[EM] Generalizing "manipulability"

Jonathan Lundell jlundell at pobox.com
Sun Jan 18 16:44:24 PST 2009 

On Jan 18, 2009, at 4:11 PM, Juho Laatu wrote:

> 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.)

"vi ranks", and vi is by definition the ballot. That's why the second  
definition introduces o.

>
>
> - 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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