[EM] Approval strategy. Approval Yee-Diagrams.

[Michael Ossipoff]

Top-Set:

I've defined the top-set already, but let me repeat the definition here:

A voter's top set is a set of candidates such that, for that voter,
electing from that set is more important than the matter of _which_ of its
candidates wins. ...and more important than the matter of which candidate
outside that set wins, if that happens.

So doing what one can do to elect from one's top-set is more important than
choosing from among that set, and more important than voting to influence
what other candidate wilns, if one does.

Because electing from the top-set is the important thing, then of course
the best Approval strategy is to approve (only) all of one's top-set.

(I've discussed agreements that voters in one wing (voters on one side of
the CWs) could make among eachother, to not approve past the CWse (expected
or evident CWs), but that's another topic.)

Existence of a Top-Set:

As I mentioned, the progressive voters always have a top-set: The
progressive candidates. The distinction & difference between the
progressives and the Republocrats is stark and obvious. even when Democrat
candidates try to sound like progressives.

Maybe the existence of top-set & bottom-set is just an artifiact of a phony
political system with disinformational media and illegitimate, probably
fraudulent, elections. But there certainly is a progressives' top-set &
bottom-set, if you believe in the current candidate-lineups.

Approval-Voting Without a Top-Set:

But what if you're participating in some Approval election in which you
don't have a top-set. That could be an election in a (fantasy
science-fiction) future in which the media are honest and open, and the
elections are legitimate, for example.

...or maybe just some other non-official Approval poll or vote, political
or otherwise.

Not zero-info:

if you don't have a top-set, and if you have a pretty good idea who the CWs
is (it isn't a 0-info election), then approve down to the CWs, and no
farther. In that way, you're voting to get the best candidate that you can
get. If the CWs is known, and everyone does that, then the CWs will win.

Zero-info:

Most people probably agree that, then, with no top-set, and with 0-info,
you want to maximize your expectation, based on the what information is
available (lbut no information is available about faction-size or
candidate-winnability).

If you have an estimate of your expectation in the election before voting,
then, if you give an approval to someome better than your expectation,
you'll raise your expectation. If you give an approval to someone below
your expectation, then you'll worsen your expectation.

So approve (only) every candidate who is better than your expectation.

How to estimate your expectation in a 0-info election:

1D election--One merit-consideration-dimension:

Suppose that it's a 1D alternatives-lineup. Just one
merit-consideration-dimension.

Our political lineup seems to be 1D...seems to have only one
merit-considerration-dimension.. At the  CIVS polling website, I've never
encountered a political poll that had a top-cycle for 1st-place. That
suggests two things: 1) No offensive strategy is being used; 2) The
political spectrum is 1D.
Or, at least 1D is the best approximation to its dimensionality.

Some suggest approving candidates whose merit is above the mean of the
candidates' merits That's based on the assumption that all the candidates
are equally likely to be elected. That assumption is false.

What if we double the number of Republorat candidates? Will that really
double the probability of electing a Republocrat. Hardly. The voters don't
suddenly want a Republocrat twice as much, just because twice as many
Republocrats run.

So don't use the above-mean approval cutoff.

Use the above-midrange approval cutoff:

Approve (only) all the candidates whose merit is above the midrange of the
candidates' merit.
.
Of course it isn't the candidates themselves who determine the best
approval cutoff. It's the voters. But the existence of a candidate at a
certain issue-space position, or (which amounts to the same thing) at a
certain merit-level,l suggest that there are voters there.   ...if someone
thinks that a candidate there could be winnable. That's why I suggest the
midrange of the extremes of candidate-merit.

Assumption about voter-density distribution:

This is a 0-info strategy, and so you don't know where the voter-median is,
in comparison to where you are. So it would be pointless to try to
determine the voter-densities at the various merit-levels, based on the
gaussian-distribution of the voters. By assumption, you have no information
about that distribution, or even where the voter-median is.

So just assume that the voter-density-distribution is uniform.

it's reasonable to say that every little element of voter-density has the
same probability of electing someone whose merit corresponds to the
merit-level at that element's position in the 1D merit-consideration space.

Because the voter-density is assumed uniform over the candidate-merit
range, then the expected winner-merit is the midrange of candidate-merit.

By the way, when finding the candidate-merit midrange, disregard the
candidates who are clearly unwinnable at the bottom-end.

(If an allegedly unwinnable candidate is at the top-end, then likely you're
not the only person who regards that candidate as tops. Then don't assume
that s/he's unwinnable.)

2D -- Two Candidate-Merit-Consideration-Dimensions:

Again, you want to maximize your expectation, by approving (only) all the
candidates whose merit is better than the expected merit of the winner.

But, for 2D, it isn't necessarily the midrange.

Find the expected merit of the winner. Find the expected distance from you,
in the 2D space, of the winner.

You know the merit of the best candidate, and of the worst winnable
candidate. From that, you determine the expected distance (demerit) of the
winner.

If the candidates are pretty much all in the same direction from you, then
of course the midrange would still be a good estimate.

Say the candidates are in all directions from you, or that they're in a
pie-piece shaped region (a section of a circle) in one general direction
from you.

In that case, the 2D-space region of interest is a circular ring, around
you, bounded by a circle at R1, the distance from you of the nearest
candidate, and R2, the distance from you of the farthest candidate.

Or maybe the candidates are all in the same general direction from you (but
not all in nearly exactly the same direction), so that a better
approximation is that they fill up a pie-piiece-shaped section of a circle,
between radii R1 & R2.

Either way--circle or section of a circle, it's the same problem.

Sum the distance from the center (you), over that entire area (the ring, or
the circle segment between radii R1 & R2).

Divide the result by the amount of area possessed by that region.

That gives you the expected distance.

It's:

(2/3) (R2^3 - R1^3)/(R2^2 - R1^2).

This post is already long, and so I'll talk about Approval Yee diagrams in
a subsequent post.

But, just briefly, Approval Yee diagrams should assume that voters use the
optimal Approval strategies that I discussed in this post:

Find out if the voter has a top-set (that can be determined from the
candidates' distances from hir.

If s/he has a top-set, then s/he approves (only) them.

If not, s/he uses the 2D, 0-info, expectation-maximizing strategy that i
described above.

Michael Ossipoff
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