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

[wclark at xoom.org]

Forest Simmons wrote:

> 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.]

Would that be the "lesser of the two most popular evils" method? :)

It does seem an awful lot like the artificial imposition of a two-party
system on a method that otherwise wouldn't tend to encourage one.  The
types of strategies that would work well in such a method aren't
immediately obvious to me, and in fact strike me as rather complicated
(although interesting, to say the least.)

Given a random distribution of preferences and equivalent strategy
assumptions among agents in a simulation of this method (and enough
trials,) I wonder if it might not simply end up being equivalent to
picking a random pair of candidates.  In that case, it might be
interesting to note whether a simulation of this modified ranked method
produced results similar to Kevin Venzke's modified Approval/CR
simulation.

> 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!

Here's one that I've been thinking about for a while, and which might
already exist under a different name for all I know, but which I've been
thinking of as "Cardinal Condorcet:"

The basic ideas is that voters rate candidates on a 0-1 scale, and then
pairwise comparisons between candidates are used to determine a winner
(following some variety of Condorcet).  Obviously if we just used the
straight sums of ratings, then this would be equivalent to regular
Cardinal Ratings.  So, before summing the ratings for each candidate, they
have to be put through some transformation.

The first such transformation that came to mind was to simply normalize
the higher ranked of each pair (for each voter) to 1 by multiplying each
rating by the inverse of the higher value (e.g. kerry:0.5, bush:0.1 would
become kerry:1.0, bush 0.2.)  A different transformation might be to
normalize by shifting rather than scaling (e.g. we add 0.5 to each rating
in our example to produce kerry:1.0, bush:0.6.)

In the case that a voter only gives 0.0 or 1.0 values (i.e. the optimal
strategy for typical cases of standard CR) this system is equivalent to
straight Condorcet (assuming any reasonable transformation function.)  But
when voters assign values in between the minimum and maximum, some
interesting things start to happen.

Forest, could you provide some pointers to where I could find out more
about some of the methods you describe?  I found this post of yours from a
few years back, which was very interesting but still left me wanting to
read more:

http://lists.electorama.com/pipermail/election-methods-electorama.com/2001-April/005741.html

I wonder if using some combination of transformation function and CR
ballots could be used to implement Dyadic Approval.  This is almost -- but
I think not quite -- what Richard Moore seems to have been suggesting
here:

http://lists.electorama.com/pipermail/election-methods-electorama.com/2001-August/006566.html

Your response in that post, that Richard's proposed implementation didn't
capture the higher expressivity of dyadic ballots, doesn't seem like as
much of a problem if we can treat ballots like:

Nader:1.0, Kerry:0.9, Bush:0.1

as a representation of something like:

Nader > Kerry >> Bush

Any thoughts?

-Bill Clark

-- 
Ralph Nader for US President in 2004
http://votenader.org/