Arrow's axioms (was Re: [EM] Re: [Fwd: Election-methods digest, Vol 1 #525 - 9 msgs])

[Forest Simmons]

On Wed, 3 Mar 2004, Steve Eppley wrote:

>
> I consider Arrow's axioms justifiable.  In the decades
> leading up to Arrow's theorem, economists and social
> scientists had struggled in vain to find a good way to
> compare different individuals' utility differences (known
> in the literature as the problem of "interpersonal
> comparison of utilities") in order to be able to calculate
> which outcome is most utilitarian.  That is, they were
> interested in being able to sum for each alternative the
> utility of that alternative for each voter, so they could
> elect the alternative with the greatest sum.  By Arrow's
> time, they'd learned that, lacking mind-reading
> technologies, they couldn't elicit cardinal utilities that
> could be compared between individuals, for instance to
> compare the utility difference between your "100" candidate
> and your "0" candidate to the utility difference between my
> "100" candidate and my "0" candidate.  Simply summing our
> reported numbers, which don't have units (such as dollars)
> attached, would not help them find which alternative had
> the greatest utility.  If each voter is constrained to
> assign numbers within a given range, such as 0 to 100, then
> the sum would not be the utilitarian sum.  Maybe these sums
> aren't worthless, but they need careful scrutiny.

I have three comments on this:

(1) Cardinal ratings don't have to be utilities.

(2) They don't have to be summed.

(3) Rankings are not the only alternatives to ratings.

Just to scratch the surface of the possibilities:

(1) Probabilities are unitless numbers that can be used as cardinal
ratings, for example the probability that candidate X would represent your
view on a random decision in his course of duty. Probabilities (and
probability distributions) can be averaged, multiplied, raised to powers,
convolved, Laplace Transformed, Fourier Transformed, etc.

(2) Cardinal Ratings make the most sense as inputs to various versions of
DSV (declared strategy voting). The best DSV methods just use the ratings
to determine which candidates should be approved on behalf of the voter of
the CR ballot, so no utilities are summed.  In fact, in the best DSV
methods it doesn't matter if all of the ballots use the same scale: above
mean candidates will still be above mean after any linear transformation.

CR ballots are also better than merely ranked ballots for voter space
methods, and methods that try to reconstruct the relative positions of the
voters and candidates in issue space, i.e. issue space methods.


(3) Dyadic Approval ballots are a compromise between CR ballots and
rankings.  They allow the voter to rank the candidates and express the
relative strength of the preferences without assigning numerical values to
those strengths. In other words it is possible to give a partial order to
the strengths of the preferences without using full blown cardinal
ratings.

Thus Nader>>Kerry>Bush and Nader>Kerry>>Bush are distinct possibilities
with dyadic ballots that would not be distinguished with ordinary ranked
ballots.

The popular ranked-ballot-with-approval-cutoff is an even simpler
improvement over merely ranked ballots. Indeed it is the first step
towards dyadic approval ballots.

As near as I know, the only deterministic method that satisfies
neutrality, anonymity, and the strong FBC (instrumentally as opposed to
merely expressively) is a method that uses additional information beyond
the rankings. [It allows voters to augment their ranked ballot with a list
of "approved pairs."  The pairwise winner (according to the ranked
ballots) of the two candidates in the most approved pair (according to the
lists of approved pairs) is the method winner.]

As I say, these examples barely scratch the surface of a neglected and
relatively unexplored territory.  The field is wide open, so don't let
Arrow or the econometricians slow you down!

Forest