Nondictatorial & Nonmanipulable axioms (was Re: York's version o

[Bruce Anderson]

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On Apr 29,  3:07pm, Steve Eppley wrote:
> Subject: Nondictatorial & Nonmanipulable axioms (was Re: York's=
 version...
> Bruce Anderson wrote:
> >According to the academically well-known Gibbard-Satterthwaite
> >theorem, any voting method essentially either is dictatorial or is
> >not strategy proof (i.e., if it is not dictatorial, then it can be
> >manipulated).
>
> Isn't this an oversimplification?

It is a simplification, that=92s why the word =93essentially=94 is included.=
  I 
paraphrased Saari here.  The direct quote is as follows:

=93The problem is to determine whether there exists a strategy proof=
 procedure.  
This issue has been resolved in a general setting by Gibbard and=
 Satterthwaite, 
where, essentially, they proved the stunning conclusion that if at least=
 three 
choices can be chosen, then either the procedure is dictatorial, or it is=
 not 
strategy proof -- it can be manipulated.=94
Saari, D.G., =93Geometry of Voting,=94 Springer-Verlag, 1994, page 262.

While I have studied Arrow=92s Impossibility Theorem at considerable length,=
 I 
have only quickly skimmed the literature concerning the Gibbard-Satterthwait=
e 
Theorem.  However, based on what I have read, I think that Saari=92s=
 statement is 
a fair characterization of that theorem.

> The theorem is derived from Arrow's theorem.  The two axioms you
> mention above (Nondictatorial and Nonmanipulable) are only two of
> five.  The theorem says that all five axioms taken together are
> inconsistent.  So it's possible for a voting method=20to satisfy the
> above two as long as it violates at least one of the other three.

True; but see, for instance,
Kelly, J.S., =93Social Choice Theory,=94 Springer-Verlag, 1988, page 104,
for some examples of voting rules that violate some of the other three.  As=
 
Kelly says:  =93Of course, these rules are just terrible.=94

> I'm sure Bruce has a much better understanding of these axioms than
> I do.  As I understand it, the Nondictatorial axiom is another one
> of these academic yes/no criteria: "Is there a possible scenario
> where some voter's ballot doesn't count?"  The existence of such a
> scenario doesn't really say much about how good or bad a voting
> method is in normal scenarios.  For example, in a large Condorcet (a
> "dictatorial" method) election, who can be said to be dictating?

I don=92t agree with the above paragraph.  In fact, I think that it is, in a=
 
sense, exactly the other way around.  It=92s not the non-dictatorial axiom,=
 but 
rather the non-manipulable axiom that is one of the academic yes/no criteria=
 of 
the form:  =93Is there a possible scenario in which ...=94  Conversely, the=
 
non-dictatorial axiom applies directly to voting methods, and its=
 prohibition 
must hold for all scenarios.  Further, according to every definition I have=
 seen 
here (or elsewhere), Condorcet=92s voting method does not fail the=
 non-dictatorial 
axiom; instead, Condorcet=92s voting method fails the non-manipulable=
 axiom.

> And the Nonmanipulable axiom doesn't appear to offer much value
> either.  If we want voters to matter, it means the method must let
> them choose how they will vote--they can't be forced to vote their
> sincere preferences since there's no way to know their sincere
> preferences.  If someday technology advances to the point where
> voters' sincere preferences can be accurately and reliably obtained
> (by mind-reading, truth serum, whatever) we'll be able to solve this
> conundrum.  In the meantime, a voting method could violate both
> axioms and still be the best method.=20

I doubt that many people would agree that a voting method that violates=
 (both 
the non-manipulable axiom and) the non-dictatorial axiom would, now or in=
 the 
future, be considered to be the best method.

> The Nonmanipulability axiom is closely related to Arrow's axiom that
> voter preferences are expressed only as rankings, disallowing the
> input of weightings (a.k.a. ratings, a.k.a. point assignments) by
> the voters.  If the voting method looks at voters' weightings, it
> creates strong incentives for voters to express strategic weightings
> instead of "sincere" weightings.  Mike York has claimed without
> proof that his Elimination and Renormalization method is much better
> than other weight methods at reducing these incentives.

Since rankings are a special case of weightings, and since the=
 non-manipulable 
axiom is one of the =93does their exist a case in which ...=94 types of=
 axioms, I 
would guess that it would be quite easy to construct a manipulable example=
 for 
York=92s method by starting with a manipulable example for Nanson=92s=
 method, which, 
by the Gibbard-Satterthwaite Theorem, must exist.  As I understand it, 
counterexamples for these =93does their exist a case (i.e., a set of voter=
=92s 
ballots) in which ...=94 types of axioms are usually constructed by limiting=
 the 
voter=92s options on the ballots, as in Approval voting, not by expanding=
 them.

> I provided a Nader/Clinton/Dole example showing ...
>
> --Steve
>-- End of excerpt from Steve Eppley

Again, since the non-manipulable axiom is one of the =93fails if there=
 exists a 
case in which ...=94 types of axioms, giving one (or even all but one) case=
 in 
which the method =93passes=94 means nothing.  In order to satisfy the axiom,=
 the 
method must =93pass=94 every case.

Bruce


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