[EM] ruminations on ordinal and cardinal information

[Russ Paielli]

Folks,

I've been busy for several days due to a family medical crisis, so I've 
been unable to reply to the many interesting messages that have been 
posted. However, I would like to present some general ideas I've been 
thinking about. Some of them may be obvious, and most if not all are 
probably unoriginal, but I would like to lay them out anyway just for 
the record if nothing else. They will give you a clue as to what guides 
my thinking, and perhaps they will provide some insight. Please tell me 
if you disagree with anything.

We've been discussing the integration of Condorcet and Approval methods. 
  As most of you know, Concdorcet is an ordinal (ranking) method, and 
Approval is the simplest form of a cardinal (rating) method.

Ordinal methods allow the voter to express the relative preferences 
among the candidates, but cardinal methods allow the voter to express an 
"absolute" rating of the candidates. I put "absolute" in quotes because 
it is probably too strong a word. The key is that they are "less 
relative." Cardinal ratings allow the voter to express a rating of the 
candidates relative, not to each other, but to the "expected value" of 
the outcome of the election itself.

Ordinal and cardinal methods complement each other -- or so it seems to 
me. Given that, I conclude that an "ideal" election, if one exists, 
method must incorporate both ordinal *and* cardinal information.

Pure Condorcet is ordinal only, and its main deficiency is the 
possibility of cycles that must be resolved by deciding which votes to 
ignore. That can get complicated, which is a serious problem for public 
acceptability. The complexity itself is not the fundamental problem, 
however. The fundamental problem is that ordinal-only methods simply do 
not obtain cardinal information from the voters.

A pure cardinal rating system, on the other hand, allows the voters to 
rate the candidates on a continuous scale. An actual continuous scale is 
not practical, of course, but it can be approximated by a high enough 
discretization, say 0-100.

The practical problem with Cardinal Ratings, as I perceive it at least, 
is that it just doesn't "seem" right for public elections. I simply 
cannot imagine a major public election in which I am allowed to "rate" 
each candidate on a scale of 0-100 or even 0-10. Well, I can imagine it, 
but I think I can safely say that it won't happen.

The other problem with a cardinal rating system is that it provides more 
"resolution" than is strategically necessary. A wise voter will realize 
that, in a "large" election at least, the best "strategy" is to give 
each candidate either the maximum or minimum rating. This is comparable 
to "bang-bang" control, in which an actuator must be either completely 
on or completely off for best efficiency (most home heating systems, for 
example). The intermediate gradations may appeal to naive voters, but 
they don't add much, if any, real value.

In other words, Approval Voting is arguably as good as any cardinal 
rating system. If that is true, then any cardinal rating system other 
than Approval is unnecessarily complicated. Hence, an "ideal" election 
method that incorporates both ordinal and cardinal information should 
use Approval as the cardinal information. This rules out any "graded" 
system that has the voters "grading" the candidates on a scale of, say, 
A-F (as in school report cards).

OK, where do we stand at this point? I claim that an "ideal" election 
method must integrate both ordinal and cardinal information, and the 
cardinal information should be simple approval (yes/no for each candidate).

I think that's enough for now, so I will continue later.

--Russ