# [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

Good!

> 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
weight
>  >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
system
>    <--->
>     (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
candidate
> elections. A hard to explain rule that voters would not see as
protecting
> 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
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
>
> 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.
Theorists
> 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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