[EM] pairwise comparisons

[Juho Laatu]

> On 19 Apr 2018, at 02:19, Curt <accounts at museworld.com> 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 ten 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? We 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.

Could be possible. There could be three candidates, A, B and C. The voter could have three areas of concern, a, b and c. The voter could rank the candidates in each area as (a: A>B>C), (b: B>C>A) and (c: C>A>B). A is better than B in two areas. B is better than C in two areas. C is better than A in two areas. The voter could form his pairwise preferences based on these (number of areas based) comparisons.

> 
> 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?

You could say that the voter is somewhat confused in the sense that his thinking process has some problems. If someone would ask him if A or B should be elected, he would answer A. If he would be then asked about A and C, he would choose C. But then he might realize that he would however prefer B to C, and would be a bit confused. At this point he might realize that he should give some weights to the areas of concern, and to the pairwise preference strengths within those areas. If he does so, he would probably be able to give the candidates ratings (or personal total utility estimates) that would put them in some linear order. It is probably more common among human brains to give total preference weights (and thereby a linear order) to different candidates or other objects. But I would not rule out cyclic pairwise comparison based preferences either. Maybe you could say that probably cyclic preferences are based on limited information, and that it is usually possible to turn them to linear preferences, if one gives the thinking process some time.

> 
> 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?

Based on the story I wrote above, this process was probably more clarifying than confusing. The initial voter preferences did not really change, but there were some changes in the weights (or better accuracy) of different preferences. The voter did not compromise and did not lie. If the linear preferences are now A>B>C, the voter abandoned the earlier idea that C>A. His thinking must be something like "C indeed was better than A in two areas, but the weights of those areas, and the strengths of pairwise beatings are such that C would be a worse choice than A or B".

Note that this kind of changes in thinking might take place also when we have only two candidates, A and C, and there is thus no cycle.

> 
> 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 Smith set means about the same as the looped opinions of the voter above. If we use only pairwise preferences, preference loops are possible. The electorate is not confused nor inconsistent. All voters may have had linear preference opinions. The Smith set is an accurate depiction of actual views (as measured in this limited way, based of rankings only). This loop can not be "fixed" since rankings is all we have, and in typical (competitive) political elections it would be difficult to collect sincere ratings.

While we may assume that individual voters can usually form a well defined linear order of the candidates, benefits to the society are a more complex topic. There are many ways to measure that. One could for example measure the maximum sum of ratings (of individual voters), one could maximize the lowest rating, or one could base the method on majorities (as usual in political elections). If we must satisfy with rankings and majority decisions, the possibility of loops is unavoidable. In this situation we just need some additonal crierion on which one of the candidates (looped ones or others) is the best.

Juho

> 
> Thanks,
> Curt