On 04/03/2012 07:22 AM, Richard Fobes wrote:
Conclusion 11: VoteFair ranking is consistent with the Condorcet-Kemeny
method because the Condorcet-Kemeny method does not specify an overall
ranking, nor does it specify who should win this election. (It only
specifies that choice A is the lowest-ranked choice, and this is
consistent with the VoteFair ranking result.)
Is that true? Speaking of Kemeny as an optimization method, the problem
is specified as:
Determine the transitive ordering of all candidates,
so that the sum of, for each pairwise preference X consistent with the
ordering, the number of voters agreeing with preference X, is maximized.
(At least in absence of ties)
That seems to make it pretty likely that Kemeny specifies a full
ordering. It has to, in order to calculate the score that is to be
optimized.
But anyway, I'll try to find an example where:
- VoteFair elects A,
- VoteFair has no ties in its social ordering,
- Kemeny finds another candidate X as the winner,
and
- There is no Kemeny-optimal ordering that puts A first.
Would that suffice to show that VoteFair isn't Kemeny?
Conclusion 12: VoteFair ranking calculates a fair result within the
limitations of the preference information available, and does so within
the context of the goal of maximizing the Condorcet-Kemeny sequence score.
It doesn't actually maximize that sequence score, however; it falls one
short. It does provide the same winner as one of the sequences that do,
I see that.
Thank you Kristofer for this interesting case!
This is an excellent example of the unclear (muddled) preferences that I
have referred to in other messages. It clarifies what I've said before,
which is that if there were a 50-candidate election that had this kind
of circular ambiguity throughout the candidates (which is what can make
it harder to quickly find the highest sequence score), and if VoteFair
ranking failed to find the sequence with the highest score (assuming
only one such sequence), then a runoff election between the fully
calculated Condorcet-Kemeny winner and the VoteFair ranking winner would
be difficult to predict.
You could say the same of other Condorcet methods. E.g. you could say
"If Ranked Pairs fails to find the winner according to Kemeny, the
outcome of a runoff election between the fully calculated Kemeny winner
and Ranked Pairs would be difficult to predict". Still doesn't save
Ranked Pairs from not being Kemeny, though! :-)
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