Democratic symmetry (fwd)
Rob Lanphier
robla at eskimo.com
Thu Mar 9 20:01:23 PST 2000
Below is a letter from Dr. Saari. Before reading the message, you may
want to read the correction:
> Oops; in my hastely written message just sent, I mentioned methods which
> elect a majority candidate. I meant to include the comment that "for
> those procedures which do not depend on binary information." Those
> methods that do depend on binary information (i.e., pairwise rankings)
> have their own list of serious difficulties due to a different data
> symmetry.
With that in mind, attached is the message:
---------- Forwarded message ----------
Date: Thu, 9 Mar 2000 16:25:51 -0600 (CST)
From: d_saari at borda2.math.nwu.edu
To: robla at eskimo.com
Cc: Donald Saari <dsaari at nwu.edu>, election-methods-list at eskimo.com,
TomStandage at economist.com
Subject: Re: Democratic symmetry
Dear Mr. Lanphier,
Thank you for the copy of your letter. Let me make a
couple of corrections to your comments. (I think the Economist did
a nice job on a difficult topic.) Also, although I normally do not use the
web, I checked one website which had a posting from you. On this site, there
were questions from others about references; correct ones are given below.
I hope you will read my two articles in "Economic Theory" (not
Jour of Economic Theory, but JET does have a related easier 1999 article
of mine on voting) which is the first ET issue of 2000. Once you do, you
will discover that rather than promoting the Borda Count, my main
goal is to develop methodology to analyze all possible procedures that
are based on weights. This includes Kemeny's Rule, Copeland's method,
Condorcet winner, runoffs, agendas, procedures using any choice of weights,
and on and on. In fact, when you read the articles, you will discover that
I discuss many of these methods in the paper. (Contrary to your comment,
I do discuss Copeland, Kemeny, etc.etc. I even find new properties of these
procedures, but mainly to illustrate my approach.) Indeed, my goal was to
find a way to explain all of the basic properties discovered about the
different procedures as well as to find a way to discover (or indicate)
all new properties. So, in contradiction to your comment below, when you
read the paper, you will learn I am deeply interested in all possible
properties of all possible procedures.
[An aside: later this year, a very long paper, with V. Merlin,
will appear in Social Choice and Welfare which characterizes properties
of Kemeny's method. Also, the two of us published a fairly complete
analysis of Copeland's method -- the paper concerning manipulative and
other properties appeared in JET in 1997 and the one concerning basic
Copeland election properties appeared in ET (Economic Theory) in 1996.
For both procedures, we found that with three candidates, they do
not seriously suffer from the Gibbard-Satterthwaite result. This is
surprising!]
Let me explain my approach. In your discussion group,
you must have experienced the frustration of having a wide number of
different properties associated with each voting method. Rather than
shedding light on the area, the debate often breaks down to claims of
"my set of properties are better than your set of properties." Indeed,
the ease of creating both new and bad properties for voting systems
caused me to facetiously volunteer (in print) to find a set of axioms for
a method of your choice showing why it is the "best," or, if you want, why
it is one of the worse. While I was being facetious, within reason,
it is not difficult to do this. Therefore, to get out of this rut, an
approach needed to be developed which would characterize *all* possible
properties for the class of procedures commonly used.
How does one do this? As I am sure happens in your discussion
group, when someone wishes to promote one procedure over another, the
standard approach is to create an example. In other words, it always is wise
to return to the data. In doing so, to my surprise, it turned out that
*all* basic properties of the kinds of voting procedures we are discussing are
based on whether the procedure does, or does not respect different kinds
of symmetries in data. For instance, using your example, procedures which
always elect a majority winner, when one exists, must violate a certain type
of data symmetry. The negative corollary of the violation is that with a high
likelihood, these same kinds of procedures have the property that they have a
a higher likelihood of electing the Condorcet loser (the candidate who loses in
all pairwise elections), of radically changing the ranking should *any*
candidate drop out, and on and on. It turns out -- returning to
my earlier facetious claim -- that just by knowing what data symmetries
a procedure ignores, it now is possible to construct several nice sounding
properties the procedure enjoys, but at the expense of violating a
host of other properties we would not want to occur.
So, when you read the papers (the first, which constitutes pp 1-53,
addresses procedures using binary information -- so this also addresses
Arrow's Theorem, and the second pp 55-103 addresses weighted or
positional methods), view them as developing tools of analysis for
voting procedures. As an analogy, think of my work as providing tools of
algebra, or of vector analysis, to be used to study all procedures. I develop
the tools; they have to be used to extract new information. (E.g. others
are using this approach to find new properties of other procedures.) As I make
clear in at least one paper, I really don't care what procedure you use,
but the approach I developed shows you what are the associated costs and
benefits. In contrast to what you state, it appears this approach can be
used to handle any form of "fairness" that has been advanced so far.
In this same spirit, please do not run into the silly mistake
(as someone on that web site did) of thinking that I am concerned about
adding or subtracting voters. My approach is equivalent to handling an algebra
problem -- to solve it, you have to factor and decompose it. To handle
the data problem, decompositions -- which involve subtracting certain
groups of data sets -- are needed. To keep the description somewhat within
reason, I construct examples by adding data. But, think of this as a reverse
decomposition of starting with a given data set. (Elsewhere, such as in
my books, I do address adding or subtracting voters -- but this is to
study strategic behavior, what happens if a voter doesn't vote, what
happens when a like-minded group joins, etc.) Also, someone was worrying
whether one of my examples had 28 or 20 voters. I don't know where
this came from, but I expect (from the numbers) I was illustrating how to
reconstruct all of Condorcet's examples. In fact, I expect this was
whn I was analyzing Condorcet's famous example to show why his conclusion
came about -- if so, it was not mine. (This example illustrates how
bright and insightful Condorcet was.)
What method do I prefer? A *modified* version of the Borda Count.
The different kinds of negative properties associated with violating data
symmetry makes the BC the *theoretical* best choice. However, there are
many practical problems with the BC. You can easily create a list: e.g., not
everyone wants or can rank all candidates, there is a tendency for
strategic voting, and on and on. But, once we understand why the BC
has such favorable properties, it becomes reasonably easy to understand
how to modify the BC to handle these difficulties. In our department, for
instance, we use a modified version of the BC.
I appreciate your comments about my analysis of Arrow's Theorem.
My opinion remains that Arrow's Theorem is one of the more important
results of the last half century. Indeed, I strongly believe that versions
of his result appear in subtle, hidden forms in all sorts of other settings
ranging from law to even engineering; I just finished a book manuscript
showing where & why this is the case. But, notice my Arrow argument shows that
if a procedure does not use all of the information from a particular
kind of data set, then peculiar, unexpected outcomes occur. The reason
is that the procedure treats certain data as coming from a nonexistent
society rather than the real one which gives the data. The *same* effect
occurs whenever procedures ignore certain other data symmetries. (By the
way, based on comments Arrow made to me, he seems to like both this
interpretation of his result and some of the extensions.)
Finally, someone asked about Steve Brams criticizing some of
my work. While Steve and I are friends, we have agreed to strongly
disagree about Approval Voting. Our exchange about Approval Voting
appears in Public Choice, (1988), vol. 59, pages 101-147. I highly
recommend this series of three papers to you -- I leave it to you to then
either endorse, or not endorse, Approval Voting. (By the way,
based on conversations with Brams and Peter Fishburn, it is safe
to say that both of them also would be comfortable in using this
series of papers as representative of our different views.)
I am delighted to learn about your discussion group. If
I had more time, I might be tempted to participate.
Sincerely,
Don Saari
>
> To whom it may concern,
>
> While I'm pleased that The Economist has chosen to tackle the subject of
> electoral system fairness in the March 4 edition of The Economist ("The
> mathematics of voting: Democratic symmetry", p.83), it saddens me to see
> an uncritical look at the work of Dr. Donald Saari. Dr. Saari's work has
> been the subject of many a discussion on the Election Methods mailing list
> [1], and the commentary has rarely been flattering.
>
> The method that Dr. Saari proports as the fairest method, the Borda count,
> presupposes a very narrow definition of fairness. While focusing on
> abstract concepts of symmetry and cancellation, he misses the boat on more
> important criteria, such as the "Majority Winner Criterion", which states
> that if a strict majority of the voters rank a particular alternative as
> their first choice, then the voting method must select that alternative
> as the unique winner [2]. Nearly all other methods proposed by electoral
> reformers pass this criterion, not to mention first-past-the-post. The
> Borda count is one of the few methods that doesn't.
>
> Dr. Saari deserves a pat on the back for persisting in the face of Kennith
> Arrow's famous thereom, which suggests that there is no perfect voting
> system. Many in Dr. Saari's field have used this thereom as an excuse for
> not finding better alternatives to first-past-the-post; Dr. Saari has
> rightly chosen to question the basis of the theroem by pointing out
> problems with it (for instance, the Independence from Irrelevent
> Alternatives Criterion has been questioned by Dr. Saari and others who
> study these matters). However, Dr. Saari goes too far by rejecting many
> other completely reasonable criteria.
>
> There are many systems which stand up to much more stringent criteria than
> Borda does, such as those methods proposed by Copeland, Fishburn, Kemeny,
> Schwartz, Smith, Hare, Coombs, Condorcet, and others. These methods were
> constructed with an eye toward many other more important criteria than
> "symmetry", and deserve a fairer treatment by your publication than the
> glib writeoff of not being the "only system that fits the bill".
>
> I'm also disappointed that this article was not selected for posting
> online. I've posted a relevant snippet to the mailing list, and a lively
> discussion is already ensuing.[3] I encourage you, Dr. Saari, and your
> readers to participate in the debate about this subject.
>
> Thank you,
> Rob Lanphier
> robla at eskimo.com
> http://www.eskimo.com/~robla
>
> [1] Election Methods Mailing list: http://www.eskimo.com/~robla/em
> Past discussions of Dr. Saari's work:
>
> [2] Anderson, L.B., "A Partial Ranking of Selected Voting Methods Based
> on Majority, Condorcet, and Monotonicity Criteria" Position Paper, March
> 17, 1996. Author Affiliation: Institute for Defense Analyses, Alexandria,
> VA.
>
> [3] There's already been many discussions about Dr. Saari's work on the
> Election Methods List. Here's links to the latest:
>
> Discussion about the Economist article, March 5, 2000:
> http://www.eGroups.com/group/election-methods-list/showthread.html?start=5117
>
> "Re: IIA Theory" - October 5, 1999
> http://www.eGroups.com/group/election-methods-list/4336.html?
>
> Quote from this thread:
> "By going through Sen's, Gibbard's and Satterthwaite's work first you
> can see how Saari's criticism of IIA as being 'absurd' (because, and
> this should already obvious, it fails to be satisfied in all cases) is
> itself problematic, especially in its "implications" towards Borda
> score systems, and at the same time how right Saari is in seeing that
> Arrow's theorem has at its heart the simple problem of IIA
> occasionally failing to be satisfied where more than two voters have
> an impact on the outcome."
> -- David Catchpole
>
> Re: Approval Voting fish (2) - March 3, 2000
> http://www.eGroups.com/group/election-methods-list/5093.html
> Quote from this thread:
> "Borda, in all the proposals that I've heard of, requires you to give
> points to all but one of the candidates, no matter how much you
> despise your 2nd to last choice, and your other lower choices. That
> doesn't happen with any other proposed method, and doesn't even happen
> with [first-past-the-post]"
> --- Mike Ossipoff
>
>
>
>
>
>
>
--
===============================
Donald G. Saari, Pancoe Prof. of Mathematics
Northwestern University, Evanston, IL 60208-2730
Ph: 847 491-5580 Fax: 847 491-8906
dsaari at nwu.edu
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