[EM] An axiomatic definition of "preference"

[Paul Kislanko]

I've argued before that the uses of the terms "preference" and "sincere
preference" on the list have treated the terms as "undefined terms" but have
been applied ambiguously, which results in some confusion about what when
and how they are applicable.
 
Within the context of election-methods list discussions, I suggest the
following:
 
 
  Let:
 
   V = { all eligible voters }
 
   A = { all possible alternatives }
 
   B = { all possible ballot configurations allowed by a method } 
 
   P(v,A)  = a subset of A that represents Voter v's "acceptable
alternatives" and 
   P*(v,A) = {A} - P(v,A), i.e. the complement of P(v,A) 
 
Then a method can be said to be "preference preserving" if it meets at least
both of these conditions:
 
   1. There exists at least one element of B that includes all members of
P(v,A) and no members of P*(v,A) for any member v of V
   2. No ballot which conforms to condition 1 contributes to the selection
of an alternative in P*(v,A)
 
Condition 1 is necessary, but not sufficient. It has implications regarding
the types of ballots that can be supported by a counting method.
 
Condition 2 is the one that is necessary as a framework for the "voting
strategy" discussions. If there is something about a method that requires a
member of V to choose a B that does not conform to condition 1, then that is
a measure of how how strategy-sensitive the method is.

Note that in this approach the only undefined term is "voter v's acceptable
alternatives'" - and this is acceptable because it is determined by the
individual voter, not by the method. We may not know what the term means,
but that doesn't matter because the voter DOES know, and we just acknowledge
that.
 
I strongly suspect that while both of these are necessary conditions, these
are not SUFFICIENT conditions. Any method more sophisticated than plurality
includes as its counting method some step that alters A and therefore B, and
it is the original A cross B that the voter used to provide input. 
 
So for the purpose of analyzing election methods, we can assume that a
method satisfying condition 1 can at least be aware of "sincere
preferences", and we can DEFINE "sincere preferences" to be the ballots cast
if the method meets condition 1.
 
Some methods meet condition 1 and not condition 2, and those are necessarily
NOT "preference preserving" (which may not be a bad thing, it is just a
fact).
 
I don't suggest that this is the only or best way to axiomatize the
definition of "sincere preference", but as far as I know it's the first
proposal that isn't ambiguous.
 
---------------------- 
Paul Kislanko
 
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