[EM] pairwise, fairness, and information content

[Richard Moore]

James Gilmour wrote:
> Maybe there's something I'm missing here, but I just cannot see
 > "the issue" with 'two candidates, one winner' elections. I'll
 > freely admit most of the maths, the complex equations and the formal
 > logic statements (to say nothing of the jargon) go straight over my
 > head, but just what is the problem?

James,

Thanks for your observations.

There are two issues. They are separate, but somehow they have become 
intertwined in Craig's mind.

First is the notion Craig submitted in response to one of my posts. 
Craig believes, or wants to believe, that pairwise results do not have 
any information relevant to the "right winner" of an election. Craig 
could not present an argument to support this notion, other than that 
pairwise comparing leads to "paradoxes". When I pointed out that 
cycles are not paradoxes, Craig conceded that "paradox" is just a 
word, thus taking any steam out of his argument, yet he still resists 
any challenge to his belief.

I suspect the reason he maintains this belief is related to the second 
issue, which is the value of pairwise methods vs. other methods. I 
think he fears that if we accept that a pairwise result contains valid 
information, then we will be forced to accept pairwise methods as the 
only valid methods -- a non sequitur.

Pairwise information exists, but it is not perfect. If it were 
perfect, we would never have cycles: If A beats B and B beats C, then 
A would have to beat C as well, in a "pairwise is perfect" world. But 
imperfect information allows that sometimes C will beat A, 
contradicting the evidence of the other two results. In other words, 
at least one of the three results, A>B, B>C, and C>A, is wrong.

The idea that cycles are "paradoxes" is rooted in the false notion 
that if pairwise comparing contains information, then it must contain 
perfect information. It is taking majoritarianism to the extreme view 
that "the majority is *always* right".

Cycles are a fact of life for 3 or more candidates when full rankings 
are provided on the ballots. They are dealt with in several different 
ways: elimination, which removes one or more candidates at a time (and 
hence removes the pairwise information, whether "good" or "bad", 
associated with those candidates); cycle-breaking (which removes 
information selectively until there are no conflicts, while keeping 
the candidates in, so that we don't remove multiple chunks of 
information at a time; ideally, such a method would try to minimize 
the amount of information it must remove to break the cycles); and 
calculating a score for each candidate based on preference information 
or pairwise information (the best known example being Borda). This is 
all related to the first issue: How much information is potentially 
thrown away in each method? Even the pairwise cycle-breaking methods 
discard information, so I don't see the "information" issue compelling 
us to accept those methods. Methods in the third class -- not usually 
thought of as pairwise though many of them could be classified that 
way -- typically discard the least information, but still must be 
judged on their other merits and problems; e.g., Borda is notorious 
for strategy problems.

Of course, you can avoid cycles altogether by only allowing two-tiered 
ranking of the candidates on the ballots (as in approval voting) or by 
using cardinal ratings.

 > I also fail to see the relevance of this boundary condition, ie "two
 > candidates, one winner", to the resolution of the real issues
 > that do arise as soon as you move away from this extreme, eg  move
 > to "three candidates, one winner".  So why is "two candidates,
 > one winner" being discussed in the context of the more general (and
 > more common) problem?

 > "Three (or more) candidates, one winner" elections present a number
 > of different problems, but still the best you can do is to
 > guarantee representation to only half of those who voted.  Of
 > course, your method of voting and counting may do much worse than
 > that.  And there will be different outcomes, depending on the
 > methods you use.  And there will be debate about which outcome
 > "best" reflects the "wishes" of the voters. And there will be debate
 > about how those "wishes" are to be assessed and how "best" is to be
 > defined.

Here's one thought I recently had on how pairwise information might be 
applied to the bigger issue:

Pairwise information could be used to evaluate each prospective 
ranking of the candidates. For instance, if we have four candidates, 
and the pairwise results

a>b  a>c  d>a  b>c  d>b  c>d

then the full ranking a>b>c>d is supported by four pieces of evidence 
(the a>b, a>c, b>c, and c>d results), and contradicted by two pieces 
of evidence (d>a and d>b). The ranking d>a>b>c is only contradicted by 
the c>d result.

This suggests a method: The strengths of each pairwise win could be 
calculated based on the numbers of votes for each candidate according 
to some (for now unknown) formula, and all the evidence for and 
against each prospective ranking could be combined to get a score for 
that ranking. For a single winner election, the top candidate would be 
the one that has the highest sum of the resulting scores from all the 
rankings that place that candidate first. This summation technique 
could be extended to any number of winners. I find this idea 
intriguing, but maybe to computationally intensive to be practical. I 
also suspect it won't be proportionally representative.

  -- Richard

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