[EM] Issue Space, Democracy Potential, and Hybrid Proxy Approval (was election-utility strategy)

[Forest Simmons]

On Mon, 29 Apr 2002, Richard Moore wrote:

> Forest Simmons wrote:
> > What if the polls could tell us (for each i and j) what percentage of the
> > voters approve both candidates i and j.  If that percentage is not close
> > to the product of the percentages of approval for i and approval for j, it
> > would tell us that that approval for i and j are statistically related;
> > perhaps the nature of this relationship might be useful information for
> > approval strategy.
> >
> > This information wouldn't require additional questionaires, only summing
> > n*(n-1)/2 combinations from each existing questionaire (where n is the
> > number of candidates).
>
> See my April 12 post. I defined Bij(X) as the probability that i will beat
> j if i has exactly X votes. If we know nothing about the relationship
> between i and j votes, then for this value we can substitute the cumulative
> probability that I called Cj(X).
>
> A correlation (or an anti-correlation) of i and j votes would skew Bij(X).
> So if you had the n*(n-1)/2 sums, then perhaps you could determine how
> to skew the Cj(X) values to get accurate Bij(X) values.
>
> I don't know what that calculation would look like, though. How good are
> you at statistical formulas?

I'm pretty rusty, but Joe Weinstein might know where to start.

BTW I've been thinking a lot about your [Richard's] Democracy Potential
ideas lately.  In fact I didn't sleep much last night because of it.

I was thinking about it in connection with ranked ballots.

Suppose that we make the simplifying assumption that all of the voters and
candidates place themselves at corners of the issue space, i.e. they
generally think of themselves as either for or against an issue, and they
think of the candidates in the same way, either for or against.

Then why does it sometimes appear that candidates and voters are strewn
along a one dimensional spectrum?

Well, suppose that the issue space is an ordinary 3D cube, and that the
greatest concentrations of voters are located at two diagonally opposite
corners.  Then the diagonal through the center of the cube connecting
those opposite corners determines the spectrum onto which the non-extreme
candidates and voters are projected.

Where a voter projects onto that line relative to the candidates roughly
determines the voter's preference order among those candidates.

The other effect in this model which can affect the preference order is
that different voters consider different issues to more important, so they
scale the three axes differently.  Geometrically, for different voters the
cube turns into boxes with various lengths, widths, and heights.

To see that this can affect the preference order, drop down a dimension.

If A, B, C, and D are the four corners of a rectangle, and A and C are
diagonally opposite, then there are two possible projection orders along
the AC diagonal, namely ABDC, ADBC (as well as their reversals CDBA, and
CBDA), depending on whether the AD or the AB dimension is considered
greater, i.e. more important to the voter.

It seems to me that if you were situated at A, i.e. you agree with
candidate A on both issues, then your preference order would either be
ABDC or ADBC  depending on whether you considered the AD issue or the AB
issue to be more important.  You would definitely agree with candidate A
on first and last place, but you might differ on the order of the two
other candidates if you placed different importance on the two issues.

So each of the four corners gives rise to two realistic preference orders,
yielding a total of eight realistic preference order possibilities out of
a total of 4*3*2 permutations, i.e.  only one third of the twenty-four
possibilities are rational according to this model.

In other words, if the issue space is truly two dimensional, then (in this
model) the candidates are divided into four clone classes, and there are
no more than eight preference orderings of these clone classes.

Here's where Democracy Potential comes in.  Suppose that in some election
we are able to discern that this model is apt, and that the issue space is
essentially two dimensional. [That doesn't mean that there are only two
issues, it just means that the issues are correlated in such a way that
there are only two effective dimensions; if you tell me your stand on the
the two key issues, then I can reliably predict your stand on the
remaining issues.]

Continuing ... suppose that there are six candidates, and that ABC are
clones at one corner of the issue square, D and E are clones at another
corner, and F is at a third corner, and no candidate occupies the
remaining corner.  To do a Democracy Potential calculation, just beef up
all the corners with virtual candidates until they all have three clones
apiece before applying Copeland.

This is still in the very rough stage.

One result that interested me was this. Condorcet Cycles are possible in
this two dimensional model, but only by taking into account that different
voters will differ on which issue is more important.

Suppose that candidates A, B, and D are not clones, so that they occupy
different corners of the issue square in such a way that the path DAB
forms a right angle.

Then place a virtual candidate C diagonally opposite A.

Suppose that the side AB is longer than the side AD.  Then the possible
preference orders along the AC diagonal would only be ADB(C) and its
opposite (C)BDA, and the only orders along the other diagonal would be the
opposites B(C)AD and DA(C)B.  So leaving out the virtual candidate, we
would be limited to ADB, BDA, BAD, and DAB. Two of these ADB and BAD are
in reverse cyclic alphabetical order, and the other two are in the other
cyclic order, so no cycle of three is possible.

But if the group of voters aggreeing with candidate A sees AD as more
important than AB, then the preference order ADB is replaced with ABD, and
a cycle ABD, BDA, DAB is formed.

Now a confession: I started thinking about this in connection with proxy
methods.  It seemed to me that if there are more than a dozen or so
candidates, and the effective dimension of the issue space is only two or
three, then there should be plenty of options for the typical voter among
the preference orders of the candidates themselves, without having to
consider all of the n factorial possible orders.

This would be true (in my model) if each voter residing at a corner of the
issue space shared the same relative sense of importance of the respective
issues with some candidate residing at the same corner.

It seems to me that this condition might be approximated pretty well in
reality, and that the exceptional voters might be willing to either go
with the flow or be satisfied with Approval ballots to express their
non-conformity.  In other words, I'm thinking hybrid Approval /
ProxyApproval would work handily in elections with large numbers of
candidates.

In a tangential thread, I would like to see Alex Small's symmetry
cancelling idea tested by Majority Potential simulation.  It seems to me
like it might do well in that setting.

Forest


----
For more information about this list (subscribe, unsubscribe, FAQ, etc), 
please see http://www.eskimo.com/~robla/em