[EM] pairwise, fairness, and information content

Richard Moore rmoore4 at cox.net
Sat Aug 17 10:37:36 PDT 2002

Craig Carey wrote:

 > Hopefully the public definition of Monotonicity allows changes to any
 > number of 'ballot papers' (preference lists) but it requires that no
 > paper change except for having the preference of a given candidate
 > either [also given beforehand]:
 > * move to the left and towards the 1st preference (this include making
 >    the preference appear on a paper), xor
 > * move away from the first preference (and this includes the cases
 >    where the preference drops off the end).
 > Your definition did not keep the other papers unchanging.

Easily fixed, just change "Replacing a ballot" to "Replacing
only one ballot".  Coverage for replacing multiple ballots
(with all replacements favoring the same candidate) is trivially
shown by induction. There's no impact on the proof.

 > Also your rule is against the style of STV


 > in that gets the treatment
 > of coalitions handled wrongly. For exampl, (and this is a rough
 > example with the other papers missing):

Without commenting on the validity of this extension, I'll
note that coalitions have no relevance to a theorem about 
two-candidate pairwise comparison.

 >  >I wonder what "perfect method" would fail to honor the unanimous
 >  >choice of the voters? What "perfect method" would give negative 
 >  >to a preference on any ballot?
 >  >
 > "honour the unanimous" is attempting to introduce a rule and so it
 > would seem to be able to remove some other rule.

Er, that isn't even a proper objection. A rule is rejected just
because it is a rule?

 > ----------------------------
 >  >3. Permuting the ballots shall not change the result. This eliminates
 >  >methods that arbitrarily weight ballots differently. With this
 >  >restriction, it is not necessary to represent the ballots as an
 >  >ordered set; it is sufficient to know the total number of ballots for
 >  >each classification.
 > ----------------------------
 > It apparently says this:  If candidate A wins the 1st then candidate A
 > wins the 2nd:
 >     (1st)   (A B C D E F G)     <-- an added paper to unspecified 
 >    <--->
 >     (2nd)   (B C D E F G A)

That's just a blatant misinterpretation. Permuting the preferences on 
a ballot is not the same as permuting the ballots.

 >  >Perhaps you would like to specify a suitable replacement for this
 >  >criterion, but I am not willing to simply drop it.
 >  >
 > No, it vanished by itself. It is also an 'undesirable' rule in 2 
 > elections. A hard to explain rule that voters would not see as 
 > their fairness interests since it is incompatible with
 > truncation resistance.

Ignoring it won't make it vanish by itself. It has everything to
do with fairness and nothing to do with truncation.

 > Why not solve the problem rather than pursue something that it is not
 >   defined and presumably not desired: "correlation".

There seems to be no basis for that presumption, especially if we want 
to know something about the subject you brought up, information content.

 >  >>  Richard still has not admitted that there is no
 >  >> need to use pairwise comparing. It is not in the text above so 
 >  >> 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
 > We ought reject it.

Your basis for rejecting it -- that it had no relevant information --
was just disproven. Do you want to introduce another basis or
are you just making a proclamation?

 >  >What I *have* been saying is that you are wrong when you say pairwise
 >  >comparison contains no information about who should win an election,
 > I am uninterested in correlations. If they are probabilities then they
 > correspond to hypervolumes which people are not interested in. 
 > write about that here, but get hindered when I ask for the formulae.

Correlations aren't always about probabilities. In this case they are 
not; I never brought in the concept in the proof I gave.

 > "Tests" of "correlation" ?. If there is an axiom there then write it
 > up as an axiom. Previously it was used to advance the idea that
 > no matter how useless pairwise comparing us, it accidentally
 > correlates with my theories and so it contains information and thus
 > pairwise comparing is not rejected. I didn't realize it was a rule too.

It's not a "rule", it's a property and I already specified the test 
for it.

Craig has not raised any sustainable objections, except for the need 
for a minor change to the wording I gave for monotonicity, and that 
has no bearing on the information issue. The rest of that post was 
well off-topic, and/or just a plain emotional rant, so I'll stop here.

  -- Richard

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