Not always impossible?
Mike Ositoff
ntk at netcom.com
Thu Jul 9 16:04:01 PDT 1998
I reply in several parts of this letter, a few parapraphs down:
On Tue, 7 Jul 1998 Saari at aol.com wrote:
> In a message dated 98-07-03 16:42:43 EDT, you write:
>
> >In literature, there are many different proves of the
> >impossibility of a non-manipulatable election method.
> >
> >All these proves have the same structure:
> >
> >1) Take some axioms, such that a tie is possible.
> > That means: Take some criteria, such that it is possible
> > that A > B > C > A or in other words such that
> > a) Candidate A would have been elected, if candidate C
> > hadn't run for office.
> > b) Candidate B would have been elected, if candidate A
> > hadn't run for office.
> > c) Candidate C would have been elected, if candidate B
> > hadn't run for office.
> >
> >2) Assume, that there is a tie.
>
> In a rated (as opposed to ranked) voting system, each candidate receives
> "votes" according to some method which allows them to be tallied. (Think of
> Olympic scoring as a simple example.) After the votes for each candidate is
> tallied, each candidate then receives a resulting "score". One of several
> methods can then be used to determine a winner:
> -Under some methods, the candidate with the highest score is automatically
> deemed the winner
> -Under another possible method, any and all candidates (if any) who receive a
> score above some pre-set threshold is declared the winner. Note that this
> method can possibly result in multiple winners, or possibly result on "no
> winner". This might be more applicable for parliamentary situations where the
> set of choices is unbounded.
>
> In either case, note that a "circular result" e.g. A > B > C > A cannot occur.
> Therefore such a voting method is not subject to the same style of
> "impossibility proof" as are other voting systems.
Arrow made it clear with his 1st criterion, the Structural Criterion,
that he was talking about rank-balloting systems. Did Gibbard &
Satterthwaite? That isn't the way I heard it. I haven't seen their
theorem in a book, myself, but I've been told that it refers to
methods in general. Riker demonstrated that, with any non-dictatorial
method, if self-interested voters have good enough knowledge about
eachother's preferences, then they will do what it takes to elect
the Condorcet winner. With most methods, including all point systems,
that nearly always involves the use of strategy. Methods differ,
however, in how often they require that strategy, and how
drastic a strategy is required. The methods this list recommended
to ER, and Schulze's method, minimize the amount & drasticness
of strategy needed, and the number of situations where it's needed.
I believe that's a good measure of a method's merit. Certainly
in terms of our majority rule & LO2E standards.
I haven't seen Riker's proof, but it's pretty obvious that his
claim is true, that a majority has the power to get what it wants,
if it shares a 1st choice, or even a pairwise preference. Markus
pointed that out in his most recent posting.
>
> (The above conclusion applies for the situation where each candidate is voted
> upon in an "accurate" manner - meaning that the addition of newer candidates
> will not result in a change in the votes for previous candidates. Such an
> "objective" voting system is still a difficult design problem - but perhaps
> not impossible. Plus it would be necessary to make sure that there were no
Based on things said so far, it does sound impossible.
> incentives to exaggerate or lie. So it is by no means a trivial problem to
> solve. I suspect it requires an "open ended" system with no fixed maximum
> vote value - this is a difficult area. But at least it cannot be proven
No fixed maximum? Ok, if you rate your favorite a trillion trillion,
then I'll rate my favorite a trillion trillion trillion trillion.
Whichever candidate's voters have the time to write the most trillions,
or googols, or the longest string of zeros wins.
Or maybe you're thinking of a point-counting system that awards
points as some function of the voter's point ratings. But, if it's
to be monotonic, then there has to be a monotonic relation between
how high you rate someone and how high the system rewards him.
If you suspect that a certain 2 candidates will be in a close
near-tie for 1st, then it's to your advantage to maximize the
difference in the points you give them. No rational point-counting
function can keep you from maximizing the difference in the
points the system awards them in your behalf, by maximizing the
difference in what you mark for them. This is just a particular case
of what Riker showed.
Someone did devise a point-counting function, so that absolute
extreme point ratings wouldn't be the optimal strategy (as they
are in direct point-assignment systems, such as the ones you
propose, and Plurality & Approval). His system gives to a candidate,
in your behalf a rating proportional or equal to the "standard score"
of your point rating of him--that is, the number of standard deviations
above or below the mean, where "mean" and "standard deviation" are
calculated from your complete set of point ratings. But, predictably,
from everything said so far, this doesn't prevent "exaggeration" from
being the optimal strategy. Though your optimal strategy is no longer
absolute extreme exaggeration, it's still exaggeration, and your
exaggeration strategy is calculated based on the probabilities of
various pairs of candidates being the top 2 contenders. As are the
optimal strategies of Plurality, Approval, Borda, and your direct
point-assignment systems.
Most people vote in Plurality for the candidate likely to be the
compromise they need, and it's usually quite obvious who that is.
In Approval they'd merely also vote also for everyone they like more.
If there isn't a real stand-out needed-compromise, then one has
to evaluate a formula involving probabilities of all the various
2-way ties for 1st, combined with your utility ratings. That's true
in Plurality, Approval, flexible point rating systems, Borda.
Runoff? Yes, but the probabilities become more complicated, and
there's much more arithmetic. IRO? Well let's just say, if it's
proposed in a jurisdiction you live in, oppose it with everything
you've got.
***
By the way, I've wanted to add this: Saari's -100 to 100 system
might have a strong appeal to the voting public, given their great
cynicism. The fact that their optimal strategy would be to give
-100 to everything that they wouldn't vote for in Pluralilty would
be very enjoyable to most voters. Maybe they couldn't pass up
adopting a system that would let them do that.
> impossible from the start as is apparently the case with all ranked voting
> systems.)
I hope that what we've said has made it clear that freedom from
strategy, including exaggeration strategy, is indeed impossible for
point systems.
Mike Ossipoff
>
> Mike Saari
>
>
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