[EM] 4-candidate CDTT LNHarm failures

[Kevin Venzke]

Hello,

I've done some work investigating the particular circumstances under
which CDTT methods (i.e., methods which elect the CDTT set member
who comes first in a ranking generated by a method satisfying LNHarm,
such as FPP, MMPO, or Random Ballot) fail LNHarm when there are four
candidates. (LNHarm is not failed at all when there are just three
candidates.)

The CDTT set, again, contains every candidate who has a majority-
strength beatpath to every candidate who has such a beatpath back
to this candidate. The purpose of this set, in my opinion, is to
satisfy Minimal Defense and as much Later-no-harm as possible. The
aim of these criteria (in my opinion) is to permit voters to rank
fully and sincerely, which in turn should permit candidates to
receive ballot support corresponding to their actual level of support.

Suppose that your sincere preference order is C>D>A>B, and you
have decided to vote C>D for sure, but aren't certain whether you
want to give your last preference, voting C>D>A, since you're worried
about harming C or D. There must be some scenarios where adding votes
for A>B can harm C or D.

First, how many scenarios are there? It's only relevant to the CDTT
whether, for each pair of candidates, one has a majority-strength
win over the other, or neither does (an indecision). So there are
3 possibilities for each pair of candidates. There are 6 pairs of
candidates in the four-candidate case: AB AC AD BC BD CD. And since
we are worried about the effect of adding A>B votes, we can assume
that there is an indecision between A-B, and that by adding our
A>B preference, we would be creating a majority-strength win for A
over B. That assumption leaves us with 5 pairs of candidates to
worry about.

The number of scenarios is thus 3^5, or 243. I have already written
a program to test all of these scenarios, and the results are
encouraging: Only 10 scenarios result in C or D being harmed by
the addition of the A>B domination, while 75 scenarios (including
2 overlap with the first category) result in C or D being helped.

(In the three-candidate case, there are only 3^2 or 9 scenarios, of 
which 3 involve the first preference being helped, and 0 involve harm.)

Now, I will give examples for these 10 scenarios, which can be
divided into three categories. The method being used in all of these
examples is CDTT,FPP, meaning that the CDTT set member with the most
first preferences is elected.

Category 1: {a,b,c,d} to {a}. (2/10 scenarios.)
20.45 CD(A)
22.41 ABC
20.04 CAD
22.05 DBA
15.03 B

FPP ranking: 40 C, 22.4 A, 22 D, 15 B.
There is a majority-strength B>C>D>B cycle initially, with no
majority-strength wins to or from A. When the A preference is
added, A has a win over B, giving A an unreturned majority-strength
beatpath to every other candidate. The winner changes from C to A.

Category 2: {b,c,d} to {a,b,c,d}. (6/10 scenarios.)
25.21 CD(A)
15.48 AD
26.88 ABC
8.24 BD
24.16 DB

FPP ranking: 42 A, 25 C, 24 D, 8 B.
The initial scenario is that there is a majority-strength cycle
among BCD, and one or both of C and D has/have majority-strength
win(s) over A, which pushes A out of the CDTT. When the A preference
is added, A obtains a majority-strength beatpath to all of the other
candidates, entering the CDTT. The winner changes from C to A.

Note that this is a scenario where Minimal Defense and LNHarm bump
heads. Initially, more than half of the voters vote D>A and don't
vote A over anyone, so A mustn't win. But when the A preference is
added, A becomes indistinguishable from the other candidates.

Category 3: {b,c} or {b,d} to {a,b,c,d}. (2/10 scenarios.)
11.65 CD(A)
28.04 AB
17.7 CBD
18.91 ABD
23.68 D

FPP ranking: 47 A, 30 C, 24 D, 0 B.
The initial scenario is that B dominates D and D dominates A, so
that B has a majority-strength beatpath to both D and A, and neither
has such a beatpath back. (C's contests are all indecisive.) When
the A preference is added, A dominates B, giving A and also D beatpaths
back to B. Again, the win moves from C to A.

This is another MD-LNHarm conflict example. Initially, more than half
of the voters vote D>A and A over no one.

This is the category which also involves LNHelp failures. It's possible
that instead of changing the win from C to A, it could be changed e.g.
from B to D, helping the CD(A) voters.

I ran 50,000 of the above type of election, and in that number there
were 483 elections where the addition of the A>B preference changed
the winner *to* C or D, and only 18 elections where the winner was
changed *from* C or D.

And this was using FPP for the ranking. FPP satisfies LNHelp. If widening 
the 483-18 gulf were seen as desirable, it could almost surely be done 
with a method that fails LNHelp, such as MMPO or DSC.

It seems to me that this is pretty good performance from the CDTT.

Kevin Venzke



	

	
		
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