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

[Bruce Anderson]

Second Try:

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's why the word "essentially" is included.  I 
paraphrased Saari here.  The direct quote is as follows:

"The 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."
Saari, D.G., "Geometry of Voting," Springer-Verlag, 1994, page 262.

While I have studied Arrow's Impossibility Theorem at considerable length, I 
have only quickly skimmed the literature concerning the Gibbard-Satterthwaite 
Theorem.  However, based on what I have read, I think that Saari's 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 to satisfy the
> above two as long as it violates at least one of the other three.

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

> 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't agree with the above paragraph.  In fact, I think that it is, in a 
sense, exactly the other way around.  It's not the non-dictatorial axiom, but 
rather the non-manipulable axiom that is one of the academic yes/no criteria of 
the form:  "Is there a possible scenario in which ..."  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's voting method does not fail the non-dictatorial 
axiom; instead, Condorcet's 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. 

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 "does their exist a case in which ..." types of axioms, I 
would guess that it would be quite easy to construct a manipulable example for 
York's method by starting with a manipulable example for Nanson's method, which, 
by the Gibbard-Satterthwaite Theorem, must exist.  As I understand it, 
counterexamples for these "does their exist a case (i.e., a set of voter's 
ballots) in which ..." types of axioms are usually constructed by limiting the 
voter's 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 "fails if there exists a 
case in which ..." types of axioms, giving one (or even all but one) case in 
which the method "passes" means nothing.  In order to satisfy the axiom, the 
method must "pass" every case.

Bruce