On Jan 18, 2009, at 5:13 PM, Juho Laatu wrote:
--- On Mon, 19/1/09, Jonathan Lundell <jlund...@pobox.com> wrote:
- Why was the first set of definitions
not good enough for Approval? (I read
"rank" as referring to the sincere
personal opinions, not to the ballot.)
"vi ranks", and vi is by definition the ballot.
That's why the second
definition introduces o.
OK. I should say that is the way I'd
like to read it.
I'd like to take another shot at that. Steve's first definition:
Let X denote the set of alternatives being voted on.
Let N denote the set of voters.
Let V(X,N) denote the set of all possible collections of admissible
votes regarding X, such that each collection contains one vote
for each voter i in N. For all collections v in V(X,N) and all
voters i in N, let vi denote i's vote in v.
Let C denote the vote-tallying function that chooses the winner
given a collection of votes. That is, for all v in V(X,N), C(v) is
some alternative in X.
Call C "manipulable by voter strategy" if there exist two
collections
of votes v,v' in V(X,N) and some voter i in N such that both of
the following conditions hold:
1. v'j = vj for all voters j in N-i.
2. vi ranks C(v') over C(v).
The idea in condition 2 is that voter i prefers the winner given the
strategic vote v'i over the winner given the sincere vote vi.
This definition is stronger than *requiring* that vi be any particular
ordering--in particular i's sincere preferences. That's very neat.
Notice also that we get away with it because the ballot in this case
is expressive enough to represent i's sincere preference ranking.
That's not true for an approval ballot, which is why the second
definition needs to introduce a separate preference order o.
Finally, the definition says nothing about how voter i might go about
*finding* v'i, or even how to discover for any particular ballot
profile whether v'i exists.
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