[EM] pairwise, fairness, and information content
Richard Moore
rmoore4 at cox.net
Thu Aug 15 23:44:44 PDT 2002
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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