Hi,
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.
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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