Also, here’s a scenario to illustrate a problem I talked about earlier (deliberate undervoting by the majority). This problem can be easily fixed with a slight change to the voter influence formula -- replace the maximum available score with the maximum score the voter in question actually uses -- and I’m not aware of any downside to such a fix.

Minority group: A=5
Majority group: B=C=D=E=1

Round 1 individual weights: Minority=1.000, Majority=1.000
Round 1 group weights: Minority=100, Majority=300
Round 1 total scores: A=500, B=300, __ C=300, D=300, E=300
Round 1 runoff: A=100, __B=300
B is elected

Round 2 individual weights: Minority=1.000, Majority=0.833
Round 2 group weights: Minority=100, Majority=150
Round 2 total scores: A=500, C=250, __ D=250, E=250
Round 2 runoff: A=100, __C=250
C is elected

Round 3 individual weights: Minority=1.000, Majority=0.714
Round 3 group weights: Minority=100, Majority=214
Round 3 total scores: A=500, D=214, __ E=214
Round 3 runoff: A=100, __D=214
D is elected

Round 4 individual weights: Minority=1.000, Majority=0.625
Round 4 group weights: Minority=100, Majority=188
Round 4 total scores: A=500, E=188
Round 4 runoff: A=100, E=188
E is elected

_Results: BCDE are elected. _

_Grade: FAIL. SRV does not elect candidate A. _

Edited to correct an error in the round 2 runoff.

Your pass/fail criteria aren't quite accurate. See the RRV proportionality theorem.

Proportionality Theorem

If some voter faction (call them the "Reds"), consisting of a fraction F (where 0≤F<1) of the voters, wants to, it is capable (regardless of what the other voters do) of electing at least ⌊(1+N)F-⌋ red winners (assuming, of course, that at least this many red candidates run, and the total number of winners is to be N).

Specifically, it can accomplish that by voting MAX for all Reds and MIN for everybody else.

To say that again: if 37% of the voters are reds, they can assure at least about 37% red winners (up to rounding-to-integers effects).

-- http://scorevoting.net/RRV.html

* PR-SRV is just "Reweighted Range Voting" with a (in my opinion pointless) runoff added.

Clay: Do you want to elaborate on how the pass/fail criterion I used differs from the one you like, and (especially) on why I should use that one instead?

Warren Smith's proportionality criterion just holds that Blues give "10" to all Blue candidates, and "0" to everyone else. Your example muddies things by making people partially Blue, partially Green, or what have you. In which case there's no unambiguous definition of proportionality.

How does IRV/STV fare in similar scenarios? Any better or worse?

I think it's quite a leap to assume that there will ever be a real scenario where all members of a "majority" vote the same way and all members of a "minority" vote the same way. Also it is quite the leap to assume that there is only one well defined and distinct and monolithic minority group, and one well defined and monolithic majority group. In reality there are always several factions and multiple different ways to slice and dice and group together various interest groups, depending on different issues, or demographics, party, neighborhood, industry, age, religion, etc.

Thanks for posting this Fillard! It's an interesting analysis and is exactly the kind of thing we need to be exploring in depth for all systems and especially the PR ones that are so complicated to analyze. We really need to have accurate simulations that can run these kinds of complex scenarios a number of different ways and a system of criteria to judge how those results are different and what is actually most desirable here. Who is currently working on that? Anyone? How do we get that happening? Host a fundraiser for PR science?

Point 1-Assumed best winner: I'm going to try and think this through so forgive me if I fail and let me know where I'm not looking at this the right way. In the above scenarios there are more people that prefer B, C, D, and E than prefer A. In this case I think the best result would be a win for BCDE. Especially if none of the other voters actually like A. Deciding that A should win seems a bit unfounded and A might even be a spoiler, but I'm not sure. 300 people prefer B, C, D and E and only 100 prefer A. Why should A win? Any argument for why A should win seems like it would be a better argument for why B, C, D, or E should win. If A voters had voted for 4 candidates like the others they would have been more fairly represented.

For me I usually prefer a 3rd party candidate so it's natural for me to prefer a system that favors and gives an extra advantage to 3rd party candidates. Affirmative action so to speak. On the other hand I'm part Jewish and I don't want to give affirmative action to fascists and bigots. We can't actually give a stronger ballot weight to people by race (though I might actually support this... actual affirmative action in the vote?!? Native Americans get 3 times the say in the Electoral collage like Wyoming does now?!) I don't think I can support giving an unfair advantage to unspecified minorities across the board. I support a system that gives all voters a fair and equal say in democracy.

Point 2- Is this a workable example case study?: Ideally in this election, in my opinion, since there will be 4 winners each voter should want to give 4 candidates all a max score. A good PR system should incentivize this kind of behavior. Since there is only going to be one looser that candidate should get a 0. In a 4 seat election we really need more than 5 candidates to do an accurate analysis. Ideally we should probably have double the number of winners running or so. If that were the case a voter should ideally be safe to give max score to your ideal winners and then also differentiate between the losers if some of them would be acceptable or better than others.

To look at who should win I think we need to not just look at the A party's favorite but also who their top 4 are. In this case a good result for Party A is more complex than just electing candidate A. Ideally they will also want to elect candidates that will help them while eliminating candidates that are their worst case scenario. By giving a full score for their top 4 (ABCD) can Party A prevent candidate E from getting elected? Can they become an important part of the coalition for Candidates BCD and then help hold them accountable after they've been elected? For actually racial and economic minorities the most effective vote outcome as judged by legislative outcome is probably to eliminate racists, corporate tyrants and war mongers. For a non racial minority they should be able to eliminate those most antagonistic towards them.
(NOTE: for simplicity here I'm assuming that A and E are opposite ends of the spectrum but in reality this is NOT always the case. The middle could be establishment and the edges could agree on small govt or any number of things).

Point 3- Do we want proportional representation by voting system or by district?
I'm a huge advocate of local self-sufficiency and community inter-dependency. We are united by place and held accountable by place. Issues at home directly effect us and we have a bond to places we consider home that is fundamental as our bonds with family. Love for the land and the need to protect it should be a driving force in government.

Additionally, I wish it were not the case but in reality people self segregate by values and culture. In the absence of a voting system that would specifically support ethnic minorities, location might be the next best thing. Moreover I would argue that in general, people are more likely to care about an issue that effects their neighbors even if it doesn't impact them directly.
See Vi Hart: http://ncase.me/polygons/
I think that there is some real value in looking at proportional representation by district. Large area's should be divided by location, natural boundaries (watersheds, bio-regions?) and population in a non-gerrymandered and neutral fashion. This holds true for boundaries ranging from neighborhoods to cities to states to countries:
Impartial Automatic Redistricting: https://www.washingtonpost.com/news/wonk/wp/2014/06/03/this-computer-programmer-solved-gerrymandering-in-his-spare-time/?utm_term=.675f1b7ec5fc
Each district should be able to elect a representative (or 3? What about a combo of district and multi winner based PR?) that will then represent them and take their local issues as their top priority and bring them to the larger group. These elected officials would then be directly accountable to their constituents in a way that at-large elected officials aren't.

For me legislative outcomes and quality of government are as important and perhaps more so than just looking at the votes, though granted this is much harder to study and impossible to "prove". Since there isn't a science yet that can judge PR systems and their long term efficacy we need to step it up with our thought experiments! Go team!

Sorry this is so long! Hope you guys find it helpful!

In reality there are always several factions and multiple different ways to slice and dice and group together various interest groups

Right. People aren't "Democrat" or "Republican". They are "21% Green, 43% Republican, 36% Democrat" or what have you.

The Proportionality Theorem is a sort of minimal test that a system is PR in the simplistic unambiguous case.

Does multi-winner IRV pass the scenarios in my above post? Yes, all of them.

As previously disclosed (or at least strongly implied), I deliberately fudged the tiebreakers in my scenarios to make the explanations simpler. For that and other reasons there’s a little room for interpretation as to how the ballot markings in my scenarios would translate to ballot markings in multi-winner IRV. But it ends up not mattering. In all three scenarios, the ballots look more or less like this:

Minority group (100 votes): AEBCD
Majority group (300 votes): BCDEA

Multi-seat IRV elects A, B, C, and D, which is the proportionally appropriate result.

None of this is to say multi-winner IRV is perfect, but in my opinion it’s the best voting system overall by a considerable margin, and it clearly qualifies as proportional representation where multi-winner SRV does not (if you accept the criteria I’ve chosen).

Adam: Yes, people and groups are extraordinarily diverse. But sometimes even a highly contrived and simplified scenario can shed light on what’s going on. (In other words I agree with Clay on that point.)

Clay: For purposes of this discussion, I consider a voting system to qualify as PR if, whenever N seats are being elected in a single contest (and N is at least 2), and 1/N of the voters unanimously prefer candidate A to every other candidate and mark their ballots accordingly, candidate A is elected regardless of how the other ballots are marked. Are you claiming that’s ambiguous?

Sara:

I wrote out a detailed explanation of what affirmative action is, then read that part of your message a third or fourth time and decided that wasn’t the problem. I think you know what affirmative action is. The problem is that for some reason you seem to think proportional representation is a kind of affirmative action. It isn’t.

Do you agree with the following statement (that I just wrote)? If not, why not?

There is nothing remotely discriminatory or unequal about saying that 1/4 of the voters should be able to elect 1/4 of the seats in a legislature or city council while the other 3/4 of the voters elect the other 3/4 of the seats. No special treatment is being given to anyone.

What about this well-known John Adams quote?

[The Representative Assembly] should be in miniature, an exact portrait of the people at large. It should think, feel, reason, and act like them. That it may be the interest of this Assembly to do strict justice at all times, it should be an equal representation, or in other words equal interest among the people should have equal interest in it.

You say: “Ideally in this election, in my opinion, since there will be 4 winners each voter should want to give 4 candidates all a max score. A good PR system should incentivize this kind of behavior.” Why do you mention PR in the latter sentence? Are the opinions you’re expressing here specific to PR? They don’t sound like it; in fact they sound like precisely the opposite. I haven’t rigorously proved this, but I’m pretty sure any system that incentivizes giving an equal maximum score to as many candidates as there are seats is automatically a winner-take-all system (i.e. not PR).

A note about "fascists and bigots"... Some people blame PR for the rise of the Nazi party, or say, “If we had PR in the USA, there’d be Nazis in the legislature!” It’s my understanding that the use of PR in Germany actually delayed the rise of the Nazi party. (I haven’t verified this.) As for allowing Nazis in the US legislature, hasn’t the current system been doing that? I’m not aware that anyone in the US legislature is a card-carrying Nazi per se, but our president campaigned on and has arguably been implementing a Muslim immigration ban. So I don’t see PR as creating that problem.

Using the phrase “proportional representation by district” to refer to districts that elect one person each is going to confuse people, because it’s not the usual meaning of “proportional representation”.

If you had to choose between geographic districts and PR (by voting system), which would you choose?
- Geographic districts do an excellent job of ensuring one specific kind of diversity -- geographic -- while completely ignoring all other forms (except insofar as they correlate to geography).
- Multi-winner IRV does a good job of ensuring all kinds of diversity.

I would absolutely choose PR over geographic districts. Though as you say, sometimes you can do both.

I consider a voting system to qualify as PR if, whenever N seats are being elected in a single contest (and N is at least 2), and 1/N of the voters unanimously prefer candidate A to every other candidate and mark their ballots accordingly, candidate A is elected regardless of how the other ballots are marked

That's wrong. If you give Green a 5 and Dem a 3, then you are not "Green". You are 62.5% Green and 37.5% Dem.

Simple example. Suppose you're electing 3 seats and there are 3 voters. The first 2 voters think Green=5, Dem=4. The last one thinks Green=0, Dem=5. It would be insane to demand 2 seats filled by Green and 1 by Dem. Group = 9/14 = 64% Democrat. So the most representative outcome would be 2 seats = 66% Democrat.