[EM] Condorcet methods - should the cycle order always determine the result order? (Toby Pereira)
Forest Simmons
fsimmons at pcc.edu
Thu Nov 6 15:18:59 PST 2014
Kristopher, Toby, and all,
Ordinal based methods"Done Right" require two things:
(1) a clone proof, monotonic way of converting rankings into ratings, and
(2) a cardinal ratings analog to the ordinal based method.
(1) One way to accomplish the first requirement is by way of any clone
free, monotonic lottery L, like the random favorite lottery. Convert the
rankings from an ordinal ballot B to ratings in two steps as follows:
(i) For each candidate X, let p(X) be the probability that lottery L would
not elect a candidate ranked ahead of X on ballot B. (ii) On each ballot
normalize these probabilities to ratings with an affine transformation so
that the extreme scores are the same for every ballot, say zero and 100.
Now for (2):
Since Borda elects the candidate with the best rank sum, the Borda analog
is the method that elects the candidate with the best ratings sum, i.e.
"Score." So Borda Done Right is just Score applied to the converted
rankings i.e. to the ratings that are obtained from the rankings by the
above conversion process.
Since Bucklin elects the candidate with best median rank, "Bucklin Done
Right" elects the candidate with the highest median rating. Ties can be
broken a la Majority Judgment, but note that ties are not as likely in this
new context, because ratings that result from the conversion process
described above will result in much greater resolution (i.e. number of
distinct rating levels).
Notice, also, that the conversion step (1) above opens up the possibility
of applying ratings based PR methods while using the more traditional (in
the context of PR) ordinal ballots as inputs. Voters vote ratings or
rankings, according to their preferences. Then the rankings are converted
to ratings so that a (possibly) superior ratings based PR method can be
applied.
For more examples search the archives for the original "done right"
messages.
Forest
On Tue, Nov 4, 2014 at 11:56 PM, Kristofer Munsterhjelm <
km_elmet at t-online.de> wrote:
> On 11/04/2014 11:46 PM, Forest Simmons wrote:
>
>> Toby,
>>
>> You mentioned Kemeny. The very purpose of Kemeny is to determine a
>> "social order," namely the one that minimizes the average "distance"
>> from that order to the ballot orders..
>>
>> The trouble with Kemeny is that the choice of metric for the "distance:"
>> is clone dependent: changing the size of a clone set changes the number
>> of transpositions of order to move a candidate past that set.
>>
>
> If you change "sum" (or average) to leximax, then that problem goes away
> and you get Ranked Pairs, right?
>
> Can River also be formulated as an optimization problem?
>
> However, if cardinal ratings (e.g. score ballots) are used, then clone
>> independent metrics can be substituted for the Kemeny distance. I once
>> posted a message to this list describing a clone free technique for
>> converting a set of ordinal ballots into a set of ratings Then based on
>> those ratings it was possible to define "Kemeny Done Right," Dodgson
>> Done Right," and "Borda Done Right." Of these three "done right"
>> methods, only the latter fails the Condorcet Criterion.
>>
>
> Does altering Kemeny in this way fix its vulnerability to cloning? I'd
> imagine that any way of turning rankings into ratings would make the
> ratings depend on the candidates in some way; otherwise, you could bolt
> that onto, say, Range, and get a deterministic ranked method that passes
> IIA (which we know is impossible).
>
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