From: Steve Eppley <sepp...@alumni.caltech.edu>
Subject: Re: [EM] Generalizing "manipulability"
To: election-meth...@electorama.com
Date: Sunday, 18 January, 2009, 7:56 PM
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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