On Jan 18, 2009, at 9:56 AM, Steve Eppley wrote:
Manipulability by voter strategy can be rigorously defined without
problematic concepts like preferences or sincere votes or how a
dictator would vote or or how a rational voter would vote given
beliefs about others' votes.
I appreciate the formalism, and I think we're on the same page. I
note, though, that concepts like preferences or sincere votes are
implicit in the formalism, or at least its consequences, indeed as you
mention below ("… vi is consistent with the voter's sincere order of
preference").
Your second definition requires a little work, or at least we qualify
ballot admissibility to be more restrictive than the voting rule
itself might be. For the purposes of applying the second definition to
approval, we need to rule out (for purposes of the definition) ballots
approval all or none of the alternatives X. Such a ballot satisfies
(3), but it makes approval trivially manipulable (though it's
manipulable anyway).
Likewise, for the generalization to apply to ranked methods, we need
to restrict truncation, perhaps by adding a condition that vi rank
C(v) and better. But that introduces difficulties for the generalizing
to approval.
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.
That definition works assuming all possible orderings of X are
admissible votes. I think it works for Range Voting too (and Range
Voting can be shown to be manipulable). The following may be a
reasonable way to generalize it to include methods like Approval
(and if this is done then Approval can be shown to be manipulable):
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 and some ordering o
of X
such that all 3 of the following conditions hold:
1. v'j = vj for all j in N-i.
2. o ranks C(v') over C(v).
3. For all pairs of alternatives x,y in X,
if vi ranks x over y then o ranks x over y.
The idea in condition 3 is that vi is consistent with the voter's
sincere order of preference. For example, approving x but not y or
z is consistent with the 2 strict (linear) orderings "x over y over
z" and "x over z over y." It's also consistent with the weak (non-
linear) ordering "x over y,z." Approving x and y but not z is
consistent with "x over y over z" and "y over x over z" and "x,y
over z." Interpreting o as the voter's sincere order of preference,
condition 2 means the voter prefers the strategic winner over the
sincere winner.
Another kind of manipulability is much more important in the context
of public elections. Call the voting method "manipulable by
irrelevant nominees" if nominating an additional alternative z is
likely to cause a significant number of voters to change their
relative vote between two other alternatives x and y, thereby
changing the winner from x to y. We observe the effects all the
time given traditional voting methods. It explains why so many
potential candidates drop out of contention before the general
election (Duverger's Law). It explains why the elites tend not to
propose competing ballot propositions when asking the voters to
change from the status quo using Yes/No Approval. I expect this
kind of manipulability to be a big problem given Approval or Range
Voting or plain Instant Runoff or Borda, but not given a good
Condorcet method.
The reason manipulability by irrelevant nominees is more important
than manipulability by voter strategy is that it takes only a tiny
number of people to affect the menu of nominees, whereas voters in
public elections tend not to be strategically minded--see the
research of Mike Alvarez of Caltech.
Regards,
Steve
--------------------------------------------------------------
On 1/17/2009 10:38 PM, Juho Laatu wrote:
--- On Sun, 18/1/09, Jonathan Lundell <jlund...@pobox.com> wrote:
On Jan 17, 2009, at 4:31 PM, Juho Laatu wrote:
The mail contained quite good
definitions.
I didn't however agree with the
referenced part below. I think "sincere"
and "zero-knowledge best strategic"
ballot need not be the same. For example
in Range(0,99) my sincere ballot could
be A=50 B=51 but my best strategic vote
would be A=0 B=99. Also other methods
may have similarly small differences
between "sincere" and "zero-knowledge
best strategic" ballots.
My argument is that the Range values (as well as the
Approval cutoff point) have meaning only within the method.
We know from your example how you rank A vs B, but the
actual values are uninterpreted except within the count.
The term "sincere" is metaphorical at best, even
with linear ballots. What I'm arguing is that that
metaphor breaks down with non-linear methods, and the
appropriate generalization/abstraction of a sincere ballot
is a zero-knowledge ballot.
I don't quite see why ranking based
methods (Range, Approval) would not
follow the same principles/definitions
as rating based methods. The sincere
message of the voter was above that she
only slightly prefers B over A but the
strategic vote indicated that she finds
B to be maximally better than A (or
that in order to make B win she better
vote this way).
Juho
Juho
--- On Sun, 18/1/09, Jonathan Lundell
<jlund...@pobox.com> wrote:
The generalization of a "sincere" ballot
then
becomes the zero-knowledge (of other voters'
behavior)
ballot, although we might still want to talk about
a
"sincere ordering" (that is, the sincere
linear
ballot) in trying to determine a "best
possible"
outcome.
----
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