# [EM] Generalizing "manipulability"

Juho Laatu juho4880 at yahoo.co.uk
Mon Jan 19 14:46:02 PST 2009

```--- On Mon, 19/1/09, Jonathan Lundell <jlundell at pobox.com> wrote:

> On Jan 18, 2009, at 5:13 PM, Juho Laatu wrote:
>
> > --- On Mon, 19/1/09, Jonathan Lundell
> <jlundell at 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
> >    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.

Yes, this is neat in the sense that
there is no need to explain what the
sincere opinion of the voter is and
how the strategic vote will be found.

A definition that would cover also
Approval and other methods with
simple ballots at one go would be
nice too.

Although it is sometimes difficult
to say what a sincere vote in
Approval is (could be e.g. to mark
all candidates that one approves) I
think it is quite natural to assume
that each voter has some preferences
(order), and that strategies mean
deviation from "simply voting as one
feels and not considering the
technical details of the method, the
impact of how others are expected to
vote and how one could get better
results out" (by e.g. voting or
nominating candidates in some
particular way).

Juho

```