[EM] pairwise, fairness, and information content

[Richard Moore]

Craig Carey wrote:
> The definition of monotonicity was wrong. In general (i.e. for some
> number of winners and candidates), all 3 rules ought be rejected for
> failing a perfect method.

Hmm, I wonder what was wrong about the definition of monotonicity. It 
certainly called for a never-negative response to any single ballot 
substitution. Perhaps there is a miswording somewhere; I didn't spot 
one on my final proofreading though. Of course, monotonicity in the 
pure mathematical sense would also allow for methods that have a 
never-positive response to a ballot substitution; but by convention 
that's not what we mean when we talk about election methods being 
monotonic. In fact, all the second case does is reverse the 
categorization of the ballots, so that what we wished to count as an 
"A>B" ballot is effectively a "B>A" ballot and vice versa.

I wonder what "perfect method" would fail to honor the unanimous 
choice of the voters? What "perfect method" would give negative weight 
to a preference on any ballot?

The third criterion I gave might be tighter than necessary from a 
fairness standpoint, and I don't have a proof for whether or not it is 
possible to construct a method that violates it, meets the other two 
criteria, and has no correlation to pairwise. At any rate, removing 
the third criterion would allow for arbitrary treatment of ballots. 
Perhaps you would like to specify a suitable replacement for this 
criterion, but I am not willing to simply drop it.

> Above you wrote, "as often as".
> 
> I request the formula that is returning the probability value that
> text is apparently involved in.

Sorry if I caused a misunderstanding. I wasn't speaking of 
probabilities. I was using "as often as" to indicate a one-to-one 
correspondence between objects in two categories, but if you prefer to 
think probabilistically, then the equivalent would be a uniform 
distribution.

My loose use of the phrase "as often as" could be applied for instance 
in a statement like, "Integers are even as often as they are odd."

> I solved the 2 candidate method easily:
> 
> A    a0
> B    b0
> 
> A fair but biased solution is:
> 
> (q < a0/(a0+b0)) implies (A wins)
> (b0/(a0+b0) < q) implies (B wins)

I didn't quite follow that until I read your second message, which (I 
believe) corrected a typo to "> q". With that correction, it passes 
the requirements I listed. It is just what my proof anticipates. (It 
isn't really biased in the sense that I use the word; rather it just 
has a higher threshold for making an "A over B" or "B over A" 
decision.) If this correction is what you intended, then the method 
agrees with pairwise whenever the threshold is met (for 1/2 <= q < 1), 
and produces no decision whenever the threshold is not met. If 0 < q < 
1/2, it agrees with pairwise whenever only one candidate meets the 
threshold, and makes no decision when both candidates meet the 
threshold. That means that there is a correlation between this method 
and pairwise.

It's a good example, but if you meant to supply a counterexample, this 
isn't one.

If you meant "(b0/(a0+b0) > (1-q) implies (B wins)", then it still 
meets all three of my requirements. In this case it is biased if q != 
1/2. As I said, my proof did not require M to be unbiased: Lo, this 
method also correlates with pairwise. It will disagree with pairwise 
some of the time, but for each case of disagreement there will be more 
than one case of agreement.

>  Richard still has not admitted that there is no
> need to use pairwise comparing. It is not in the text above so Richard
> either is wrong or will be expecting that the text above is wrong,
> unless that dictator idea somehow contradicts.

We really do have a communications problem here! I haven't said there 
*is* a need to use pairwise comparing. The closest I came to saying 
that was in my last post, where I wrote that I believe "P plus a 
random tiebreaker" to be the best method for getting a ranking in the 
two-candidate case. That's a far cry from saying that I believe the 
method *must* be used, or that I believe it is still the best method 
for more than two candidates. Certainly your method above is one 
possibility, though it might be rejected by many organizations for 
practical reasons.

What I *have* been saying is that you are wrong when you say pairwise 
comparison contains no information about who should win an election, 
unless you want to completely disregard one or more of the criteria I 
listed. Of course you are free to disregard them for your own purposes 
but I have no interest in the sort of method that could result, and 
with the exception of the IRV people (person?), I doubt anyone else on 
this list will be interested, unless only from an academic standpoint.

And I'm not sure why you still think "that dictator idea" has any 
bearing on the matter. No sense beating a red herring after it's dead. 
Once more for the record: A counterexample would have to fail the 
correlation test and pass requirements 1 through 3. Dictatorship 
passes the correlation test (weakly), but fails requirement #3 since 
we need to identify the dictator's ballot. If we alternately define 
dictatorship so that the dictator's choice isn't anywhere in the set 
of ballots, then it fails requirement #2.

  -- Richard

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