[EM] Condorcet vs Approval

[Forest Simmons]

Here's a simple example that I hope will be of interest to those who
believe that there can be valid distinctions in levels of approval: 

Utilities:  3     2    1    0
----------------------------------
Faction 1:  A  >  B   >>    C
Faction 2:  C  >  B   >>    A

Assume zero information from polls, etc.

Under (ordinary) Approval, B wins with 100% approval, no matter which of
the two factions is in the majority. (This would probably not be the case
in real life where there is no such thing as zero information: we always
have various amounts of information and disinformation.)

Under Condorcet, A or C wins, depending on which faction is in the
majority.

Borda Count agrees with Condorcet in this example (assuming honest
preferences).

Under both Dyadic Approval and "honest CR", the leading candidate (A or C) 
for one of the factions wins only when that faction has a two-thirds plus
majority, otherwise B wins.

This seems like a reasonable result to me ... a nice middle ground between
Condorcet and ordinary Approval, and a low stakes resolution of the
tension between those two great methods.

People that don't like this result don't seem to accept the validity of
the voters' utility judgments.  But when a whole major faction agrees on
these utilities, that consistency argues for some kind of validity.
(Perhaps there are three major issues, and each voter in the first faction
believes that candidate B does well on the first two, while each voter in
the second faction believes that he does well on the second two issues.)

On the other hand, when consistency is lacking (as it might be in some
other example), then the inconsistencies in utility levels tend to cancel
each other out statistically, and no harm done. Note that pure rankers
also give a statistical cancellation justification for ignoring strength
of preference, thereby implicitly admitting that there might be some
validity to the idea.  But their cancellation is from one election to
another (as Bart Ingles recently pointed out) whereas the cancellation we
are talking about takes place within each and every election, so you don't
have to wait for "fairness on the average".

At any rate the candidates are all being graded by the same voters, not
some by one subset, and others by another. 

For those who do view the Dyadic Approval and "honest CR" result (in the
above example) as the most reasonable, why would we prefer one over the
other?  Well, there's no strategic reason to totally disregard a
preference in Dyadic Approval, but (on the other hand) there is no
strategic incentive for honesty in "honest CR" . 

I maintain that Dyadic Approval is the highest resolution method that we
currently have that does not have strategic incentives for disregarding
preferences. It may strategically distort (or refine?) a priori utilities
in light of information from polls, but not to the point of reversing or
erasing any preference. 

Minor strategic refinements of utilities should not necessarily be
considered bad. It's like when you start to take your job for granted and
your boss tells you to produce more or find a new job. That information
changes your course one way or the other.  Either you start working harder
or else you decide it's not worth it, and start looking for a new job. The
information about costs and benefits could be valuable for improving your
course of action, one way or the other. The distortion from your
ignorantly blissful course was probably a good thing.

Anyway, let's say that being forced to weigh the costs and benefits of
approval/disapproval at each level helps "refine" (as opposed to distort) 
our utilities, since our utilities should take into account the expected
outcome of the election as well as our tiny influence on it.

By the way, in a neck-to-neck plurality race, the probability that the
election will be decided by one vote, is less than eighty percent of the
reciprocal of the square root of the number of voters. So if there are
6400 voters, there is less than one percent probability that the election
will be decided by a single vote (and that much probability only in the
case of neck-to-neck).

Forest