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

[Richard Moore]

Forest is referring to an idea I had last year, Majority Potential. This
isn't an election method but a way of rating a candidate's ability to win
majorities in head-to-head contests. I don't see way to do this in the
real world but it can be used in simulations for evaluation of methods.

To start with, consider Copeland's method, in which each candidate's score
is equal to the number of pairwise wins by that candidate over other
candidates. This is a Condorcet-compliant method but it has a few problems
when there is no CW: A high frequency of ties, sensitivity to clones, and
non-compliance with IIAC.

But suppose we add a large number of hypothetical candidates to the mix,
and these hypothetical candidates are uniformly distributed in the issue
space. Now run the Copeland election, but after the pairwise wins are
counted, observe the scores of the real candidates. It turns out the
problems I listed are greatly diminished. What's more, if the number
of hypothetical candidates is large enough, then the scores represent
the probability that the candidate would beat a randomly chosen candidate
in a two-way race. It's the best way of quantifying (in a theoretical or
simulated context) a candidate's mass appeal that I can imagine.

This also relates to the expressivity of ranked ballots. Imagine there
are five real candidates whom I rank as follows: C1 > C2 > C3 > C4 > C5.
You can tell from this which candidate I would prefer in any two-way
race. But suppose we throw in a bunch of hypothetical candidates (H1,
H2, etc.) and my ranking looks like:

H1 > C1 > C2 > C3 > H2 > H3 > H4 > H5 > H6 > H7 > H8 > C4 > H9 > C5 > H10

Suddenly a lot more information about my preferences, in particular the
large gap between C3 and C4, becomes apparent. This information is
missing from conventional ranked ballots.

  -- Richard


Forest Simmons wrote:
> 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
> 
> 


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