On 4/23/2012 12:05 PM, Kristofer Munsterhjelm wrote:
On 04/22/2012 05:07 PM, Richard Fobes wrote:
The core of the system is VoteFair popularity ranking, which is
mathematically equivalent to the Condorcet-Kemeny method, which is
one of the methods supported by the "Declaration of Election-Method
Reform Advocates."
You said there are ballot sets for which the Kemeny method and VoteFair
provides different winners. How, then, can VoteFair be /mathematically/
equivalent? You say the differences don't matter in practice, but for
the method to be mathematically equivalent, wouldn't the mapping have to
be completely identical?
First of all, in the context of a publication that is read by
non-mathematicians (which is what the Democracy Chronicles is) the word
"equivalent" does not refer to a rigorous "sameness."
Second, both methods identify the same winner, regardless of the number
of candidates, if the Smith set is not larger than 6. This
"qualification" (of the Smith set not exceeding 6) is true of every
election ever held in the United States even in municipalities that use
non-plurality methods, and is likely to be true of every election ever
held in any country using any voting method. (If you really want to
take it one step farther, it would be difficult for a small town of
voters to produce a Smith set larger than 6 even if they tried!) The
mathematical possibility of a larger-than-six Smith set is well beyond
what the readers of the article care about.
Third, the reinforcement issue -- which has no effect on which candidate
wins (if the Smith set does not exceed 6) and which no other Condorcet
method can even achieve -- is the area in which it can be said that
VoteFair ranking calculations can differ (but would rarely differ) from
the results of using the Condorcet-Kemeny method, but that difference is
too subtle to bring up in an article about basic voting concepts (vote
splitting, strategic voting, etc.).
Fourth, to repeat an important point for Adrian's sake, the cases in
which it is possible for the two methods to differ involve highly
convoluted (muddled) voter preferences that have no clear preference
pattern. To clarify this concept with an analogy, if the purpose of a
voting method were to identify the highest mountain peak, then
situations in which it is possible for the Condorcet-Kemeny method and
VoteFair ranking to identify a different winner amount to attempting to
find the highest sand dune in a desert -- which means that if the two
methods identify different sand dunes as the highest, the difference is
not significant.
Of course here in this forum we will continue to discuss the
circumstances that can cause a difference between the Condorcet-Kemeny
method and VoteFair popularity ranking, but from the perspective of
real-life elections in which the goal is to identify which candidate
wins, the two methods are "mathematically equivalent."
Richard Fobes
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