[EM] Dodgson and Kemeny "done right"?

[Warren Smith]

Dodgson and Kemeny done right (F.W.Simmons)
---------Warren D. Smith, Sept 2011----------------------

Simmons claims he had posted something called "Dodgson done right"
which gets around the problem that with Dodgson voting it is NP-hard
to find the winner, and supposedly Kemeny has a similar fix.

I failed to find his post, but reading between the lines am attempting
to try to determine what Simmons probably had in mind by reverse
engineering, and/or the fact I had similar thoughts of my own a long
time back.

DODGSON:
votes are rank-orderings of the N candidates.
Output ordering: the one such that the smallest total
candidate motion (distance moved, summed over all
candidates on all ballots) is required to convert the input orders
into the output order.

KEMENY:
votes are rank-orderings of the N candidates.
Output ordering: the one with the fewest total number of
disagreements with the input orders about candidate-pairs (summed over
all candidate-pairs on all ballots)

ATTEMPT TO REPAIR/REDEFINE:
Make the ballots instead be range-voting style RATINGS ballots.
Output now also is a ratings-style "ballot."
   For ratingified Dodgson, output is a ballot with
minimum total candidate motion required to convert all
the input ballots into the output ballot (total distance traveled
along the ratings axis, summed over all candidates and all ballots).
  For ratingified Kemeny, output is the ballot with
minimum total 2-candidate travel distance summed over all
candidate-pairs on all input ballots, where that candidate pair has to
travel along the ratings axis so instead of their original directed
separation, they now have the same separation (in same direction) as
the output ballot.

THEOREM:
Ratingified Dodgson is no longer NP-hard to find, in fact it is easy,
in fact it is just this: each candidate's output score is the median
of his input scores.
PROOF: Easy.

REMARK:
If instead we were minimizing the sum of the SQUARES of the travel
distances, then ratingified Dodgson would just become average-based
range voting, the output score is the mean of that candidate's input
scores.

THEOREM:
Ratingified Kemeny is by this definition the same thing as ratingified Dodgson.

-- 
Warren D. Smith
http://RangeVoting.org  <-- add your endorsement (by clicking
"endorse" as 1st step)