[EM] Finding "defining" scenario results for a given method

[Kevin Venzke]

Here is a simple proof of concept, attempting to find a smallest set of scenarios
which can differentiate a set of election methods. Here I've used full, strict
rankings with three candidates and four blocs, and I've used a rather short list
of 17 methods. The result is that 7 scenarios will differentiate them all.

Note that certain methods become identical with such parameters:
MinMax includes MMPO (of course), MAMPO, and numerous others.
DSC includes DAC.
IRV includes King of the Hill (KOTH).
C//A includes MinMax(AWP) and MDDA.
Approval elimination runoff (AER) includes DMC.
C//FPP includes BTRIRV.
Antiplurality is equivalent to implicit Approval.
fpA-fpC includes C//IFPP.*

(* But actually, I think these two may be the exact same method. That would be
interesting not just because the formulations of the logic look very different,
but also because the motivations behind them weren't the same. fpA-fpC was
discovered based on a search for low-manipulability methods (if I remember
correctly what Kristofer did) while IFPP seems intended to fix the monotonicity
issue of IRV and nothing more.)

The scenarios are evaluated in order, so that as we go on fewer and fewer
distinctions can be made. Pairs of methods split at any step aren't considered
further.

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Scenario 1

0.4770: B>C>A
0.2471: A>B>C
0.1960: A>C>B
0.0796: C>A>B

Elect A: Stensholt BPW, IRV, C//IRV, IFPP, fpA-fpC, declonedCopeland
Elect B: Approval elimination runoff (AER), Borda, FPP, C//FPP, DSC, MinMax
Elect C: Antiplurality, Bucklin, C//A, ChainRunoff, TACC

It's hard to completely generalize these groups, but a number of "anti-burial"
methods pick A. Many approval-oriented methods pick C. The B group is rather
diverse, but includes several FP-emphasizing methods.


Scenario 2

0.3222: A>C>B
0.3048: B>A>C
0.2100: C>B>A
0.1628: C>A>B

Elect A: AER, C//A, Antiplurality, Borda, Bucklin, BPW, IRV, C//IRV, dc-Copeland, MinMax
Elect B: ChainRunoff
Elect C: FPP, C//FPP, DSC, IFPP, fpA-fpC, TACC

The C group unites some intuitively FP-emphasizing methods, and almost every
other method is in the A group.

Chain Runoff's result of B is unique. Any unique result would differentiate a
method in a single step.

TACC is also grouped with completely different methods from before, and so is
completely differentiated.

Of note, this scenario lets us differentiate now between BPW, IRV, and C//IRV on
on one side, and IFPP and fpA-fpC on the other.


Scenario 3

0.4119: A>C>B
0.2785: B>C>A
0.2431: C>A>B
0.0664: A>B>C

Elect A: Borda, ChainRunoff, DSC, FPP, IFPP, IRV
Elect C: AER, Antiplurality, BPW, Bucklin, C//A, C//FPP, C//IRV, fpA-fpC, dc-Copeland, MinMax, TACC

Here C is the Condorcet winner.

This establishes splits especially between non-Condorcet methods and their
Condorcet versions. Additionally Borda gets split off from MinMax and AER; DSC
gets split from C//FPP; and IRV splits from BPW and dc-Copeland.


Scenario 4

0.4012: B>C>A
0.2855: A>B>C
0.1776: A>C>B
0.1355: C>A>B

Elect A: FPP, C//FPP, ChainRunoff, IRV, C//IRV, dc-Copeland, DSC, IFPP, fpA-fpC, MinMax  
Elect B: AER, Borda
Elect C: Antiplurality, BPW, Bucklin, C//A, TACC

This scenario splits AER (elect B) from MinMax (elect A).
And it splits BPW (elect C) from C//IRV and dc-Copeland (elect A).


Scenario 5

0.3202: A>B>C
0.2788: C>B>A
0.2145: B>A>C
0.1863: C>A>B

Elect A: most of the methods
Elect B: Antiplurality, Bucklin    
Elect C: FPP, IFPP

This splits off C//A from Antiplurality and Bucklin.
And it splits up FPP and DSC.


Scenario 6

0.3904: A>B>C
0.2525: C>B>A
0.2054: A>B>C
0.1516: B>A>C

Elect A: everything except Antiplurality
Elect B: Antiplurality

A is majority favorite.
All this does is split Antiplurality and Bucklin.


Scenario 7

0.3911: C>A>B
0.2984: B>C>A
0.2814: A>B>C
0.0289: A>C>B

Elect A: TACC
Elect B: BPW, dc-Copeland  
Elect C: all other methods

There is no CW, but almost all methods agree on C winning.

This scenario just splits up C//IRV and dc-Copeland.

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And that's the end.

If we add more methods and allow truncation or equal-ranking, the number of
scenarios needed doesn't grow very fast.

What would have been quite useful is if the small number of scenarios could be
translated into a "method DNA" setting, where the number of possible scenarios
is relatively large despite offering less flexibility in expressiveness.
Unfortunately that doesn't seem very feasible because in the DNA setting we can
define the "movement" from one scenario to another, and thereby evaluate
criteria, which we can't do here.

A slightly different approach could be to focus on finding a few scenarios that
each independently differentiate as many methods as possible, while also
correlating as little as possible with the other scenarios' differentiations.
Then these several scenarios could be used as a litmus test to gauge what a
proposed rank method is similar to, without having to implement it.

Kevin