In article <[EMAIL PROTECTED]>,
Donald Burrill <[EMAIL PROTECTED]> wrote:
>On Sat, 22 Dec 2001, Ralph Noble asked:
>> How would you have done this?
>> A local newspaper asked its readers to rank the year's Top 10 news stories
>> by completing a ballot form. There were 10 choices on all but one ballot
>> (i.e. local news, sports news, business news, etc.), and you had to rank
>> those from 1 to 10 without duplicating any of your choices. One was their
>> top pick, 10 their lowest. Only one ballot had more than 10 choices,
>> because of the large number of local news stories you could choose from.
>> I would have thought if you only had 10 choices and had to rank from 1 to
>> 10, then you'd count up all the stories that got the readers' Number One
>> vote and which ever story got the most Number One votes would have been
>> declared the winner.
>That is certainly one way of determining a "winner". But if one were
>going to do this in the end, there is not much point to asking for ranks
>other than 1, because that information is not going to be used at all.
>(Unless, of course, one uses a variant of this method for the breaking of
>ties, or for obtaining a majority of votes cast for the "winner".)
>> Not so in the case of this newspaper. So maybe I do not understand
>> statistics.
>Non sequitur. You are not discussing statistics, you are discussing the
>choice of methods of counting votes.
>> The newspaper told the readers there were several ways it could have
>> tallied the rankings.
>This is true. "Several" may be an understatement.
This problem is the type of problem which results in Arrow's
Paradox. The problem is, given the opinions of all individuals,
to form the opinion of the group. Arrow showed in his thesis
that if certain reasonable conditions are satisfied, there is
no way of doing this other than dictatorial or conventional.
Now Arrow did not consider consistency under randomization.
With this included, the proof becomes very short, and can
be found as a comment in my paper showing that self-consistent
behavior must be Bayesian. Briefly, the argument is that the
group evaluation must be a positive linear combination of those
of the individuals. However, how do we compare the scales?
Any method to do this leads to paradoxes.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
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