[EM] Generalizing "manipulability"

[Jonathan Lundell]

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 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.

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.