[EM] pairwise comparisons

[Kristofer Munsterhjelm]

On 2018-04-19 01:19, Curt wrote:
> Hi,
> 
> I’ve been chewing on some questions as part of exploring my views of ranked voting, and thought I would share here. For those of you who prefer not to think more philosophically about this, please excuse the missive. :-) But for the rest of you, I’m interested in your thoughts.
> 
> Imagine a set of [six] candidates, and one voter. The voter is asked to determine their views of these candidates. But instead of just being asked to rank them in order, the voter is asked to judge them pairwise.
> 
> For ten candidates, this means 15 questions. Each question being a comparison of A and B, with the voter picking their favorite of the two.
> 
> First question, is it possible for a voter to generate a cycle? W > know it is technically possible, trivially demonstrated. But is it
> possible that a voter, using some internal set of principles, would also
> generate a cycle? I would argue yes. >
> If the voter does generate a cycle via these pairwise comparisons,
> what does this mean? Does it mean the voter is confused? Does it mean
> the voter is inconsistent? Does it mean that this cycle or cycles are an
> accurate depiction of the voter’s actual views?

One way to look at voting is as a substitute for deliberation. The idea 
is that you don't have a way to get a million people into a town hall 
and discuss the issue; or that there's a deadlock in a small assembly 
and you have to decide somehow.

Then the presence of cycles is an effect of the different parts of the 
electorate prioritizing different things. E.g. if it's a presidential 
election, one part may be prioritizing defense and rank the candidates 
A>B>C, another part may be prioritizing not having been involved in 
corruption and rank the candidates B>C>A, and yet another part may be 
prioritizing domestic credibility and rank the candidates C>A>B.

(If the voters share the same 1D view of politics and prefer candidates 
closer to their point on the line, then there will always be a Condorcet 
winner.)

In the substitute-for-deliberation model, a single voter *could* have a 
cyclical preference, but it would be very rare. Since a voter knows his 
own mind very well (or so we would assume), he would "deliberate" with 
it pretty easily and come up with a noncyclical ranking.

Even if he were to consider some candidates to be tied in a cycle, he'd 
probably just rank them all equal. I think it's more realistic that a 
voter would say "this candidate is better at defense, that candidate has 
not been involvled in a corruption scandal, and that candidate has a 
more convincing domestic program, but I can't decide which is more 
important to me, so I'll rank them all equal" than "I'll switch my 
mind's point of view in between each pairwise comparison and thus set up 
a cycle".

On a side note, I've sometimes thought of having a pairwise-only ballot 
for the purpose of discouraging strategy. When the voter is to vote, he 
is given a "do you prefer X to Y" yes/no ballot where X and Y are given 
at random. It's harder to coordinate strategy this way because not the 
entire ballot is available to the voter. But in any case, under such a 
protocol, I think it would be more likely to "see" cycles happening for 
a single voter.

Imagine the universe splitting into three parallel universes at the time 
the voter picks up one of these ballot cards at random. On one, the 
ballot says "Do you prefer A to B?", on the other "Do you prefer B to 
C?", and on the third, "Do you prefer C to A?". It's possible that the 
voter would think of defense in the first universe, corruption in the 
second, and domestic issues in the third, and so in all of these 
universes, he would say yes. The difference between this and the usual 
setup is that the voter isn't provided the whole question at once.

> Say that we then ask the voter to create an actual ranked ballot out of these ten candidates, and the voter manages to do so. What happened in that process of the voter deciding the rankings? Was it a clarifying experience? Did the voter’s preferences change? Did the voter compromise? Did the voter lie?
> 
> And finally, say that a collection of these voters submit their ranked ballots (not just their pairwise comparisons), and the votes are tabulated, and the result is a three-candidate Smith Set, where each candidate defeats all other candidates outside the Smith Set.
> 
> What does that Smith Set mean? Is the electorate confused? Is the electorate inconsistent? Or is the Smith Set an accurate depiction of the electorate’s actual views?

The model above would say that, assuming there's no strategy going on, 
the electorate is inconsistent. They value different things. The Smith 
set being a subset of the candidates would be analogous to the 
outside-of-Smith set being judged worse on all three axes by a majority 
of the voters (not necessarily the same majority for each).

Other possibilities are:

- Strategic cycle. A party judges that its candidate will win if it can 
set up a particular cycle, and so it tells all its voters to vote in a 
way that produces this cycle. This could even happen if the voting 
method is runoff: the party could judge that its candidate is a 
particularly good public speaker and so would have a better chance of 
winning in a small runoff with increased attention on the few remaining 
candidates.

- Strategic cycle backfire. As above, but with multiple parties and the 
result is that someone neither party wants to win, wins.

- Noise: The voters aren't perfectly sure of what they want, e.g. each 
voter has a "true ranking" of the candidates but it would take way too 
long to come up with it, so the actual ranking a voter submits is 
similar but not identical to the true ranking. If the perturbation from 
this noise comes out just right, you could get an accidental cycle.