distinct parties has 
been limited to 10. That is, no more than 10 parties may be represented 
in that large council.

Furthermore, assume that the voters' ballots are completely identical in 
the two settings. So if a voter in setting two ranks party A > party B > 
party C, then in setting one he would rank candidate A > candidate B > 
candidate C.

Now, my question is: is there any situation where we would expect the 
candidates elected in setting one to differ from the parties that get at 
least one seat in setting two?

I can't think of any off-hand, but if there are none, what implications 
do the fact that different parties have very different numbers of seats 
in party list PR have for ordinary multiwinner elections? Is it simply 
an artifact of parties being a lot harder to start than to run as an 
independent candidate in ordinary multiwinner PR? Or is it a consequence 
of party list methods being based on Plurality?

Or are there cases where the party composition would differ from the 
candidates elected in an analogous multiwinner election? Perhaps party 
composition should be more like the results of Minmax Approval (i.e. 
should obey Warren's "representativeness" criterion) and one should use 
relative seat counts to even out the power imbalance? What do you think?
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Toby Pereira | 8 Oct 20:50 2014

Re: General PR question (from Andy Jennings in 2011)

From: Kathy Dopp <kathy.dopp <at>>
>> My
>> system, for example, uses squared rather than absolute deviation (and uses a
>> different measure of deviation anyway) and it gives the results that I
>> wanted it to when I tested it, including stable results for the largest two
>> factions when the size of the third tiny faction changes, and the three-way
>> tie from the other example. It doesn't work by ignoring or eliminating
>> smaller factions;

>Neither does mine (in case you are implying such)  Some party list
>systems do work that way however.

I wasn't implying that but you suggested in one of your posts that it might be desirable. I was just saying that my system deals with it "naturally" - i.e. without manually taking out factions.

>> And I'm still unsure how to translate your method into approval voting with
>> overlapping factions.

>It works exactly the same way with overlapping candidate support in
>different factions. (i.e. v_i and s_i have exactly the same meanings,
>the number of voters in the group and the number of winning candidates
>each group contributes to electing.

But what I mean is that if a large faction (with say 50% of all voters) is divided into two (say 25% each) because of a single controversial candidate who appears on half of that faction's ballots but not the other half, then if that faction receives half the candidates (and the one controversial candidate is not elected), then it will be measured as unproportional because each faction will have each contributed to 50% of the candidates but will only be 25% of the electorate each.

>What is the logic of using squared rather than absolute deviation? and
>are you also selecting the slate of candidates minimizing your formula?

Squared deviation gave better and more consistent results when I tried it. I always come armed with election scenarios where I have an intended result, and I see if the method being tested gives the intended result. My method with squared deviation gave every result I wanted it to. Absolute deviation didn't.

And yes, in my method the winning set would be the one with the lowest sum of the squared deviations. Well, not necessarily, because if candidates could be elected sequentially, which could give a different result.


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Toby Pereira | 7 Oct 23:23 2014

Re: General PR question (from Andy Jennings in 2011)

From: Kathy Dopp <kathy.dopp <at>>

>So, perhaps there is an improved alternative over a truly
>proportionate allocation of seats to voters. Perhaps the voting groups
>who cannot win seats should be taken out of the equation as, I
>believe, some party list systems do when calculating winners.

>Kathy Dopp

Perhaps there's a more proportional method than a "truly proportionate allocation of seats to voters"! But what I would say is that there are several methods that people might deem to be proportional, and so to say that yours is *the* method would probably cause some disagreement.
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Richard Fobes | 7 Oct 22:42 2014

Re: How choice of voting systems depend on amount of participants

On 10/6/2014 11:47 PM, Kristofer Munsterhjelm wrote:
> ...
> I seem to recall someone mentioning a US region that used majority
> voting: there were rounds of voting until someone got an outright
> majority, and the rounds kept on for as long as needed. I don't recall
> the details, though.

This is how the "electoral college" works when voting for U.S. 
President, or at least the way it was set up to work.  Under current 
conditions, with each state giving all its electoral votes to either the 
Republican candidate or the Democratic candidate (and never any 
electoral votes to a third-party candidate), in recent decades there has 
always a winner on the first round of voting.  If there were not a 
majority winner, then the contest would be (and has been, three or four 
times) transferred to the U.S. House of Representatives, with each state 
getting one vote (but with no indication as to how that one vote (per 
state) is assigned).

Richard Fobes

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Toby Pereira | 28 Sep 01:28 2014

Re: General PR question (from Andy Jennings in 2011)

I was thinking recently again about Andy Jennings's PR question (below) and available here, which is about the trade of between proportionality and having candidates with strong support. Warren Smith ( gave the extreme example of a 500-member parliament where two candidates each get 50% approval, and the others each get 0.2% approval. Perfect proportionality could be achieved by electing 500 candidates with 0.2% approval, but in many ways this would seem a perverse result.

But the more I think about it, the more I think there isn't a non-arbitrary solution to the problem. What's the exchange rate between proportionality and support? There isn't an obvious answer.

I proposed my own proportional approval and score system a few months ago (, and it purely bases result on proportionality, so would elect CDE in Andy's example but would also elect 500 candidates with 0.2% support in Warren's example. However, this also assumes that every possible winning set of candidates would be looked at and the most proportional one found. In practice, the system might be used sequentially. This would force through the most popular candidate, and then each subsequent candidate would be elected to balance it proportionally. This would elect the two most popular candidates in Warren's example, but would fail to elect CDE in Andy's example. But given that there may be no non-arbitrary solution, electing sequentially may be the simplest and least arbitrary way around the problems we have. It is also a solution that would likely be forced upon us due to limits on computing power when it comes to comparing all possible sets of candidates. Necessity may force the pragmatic solution upon us.


>Forest and I were discussing PR last week and the following  situation came
>up.  Suppose there are five candidates, A, B, C, D, E.  A and B evenly
>divide the electorate and, in a completely orthogonal way, C, D, and E
>evenly divide the electorate.  That is:

>One-sixth of the electorate approves A and C.
>One-sixth of the electorate approves A and D.
>One-sixth of the electorate approves A and E.
>One-sixth of the electorate approves B and C.
>One-sixth of the electorate approves B and D.
>One-sixth of the electorate approves B and E.

>It is obvious that the best two-winner representative body is A and B.  What
>is the best three-winner representative body?

>CDE seems to be the fairest.  As Forest said, it is "envy-free".

>Some methods would choose ABC, ABD, or ABE, which seem to give more total

>Is one unequivocally better than the other?

>I tend to feel that each representative should represent one-third of the
>voters, so CDE is a much better outcome.  Certain methods, like STV, Monroe,
>and AT-TV (I think) can even output a list of which voters are represented
>by each candidate, which I really like.

>I also note that if there was another candidate, F, approved by everybody,
>it is probably true that ABF would be an even better committee than CDE.  Is
>there a method that can choose CDE in the first case and ABF in the second


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dikov dikov | 25 Sep 01:59 2014

How choice of voting systems depend on amount of participants


One of my friends wants to implement contests to his website and learning on the topic brought me to the voting systems. Since amount of voters (participants+followers) may reach even thouthands I suspect that different voting mechanisms would be appropriate, depending on amount of votes.
Is that right or he can stick to some systems that would be efficient on large and small scales?

Thank you in advance,


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Dick Burkhart | 16 Sep 05:40 2014

A "top 3" to replace the "top 2"

Oregon is considering a "top 2" primary, a very regressive move in my opinion.

So I thought what would be simplest "top 3" method that would give voters more choices, especially in regard
to independent or minority candidates or parties. Less partisanship would be an added bonus.

Below is a draft answer. Effective and simple, except for the modest complexity needed to counter
strategic voting.

Dick Burkhart, Ph. D., President, Democracy Works
4802 S Othello St,  Seattle, WA  98118
206-721-5672 (home)  206-851-0027 (cell)
dickburkhart <at>

There is a better way than the top two primary. The top two effectively eliminates third party and
independent candidates, sometimes even the second party candidate, as in Washington State’s 4th
congressional district in 2014. In fact we could get rid of the primary altogether. Just rank your top 3
candidates and tally the corresponding points.

How would this work? In the standard case, a voter’s top candidate gets 3 points, the second place
candidate 2 points, and third place 1 point. So imagine a Democrat, a Republican, and an Independent
running against each other. Then a well regarded independent could win with only 25% backing. 

How?  Democrats would rank the Democratic candidate first, the Independent second, and the hated
Republican third. Republicans would do the same, except putting their candidate on top, with the hated
Democrat last. Independents would rank their candidate first and split their second and third choices
between the Democrat and the Republican (so each would get an average of 1.5 points per ballot). Then if the
Democrats and the Republicans split the vote with 37.5% each, the number of points counted for the
Democrat would be 3*.375 + 1*.375 + 1.5*.25 = 1.875 times the size of the electorate. The Republican would
have the same point total, whereas the Independent would win with 2*.375 + 2*.375 + 3*.25 = 2.25 times the
size of the electorate.

This kind of calculus would provide a strong incentive for the political parties (or advocacy groups) to
lessen the mud slinging or form alliances. The result: more moderate candidates, less political
gridlock, and more minority candidates.
However, this standard form, which is called a restricted Borda Count, may be subjected to clever voting
strategies. The first strategy is for a candidate to ask his or her supporters to rank no other candidates,
so that the whole thing degenerates into ordinary plurality voting. To counter this tactic we adopt a
penalty method, called the Modified Borda Count: If a voter ranks only 1 candidate, then that candidate
gets only 1 point, instead of 3. If only 2 are ranked, they get 2 points and 1 point respectively, instead of 3
and 2.

A second strategy would be for candidate A to recruit 2 friends or allies to put their names on the ballot
without actually running campaigns. Then candidate A would tell supporters to rank A first and the 2
friends second and third. This tactic may be nullified by an elimination strategy. When there are more
than 3 candidates, we first eliminate the one with the fewest points, then next fewest, etc., until we are
down to 3 candidates. In the process the number of points assigned by each ballot to each candidates is
fractionally adjusted by the computer. 

Here the algorithm gets more intricate. If the eliminated candidate was third ranked by a ballot, then that
1 point would be divided up equally among the remaining candidates, except for the 1st and 2nd ranked
candidates. If the second ranked candidate is eliminated, then the third ranked candidate takes its
place, getting 2 points, with the extra point being distributed equally among the unranked candidates as
before. If the top ranked candidate is eliminated, then the second and third place candidates are moved
up, getting 3 and 2 points respectively, with the extra point again distributed equally among the
remaining unranked candidates. If an unranked candidate with a fractional number of points is
eliminated, then that fraction is divided up equally among the remaining unranked candidates. 

The end result of all this is that the recruited friends of candidate A will usually be eliminated, unless
the other voting blocks are just too small, with the other 2 surviving candidates getting an equal number
of points from candidate A’s voters. Thus candidate A will have lost all leverage that might have come
from forming alliances or being strategic about ranking the other viable candidates.

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DNOW1 | 6 Sep 07:56 2014

Top 5 P.R. method 26 Aug 2014

The gerrymander CRISIS in the U.S.A. is N-O-W.
Advanced P.R. will require a bit of public education -- regardless of the armies of math MORONS in the courts, in legislative bodies and esp. in the brain dead media and polisci depts in colleges and universities.
The top [5] below is of course arbitrary -- BUT something MUST BE done to get SOME sort of P.R. into the U.S.A. before the gerrymander robot party hacks [aka powermad OLIGARCHS] start Civil War II and/or World War III.
Reality check -- The U.S.A. is as *democratic* as a rock on Mars --
i.e. Democracy is *ALMOST* gerrymander D-E-A-D in the U.S.A.
AUTOMATIC ANTI-Democracy indirect minority rule in all legislative bodies having single member districts (subareas) since 4 July 1776 -- with or without intentional gerrymanders --

-- BOTH houses of the U.S.A. Congress, ALL houses of ALL 50 State legislatures and many, many local govt legislative bodies (whole or part). 

INTENTIONAL Gerrymanders (based on prior election results) =

(A) The most political enemies possible are PACKED into the fewest gerrymander districts possible = political concentration camps.

(B) The rest of the area is CRACKED into friendly gerrymander districts -- with the friendly party hack candidates trying to get a majority / plurality of the votes in such districts -- at least 55 percent to be in a *safe seat* district -- i.e. a *safe* 10 percent winning margin [NOT so safe after the 2006-2008-2010 *wave* elections ???].

Results --

1. Half [a plurality - near 1/2] of the votes in a bare majority [1/2 plus 1] of the gerrymander districts [political concentration camps] for 1 party control = about 25 percent ANTI-Democracy indirect MINORITY RULE in ALL 50 States.

1/2 votes x 1/2 gerrymander districts = 1/4 CONTROL = pre-school STONE AGE minority rule math -- with or without intentional gerrymanders.

Much, much worse minority rule math in primary elections -- with the winners elected in the gerrymander districts in general elections.

2. UNEQUAL votes for each gerrymander district winner -- i.e. a de facto POWERMAD legislative monarch in each political concentration camp.

3. UNEQUAL total votes in each gerrymander district -- i.e. political concentration camp.

See ALL of the SUPER-MORON gerrymander math opinions in Vieth v. Juberlier, 541 U.S. 267 (2004) [PA] and L.U.L.A.C. v. Perry, 548 U.S. 399 (2006) [TX] due to MANY, MANY, MANY lawyer and amicus prof MORONS

-- unable to detect the above 3 math items in their New Age ignorant MORON brains

-- and especially the district AREA FIXATION in gerrymanders -- i.e. regardless of the number of ACTUAL voters in each gerrymander district ON ELECTION DAYS - regardless of any obsolete Census stats.

Each gerrymander is an Act of W-A-R against REAL Democracy.

The persons making the gerrymanders are ANTI-Democracy W-A-R criminals.

Again -- there is AUTOMATIC 25 percent indirect minority rule just by having districts.

Result -- the business-as-usual nonstop brain dead ANTI-Democracy minority rule gerrymander *politics* in the Congress and ALL States in the U.S.A. since 4 July 1776 -- i.e. gerrymander monarchs / oligarchs making special interest gang monarchy / oligarchy laws.
Computerized minority rule gerrymanders 2010 Census

Anything new and different since the 1964-1966, 1972, 1982, 1992 and 2002 minority rule gerrymanders -- except even better and better ANTI-Democracy computerized gerrymander programs to even better and better identify the political prisoners (i.e. Electors-Voters) in each political concentration camp (i.e. gerrymander district) using the 2010 Census stats in 2011-2012 ???
ATTACK the ANTI-Democracy gerrymander systems in the media and the courts.

U.S.A. Const. Art. IV, Sec. 4 [Republican Form of Government -- NO monarchy / oligarchy (minority rule) regimes allowed to control a State directly or indirectly] and 14th Amdt, Sec. 1 [Equal Protection Clause].

See about 25 paragraphs in the 1787-1788 Federalist about the RFG in Art. IV, Sec. 4 --

I.E. RFG starts with D and ends with Y -- direct or indirect.

REAL Democracy begins at home.

SAVE Democracy NOW 26 Aug 2014
   Current ANTI-Democracy gerrymander math in many governments in the U.S.A. --
1/2 (or less) votes x 1/2 (bare majority of) rigged pack/crack gerrymander districts (concentration camps) = 1/4 (or less) CONTROL indirectly.
   Blatant subversions of USA Const. Art. IV, Sec. 4 (Republican Form of Government) and 14th Amdt, Sec. 1 (Equal Protection Clause).
Basic Democracy Reforms
26 Aug 2014  [optional language]
1. ONE election day. NO caucuses, primaries and conventions.
2. Proportional Representation in all legislative body elections. ALL voters get represented with BOTH majority rule and minority representation = DEMOCRACY.
3. NONPARTISAN nominations and elections of all elected executive officers and all judges.
Sec. 1. An election shall only be held on [date] of each year [(except for recall elections)].

Sec. 2. (1) All elected officers of any type shall be registered Electors in the election area involved in addition to any other qualifications in this constitution.
(2) All election laws and districts shall exist by [210] days before the election involved.
(3) No change in any election area shall reduce the terms involved.
(4) Any incumbent seeking re-election to the same office shall file an affidavit of candidacy provided by law by [203] days before the election.
(5) The names of other candidates shall by law be put on the ballots by (a) nominating petitions signed by Electors in the area involved equal to [0.2] percent of the number of Electors who voted at the last regular election [for governor] in such area and filed by [98] days before the election, or (b) filing fees equal to such [0.2] number multiplied by a uniform money amount and filed by [196] days before the election.
(6) Each Elector may sign 1 or more petitions for the same office.
(7) No filing shall be withdrawn.

Sec. 3. (1) The Electors shall elect the members of each legislative body for [1] year terms.
(2) As nearly as possible, each legislative election district shall be 1 or more [local governments] or a part of 1 [local government] (using major roads and rivers for internal boundaries), have the same number of Electors who voted at the last regular election [for governor] and be contiguous and square.

Sec. 4. (1) Each legislative body candidate shall receive a list of all other candidates in all districts by [56] days before the election.
(2) Each candidate shall rank such all other candidates (using 1 (highest), 2, etc.) and file such list by [49] days before the election.
(3) The lists shall be made public on the next day.
(4) If a valid list is not filed, then the candidate’s name shall be removed from the ballots.

Sec. 5. (1) Each Elector may vote for 1 candidate for each legislative body.
(2) The [5] candidates getting the most votes in each district shall be elected.
(3) The votes for each losing candidate shall be moved to the elected candidate in any district who is highest on the losing candidate’s rank order list.
(4) Example- 100 Votes. Elect 5.
A 20 Elected
B 18 Elected
C 15 + 9 = 24 Elected
D 13 + 8 = 21 Elected
E 10 + 7 = 17 Elected
X 9 - 9 = 0 Loses, Votes to C
Y 8 - 8 = 0 Loses, Votes to D
Z 7 - 7 = 0 Loses, Votes to E
Total 100
(5) Each member shall have a voting power in the legislative body equal to the final votes that he/she gets.
(6) A fractional part of a legislative body shall mean the members having such fractional part of the total voting powers.

Sec. 6. (1) A legislative body candidate or member may file a written rank order list of persons to fill his/her vacancy (if any) which shall be made public immediately.
(2) The qualified person who is highest on the list shall fill such vacancy.
(3) If the preceding does not happen, then the legislative body shall fill such vacancy with a person of the same party (if any) immediately at its next meeting.
[i.e. NO more vacancy special elections for legislators.]

Sec. 7. The Electors shall elect all elected executive officers [(at least a chief executive, a clerk and a treasurer)] in each government for [1] year terms and all judges [for [6, 4 and/or 2] year terms as provided by this constitution and law] at nonpartisan elections.

Sec. 8. (1) Each Elector may vote for 1 or more candidates for each executive or judicial office (including 1 write-in vote for each position).
(2) The candidate(s) getting the most votes shall be elected (for the longest terms respectively).
[This is the Approval Voting method.]

Sec. 9. Terms of office involved shall end on election days and start the next day.
[To END after election lame duck stuff. Elections would use ALL paper mail ballots to be counted not later than Election days.  See Oregon elections.]
Advanced (more complex) P.R. would have Condorcet head to head math (using Number Votes) and YES/NO votes as a tiebreaker.
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Kristofer Munsterhjelm | 3 Sep 20:10 2014

Whoops! Correction to Statistical Condorcet

Apparently the unconstrained MLE for the multinomial distribution with 
given probabilities isn't the Sainte-Laguë or Webster apportionment, but 
the D'Hondt or Jefferson apportionment[1].

This came as quite a surprise to me, given that the chi-squared and 
G-tests are claimed to approach the exact test as n goes to infinity. I 
may look into this in detail later, but I suspect the situation is that 
although the exact test value, call it x, and the G-test value, call it 
y, obey lim n->inf x_n-y_n = 0, for any finite n, the exact test will 
give the D'Hondt assignment a greater value (more likely draw) than the 
Sainte-Laguë one, whereas it's the other way around for the G-test (or 
chi-squared test).

I am not a formal statistician, though! And since I got the implications 
of the convergence wrong, I might be wrong about this as well.

For clarification purposes, letting s be the number of seats, the exact 
test for the multinomial distribution is: given a draw vector x, (i.e. 
so many of x_1, so many of x_2, etc) and a probability vector p = (p_1, 
p_2, ...), and the multinomial pmf P

Pr(x) = sum [for all y so that P(y; s, p) <= P(x; s, p)] P(y;s,p)

And letting s be the number of seats, define chi-squared test statistic as

chi(x, p) = SUM i=1...n (x_i/s - p_i)^2/(p_i)

Then, say, for the following votes: (10, 9, 8, 5, 4) and 5 seats, we have:

p-vector: 0.28, 0.25, 0.22, 0.14, 0.11

Sainte-Lague: (1, 1, 1, 1, 1)
D'Hondt:      (2, 1, 1, 1, 0)

Exact test value for Sainte-Lague: 0.899
Exact test value for D'Hondt: 1.0

multinomial pmf for the Sainte-Lague assignment: 0.028
multinomial pmf for the assignments with greater probability than this:

0.03234 for [1, 2, 1, 1, 0]
0.03622 for [2, 1, 1, 1, 0]
0.03234 for [2, 2, 1, 0, 0]

But the chi-squared statistic for Sainte-Lague is 0.13403 while the one 
for the D'Hondt apportionment is 0.19896, thus ranking the former higher 
than the latter.

This is true even for large s, e.g.:
	p = (0.3786, 0.245265, 0.1846, 0.06637, 0.06583, 0.059335)
	150 seats
	Sainte-Lague: [56, 37, 28, 10, 10, 9]
	D'Hondt: [57, 37, 28, 10, 9, 9]
	pmf for Sainte-Lague: 1.64*10^-5
	pmf for D'Hondt: 1.66*10^-5
	chi-square for Sainte-Lague: 0.00012
	chi-square for D'Hondt: 0.00058


Of course, if you like D'Hondt (for stability reasons or otherwise), you 
don't need to do anything to Statistical Condorcet to fix the above. 
Because it's Condorcet-based, it should also favor compromise parties 
rather than parties that get large numbers of first preference votes, so 
it is better than ordinary D'Hondt in that respect.

But if you don't, then the elegance of maximizing the pmf falls. So we'd 
have to find some way of using, say, the global optimality properties 
mentioned in directly. But this is 
tricky because they are all minimization properties, which means that 
the optimizer might just decide to set zero voters to participate and 
thus get a perfect zero every time.
That is again something to investigate later. Perhaps taking the area of 
the chi-squared distribution above the point given by the chi-squared or 
G-test would work: that turns it into a maximization problem again. But 
since the cdf for chi-squared involves gamma functions, optimizing that 
might be rather difficult.

Alternatively, we might go deeper. Why choose Sainte-Laguë to begin 
with? Because it's unbiased: it doesn't consistently favor small or 
large parties (and, because it's a divisor method, it has certain 
favorable properties we'd like to carry over). So find something that is 
unbiased. But the problem with that is that we might lose the "reduction 
to Webster when everybody plumps" property.


[1] I uncovered this when reading "A fast and simple algorithm for 
finding the modes of a multinomial distribution" by White and Hendy. It 
gives an algorithm for finding a mode of the multinomial, i.e. an 
apportionment that maximizes the exact test value. The paper is 
paywalled, but the algorithm is essentially a combination of Jefferson 
and D'Hondt: first they get a Jefferson solution for a number of seats 
that's close enough to the number of seats specified, and then they run 
D'Hondt either forwards or in reverse until they get the number of seats 
you want. The authors don't appear to recognize the solution as D'Hondt, 
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Kristofer Munsterhjelm | 31 Aug 12:08 2014

A simple Random Ballot proportional method

And here's a simple method I thought of for a multiwinner method. Would 
it be any good?

- Draw k assemblies using Random Ballot.
- Gather everybody who were selected in at least one of these draws, and 
set a deadline.
- If, after negotiations, at least 90% (or some high supermajority) of 
the gathered people agree on one of the k assemblies drawn, that outcome 
is chosen.
- Otherwise, if the deadline passes without agreement, draw another 
assembly using Random Ballot. That outcome is then chosen.

The idea is to exclude unlucky draws by using a negotiation phase. Yet 
there's little incentive to stall in the hope of getting a favorable 
outcome to oneself, because the post-deadline default isn't status quo 
but simply another random ballot council.

I suppose the main risk would that not enough people agree and then the 
final outcome turns out to be quite disproportional due to bad luck. The 
risk would be greater for smaller assemblies. Perhaps it could be 
lessened by running n different instances of the above in parallel, and 
as long as one of them decide upon an outcome, that outcome is picked. 
(If multiple do so, draw one at random.) But I'd have to consider it in 
greater detail to find out if that would be proportional.
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Kristofer Munsterhjelm | 30 Aug 18:06 2014

Statistical Condorcet: Combining Schulze STV and Webster

Some days ago, I thought more about the relation of chi-square to the 
Sainte-Lague index and Webster as a MLE for the multinomial (for large 
seat assemblies), and after considering this further, I think I have 
found a way to combine Schulze STV and Webster to make a more fair party 
list method. Unfortunately, unless the structure of the thing permits 
massive pruning, it will be much too slow to run on a national scale. 
But perhaps it could be used in local elections.

The method goes like this: The meta-system is like Schulze STV. Define a 
contest function c(sA,sB) where sA and sB are sets of candidates that
differ from each other by just one candidate. Contests between sets that 
differ by more than one candidate are handled through strongest path, as 
in ordinary Schulze STV.

Since voters vote for parties, a contest may also contain numerous 
copies of the same "candidate". For instance, [AAAB] is a set with 3 
members from party A and one from B.

The main difference between this method and Schulze STV is in what c(sA, 
sB) is. Let sA and sB be council assignments that differ by a single 
candidate which we'll call C. C is present in sB but not in sA, so c(sA, 
sB) returning a higher score means sA is better than sB.

As in Schulze STV, in a given contest, each voter may give his vote to
any candidate he prefers to C. But how is this assignment to be guided? 
Well, that's where the difference comes in. Let x_j be the number of 
seats given to the jth party in the set sA.
Let f(x_1, ..., x_n, n; p_1, ..., p_n) be the multinomial pmf for n 
draws where the probability of drawing an item from category j is p_j, 
and x_j is the number of items drawn. Let p_j be the fraction of the 
voters' support that goes to the jth party. Then the voter-party support 
assignment is done so as to maximize f. So in this particular contest, 
the voters are engaging in a kind of maximum-likelihood cumulative vote 
DSV against C.

Some voters may not be able to contribute to this - for instance, if
they rank C on top. Let nv be the number of voters who did contribute.
Then c(sA, sB) = nv * f(x_1, ..., x_n, n; p_1, ..., p_n).

There is one other quirk to iron out. Since parties may have multiple
candidates in the running, C might be a member of a party that's also 
present in sA. In that case, since voters directly vote for parties, 
they are also allowed to support the party C is a part of if sA contains 
at least one member of that party. This doesn't directly help C because 
if the supporters of C's party were to dump too much support into that 
party, the fit that is optimized by f would decrease. Yet it's only 
permitted if at least one candidate of that party is present on the left 
side; otherwise, single-winner won't reduce properly. (It can also be 
seen by considering party support for A>B>C as 
A1>...>An>B1>...>Bn>C1>...>Cn and then using ordinary Schulze STV rules)


Note that it's conceptually easy to alter this system to take other
factors into account. Just alter the distribution and the pmf. For 
instance, if you want to incorporate that some parties may only have a 
few members (not enough to fill the council), then replace the 
multinomial with the multivariate hypergeometric distribution.

Also, the method above converges to Sainte-Lague/Webster when everybody
plumps for one party each, and the council size is large. This is 
because Sainte-Lague/Webster optimizes the Sainte-Lague index, which, 
through its connection to the chi-squared test, finds the council that 
optimizes f above when the council size is large enough.

The mathematical simplicity of the thing might mean that one can 
optimize it to run on large instances. In particular, dealing with it on 
a candidate by candidate basis rather than a party-by-party basis is 
designed to make full use of excess votes so they may add up to give a 
seat to a compromise party. Perhaps one could determine the number of 
safe seats for each party in a per-party manner and then hone the result 

One could also imagine an Approval variant of the concept: voters 
approve of one or more candidates or parties. Then the method picks the 
council that maximizes nv * f as above where each voter's support is 
distributed over the candidates or parties he approved, cumulative 
voting style. But I have a hunch that such a method would reduce to 
Monroe's method or a variant of it.

As for applying MJ to this idea... perhaps robust estimators? I don't know.

I could write something about how to find the optimal vote supports as 
well, but this post is long enough. Tell me so if you think I should. 
The short of it is that it comes down to optimizing a weighted sum of 
logarithms under linear constraints.
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