[EM] Method space of 3-bloc, strictly-ranked scenarios

[Kevin Venzke]

I'll define a simple "ballot space," meaning a range of scenarios that can occur given voter, candidate, and ballot limitations. Then I'll talk about the "method space" for that ballot space, meaning the range of methods that are possible to define.
Let there be three factions, each preferring a different one of candidates labeled A, B, and C. The "A bloc" is the largest, with B being second-largest, and C being smallest. No bloc is a majority, but any two blocs would be a majority. Every voter within a bloc votes exactly the same (so that we see, total, only three unique ways of filling out the ballot). Every ballot is required to be a full, strict ranking. Additionally, each bloc will always rank their preferred candidate at the top.
Since each bloc can only fill out the ballot in two ways (e.g. A>B>C or A>C>B for the A bloc), this ballot space has only 2^3 = 8 scenarios:1. A>B B>A C>A ballots. A+B mutual majority, A is CW.2. A>B B>A C>B. A+B mutual majority, B is CW.3. A>B B>C C>A. A>B>C>A preference cycle.4. A>B B>C C>B. B+C mutual majority, B is CW.5. A>C B>A C>A. A+C mutual majority, A is CW.6. A>C B>A C>B. A>C>B>A preference cycle.7. A>C B>C C>A. A+C mutual majority, C is CW.8. A>C B>C C>B. B+C mutual majority, C is CW.
If we put a sequence of eight As, Bs, or Cs in a row to show who wins in each scenario, that sequence can define a method for this ballot space. I call this a DNA code.
I have 343-digit codes, covering a larger ballot space, for a few dozen election methods. But in this eight-scenario space these reduce to a quite small number of codes. For methods that use an approval concept, I use the assumption that only the bottom candidate is unapproved. Here then is a full list (so far) of the 8-digit codes resulting from methods that are at least mostly serious:
AAAAAAAA: FPP has its own code. Notice how it just elects A no matter what any lower preferences are.
ABABAACC: The MinMax group. This includes most MinMax methods (including also Schulze, MAM, etc.), DMC, Approval-Elimination Runoff, MAMPO, some scenarios of Kristofer's "linear" Condorcet method, a couple of obscure Woodall methods (MinGS and 3MaxGS), Condorcet//FPP, and probably a lot of other methods.
ABBBAACC: The Bucklin group. This includes Chris' IBIFA, Condorcet//Approval and ICA, MDDA, CdlA, MinMax(AWP), and probably many others. Note that, within this ballot space, these are a Condorcet method!
AAABAAAC: The "descending coalitions" group. It includes DAC, DSC, and a couple of my methods (SPST, and both pairwise and Bucklinesque versions of "Venzke Bucklin Variant" aka VBV). 
ABABABAB: IRV group, including my "King of the Hill" (KH) method and IRV-without-elimination.
ABABABCC: Condorcet//IRV group, including Raynaud, Condorcet//KH, and most scenarios of Kristofer's "linear" method. Off the top of my head it should also include the "Woodall" and "Benham" methods.
ABABACAC: my QR, aka Chain Runoff.
ABABACCC: Condorcet//QR.
ABBBACCC: Forest's TACC method.
ABCBABCC: BPW (Eivind Stensholt's "Beats Plurality Winner" Condorcet method).
AAABAAAB: VDP (Venzke Disqualified Plurality, aka VFA). 
That's really it... Other known codes are limited to experiments like "IRV where you eliminate the wrong candidate" or "just elect the winner of the strongest pairwise contest." (Let me know if you think there's an unusual method I've missed, and that can operate on these ballots.)
Here is an attempt at plotting the Hamming distances of these codes, so that more similar methods are plotted closer together:
. . . . . . . TACC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BPW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bucklin . . . . . . . . . . . . C//QR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ------- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ./. . . . . . . . . . . . . . . . . . . . . . . . . . . . . C//IRV. . . . . . --------- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ./. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .------ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . / . . . . . . . . . . . . . . . . . . . . . MinMax. . . . . . . . . . . . . . . . . ./. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . --Condorcet-Boundary--- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ./. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ------. . . . . . QR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ./. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .---------------------. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IRV .----. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DAC/DSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
That is 11 methods in a method space containing 3^8 = 6561 methods. Can we find any more methods?
In all of those 6561, only nine of them are Condorcet methods. I can describe all of them:
ABABAACC: MinMaxABABABCC: C//IRVABABACCC: C//QRABBBAACC: BucklinABBBABCC: (???) always elect B in a cycleABBBACCC: TACCABCBAACC: Condorcet//"IRV where you eliminate second-lowest tally"ABCBABCC: BPWABCBACCC: (???) always elect C in a cycle
Only two scenarios lack a CW, leaving not a lot of room to experiment.
How about non-Condorcet methods? Where is their diversity located? Let's ignore FPP and remove the cycle scenarios, so that we can just see where and why methods choose not to elect a CW:
AB_BA_CC: Condorcet methodsAA_BA_AB: VDPAB_BA_AB: IRVAB_BA_AC: QRAA_BA_AC: DAC/DSC 
The disagreements come in scenarios 2, 7, and 8. If you check you'll see that they can be summarized in one statement: Non-Condorcet methods may prefer to elect the candidate of the larger bloc of a mutual majority, rather than the candidate of the majority who is the CW. (Note that the CW is the second preference of the faction excluded from the majority.)
In the scenarios where the CW is also from the larger bloc of the mutual majority, all methods agree (except for FPP in scenario 4). And no method outside of FPP will elect the Condorcet Loser (in this ballot space), as it would violate mutual majority. So it seems likely there are few possibilities for reasonable methods. If you don't think it's reasonable to elect C in the cycles, it might only be 2^5 = 32 possible methods.
That's all for now. I hope in the future to discuss scenarios with more ballot types.
Kevin Venzke
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