[EM] Generalizing "manipulability"

[Juho Laatu]

OK, roughly agreed.

Some problems that I had:

- 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.)

- Also Condorcet is *slightly*
vulnerable to "irrelevant nominees".
Imagine an election with 100 candidates
from one party and voters that prefer
to mark only a limited number of
candidates in the ballot.

Juho


--- On Sun, 18/1/09, Steve Eppley <SEppley at alumni.caltech.edu> wrote:

> From: Steve Eppley <SEppley at alumni.caltech.edu>
> Subject: Re: [EM] Generalizing "manipulability"
> To: election-methods at 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
> <jlundell at 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
> >>>       
> >> <jlundell at 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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> > 
> >   
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