[EM] Comparing ranked versus unranked methods

[Forest Simmons]

On Tue, 5 Feb 2002, Adam Tarr wrote:

> 
> I just looked up your posts on this [PAV], with a little effort I was
> able to 
> find them.  This seems like a really excellent method, indeed.  The only 
> drawbacks it has, in my mind, are shared by Approval; you can't really 
> express your preferences within your approved list.  But the fundamental 
> idea of using good single-winner methods, but decaying the voter's 
> preference in a proportional fashion, is a sound one.  It seems obvious in 
> the face of fractional STV, but I hadn't thought of it before.
> 
> It seems to me that rather than talking about "satisfaction points" it 
> makes more sense to decay the value of the ballots that voted for a 
> winner.  The effect is the same, it just makes more sense by my way of 
> thinking.  For example, if we wanted to use Webster's method, then divide 
> the value of ballots that voted for the most recent winner by (2n+1), where 
> n is the number of candidates that ballot has elected.  Or multiply by 
> (2n-1)/(2n+1) if you prefer a recursive definition.  For d'Hondt, use 
> 1/(n+1) at each stage, or n/(n+1) to work recursively.
> 
> I have not checked the above, but it seems to work on toy examples in my 
> head.  Am I right about this?

This approach is what we have called Sequential PAV, and is a special case
of Conditioned PAV.  An example of Conditioned PAV's usefulness would be
in the case of electing members of a council or governing committee that
was to be headed up by a triumvirate of a council head and two counselors.

The council head H is taken to be the Approval winner from the Approval
ballots.  Then (using the same ballots) the PAV procedure is applied to
all three member subsets of the form {H, X1, X2}. The subset with the
highest PAV score determines who the two counselors C1 and C2 will be.

Then (using the same ballots) the PAV procedure is applied to all subsets
of the form {H, C1, C2, Y1, Y2, ... Yn} where n is the number of other
members to be chosen for the council.

In sequential PAV the successively conditioned subsets are of the
respective forms {C1, X}, {C1, C2, X}, {C1,C2,C3,X}, etc. where one member
is added at a time.

This is appropriate if there is supposed to be a pecking order or
seniority system in the elected body.

It can also be used when the number of candidates becomes so large that
regular PAV requires too much computation or when simplicity and
transparency of the method is important, since the computation can be done
by the simple vote decay method that you mentioned.

> 
> I'm currently wracking my brain trying to figure out if there is a way to 
> apply this insight to create a generalized Condorcet voting technique.  How 
> to determine a winner at each stage is easy.  The problem lies in 
> determining whether someone has voted for a winner or not.  In PAV and 
> (fractional) STV, it is obvious; the candidate is either in first place on 
> the ballot or isn't.  But in Condorcet voting, it is unclear how to best 
> decay (or add satisfaction points to) a ballot to accurately reflect the 
> voter's choice.
> 
> One option might be to decay a voter's ballot based on the position of the 
> elected candidate on the ballot... a sort of Borda-based decay setup.

Right.  It turns out that since the sequence 1 + 1/2 + ... + 1/n  is
asymptotic to ln(n) that there is a simple way of getting a Borda version
of PR:

Suppose that there are to be three winners, and that {X,Y,Z} is a set of
candidates to be scored by ballot B.  Just take the logarithm of the sum
of the Borda scores for the three candidates on ballot B.

The three member subset with the highest sum of logarithms over all
ballots is the winning subset.

Since the sum of the logs is the log of the product we can avoid
logarithms altogether in the statement (although computationally we would
still want to use them if there are more than a few dozen voters):

The three member subset with the highest product of ballot totals is the
winning combination.


>  My 
> intuition is that this would lead to a huge strategic mess, but I could be 
> wrong.  Is there any way to reliably (i.e. accurately, fairly, 
> proportionally, in a way that discourages insincere strategy) decay 
> Condorcet ballots in multi-winner elections?  If someone could, well, that 
> would just about settle the whole "best election method" thing for me.
>

I posted a few attempts at Condorcet PR last spring or summer.  The best I
could come up with was a sequential method analogous to sequential PAV
that went something like this:

When comparing two candidates X and Y head-to-head to see which one should
be added to the list of winners, discount the influence of ballot B by
giving it a weight W to be determined as follows:

Let K be the number of already determined winners above the
preferred of X and Y.  Let J be the number of already determined winners
between X and Y in preference.  Then take W to be 2/(2+J+2*K).

In other words, the more satisfaction a voter already has relative to the
two candidates in the current contest, the less the voter has to say about
that current contest.

I had mixed feelings about putting the J term into the formula. I would
like to hear your ideas, perhaps stimulated by this partial progress.

Demorep has posted cryptic messages about PR Condorcet from time to time,
but I haven't been able to decipher them. 
 
> On an unrelated note, congratulations on finding a way to appropriately 
> introduce the word, "hysteresis" into a discussion of voting theory.  It 
> got me thinking back to PN junctions and other pieces of electrical 
> engineering trivia that I have mostly forgotten, but will have to remember 
> again for my qualifying examinations in the fall.

Sometimes it is convenient to think of election methods as non-linear
filters. They process a signal or set of signals from the voters and
output a candidate or set of candidates.

More generally in analyzing methods we may consider as input the set of
possible or likely election scenarios, including all the polling data, the
voter utilities, etc.

I majored in math, minored in physics, and worked with a bunch of
engineers and computer scientists, which has given me lots of metaphors to
work with. "I never metaphor I didn't like."


> 
> Good luck in the Oregon IRV thing... is it state-wide or just Portland?

It's a state-wide initiative.  I have mixed feelings about it, but I don't
think it has enough of a chance of succeeding to waste any time worrying
about it this time around. 


Forest

> 
> Adam Tarr, Ph.D. Student
> Purdue University
> School of Electrical and Computer Engineering
> atarr at purdue.edu
> (765)743-7287
>