[EM] request for reading matter

[Bart Ingles]

"John B. Hodges" wrote (way back on 7/8/03):
> 
> Bart Ingles wrote:
> >
> >"John B. Hodges" wrote:
> >  > I've read a book by Donald Saari, CHAOTIC ELECTIONS, in which Saari
> >>  finds unique virtues in the Borda Count and many weaknesses in
> >  > Approval. (snip) Can y'all recommend some text that
> >  > replies to Saari regarding Approval?
> >
> >Try the following four-article debate in Public Choice
> >http://www.kluweronline.com/issn/0048-5829
> >(I recommend checking your university library for these):
> >
> >(1) The problem of indeterminacy in approval, multiple, and truncated
> >voting systems
> >Donald G. Saari and Jill Van Newenhizen
> >Public Choice 59: 101-120 (1988)  (c) Kluwer Academic Publishers
> >
> >(2) The responsiveness of approval voting: Comments on Saari and Van
> >Newenhizen
> >Steven J. Brams, Peter C. Fishburn, Samuel Merrill III
> >Public Choice 59: 121-131 (1988)
> >
> >(3) Is approval voting an 'unmitigated evil'?: A response to Brams,
> >Fishburn, and Merrill
> >Saari, Van Newenhizen
> >Public Choice 59:133-147
> >
> >(4) Rejoinder to Saari and Van Newenhizen
> >Brams, Fishburn, Merrill
> >Public Choice 59:149
> 
> JBH here- I read the journal articles above. (Shelved under H35 P8X )
> I'll have to read them again, but my first impression is that Saari
> wiped the floor with Brams&company. Their final rejoinder is that he
> missed the point of their first rejoinder, and I'll have to reread
> the whole exchange to judge that.

I finally re-read the articles myself, but was even less impressed than
I remembered being a few years ago.  Article #3 in particular was
strange, more like a flame war that had gone on too long than something
in an academic journal.  The theme seemed to be "we changed our
mind--approval voting is even worse than we claimed before."  Using this
as the main starting premise, it not surprisingly arrives at the same
conclusion.  Maybe the authors were just pissed off at the previous
article's use of "BS" as an acronym for "Borda system".

After this, the one-paragraph response in article #4, returning to the
original subject matter (indeterminacy), seemed appropriate.

In spite of this, one possible insight into Saari's thinking emerges in
article #3, in that that he seems to view every possible voter behavior
as equally likely, so that for him an indeterminate system means that
voters will choose every conceivable strategy consistent with their
ordinal preferences, and without regard to individual utility
maximization.  This thinking also shows in some of his analytical
methods, which seem to be to plot all possible outcomes into
multidimensional space, as though each outcome were equally likely.


One observation of mine after reading the articles is that Saari appears
to be comparing sincere ordinal preferences in the case of AV to actual
voted Borda ballots.  This undermines his argument in two directions:

(1) In the case of AV, if you can read the voters' minds well enough to
determine sincere ordinal preference, you can also determine cardinal
preferences.  This additional knowledge constrains the number of
possible voter strategies, even more so when excluding strategic
information as implicit in Saari's discussion of Borda.

(2) In the case of Borda, if you allow for the possibility that voted
rankings may differ from sincere ordinal preferences, then the claim
that the method is deterministic has no particular meaning. 
Already-voted approval ballots are every bit as deterministic as Borda
ballots.


A second observation concerns truncation.  As Saari points out in the
article #1, any implementation of Borda that allows voters to truncate
is also indeterminate.  What he seems to miss is that there is no way to
prevent or penalize truncation if the voter is willing to do so
stochastically.  This is very easy to do with Borda.  All you need is a
single coin toss (whose result you can memorize and use for your entire
voting career if desired).  Then, for every group of candidates that you
wish to rank equally, either vote them in ballot order (heads) or
reverse ballot order (tails).  With multiple voters following the same
strategy, the two types of ballot will cancel out and each of the
"truncated" candidates will receive (approximately) the same overall
score.

Note that with ranked systems other than Borda, a single coin toss is
not sufficient, since A>B>C>D may not exactly cancel D>C>B>A.  Instead
you would need to do some sort of repeated coin toss or dice roll for
each candidate.  In effect Borda's symmetry makes it an indeterminate
system in a very practical sense.


> The essence of his argument is in the math, the theorems he claims to
> prove and his interpretation of what they mean. Isolated examples and
> "nightmare scenarios" don't really prove anything, because no voting
> system is perfect. But I'll pass on one of the two examples he gives,
> as an "approval voting nightmare". You have a town of 10,000 people,
> choosing a Mayor. 9,999 people regard A as excellent, B as mediocre
> but passably competent, and C as a disaster. One voter (possibly C
> himself) regards C as excellent, B as passably competent, and A as a
> disaster. Everyone follows the recommended strategy for Approval
> voting of voting for all candidates that offer "above average"
> utility for the three candidates; so everyone votes for their top
> two. Tally is C gets one vote, A gets
> 9,999 votes, and B gets 10,000 and wins.

I think worst-case analysis is important, it's just that it shouldn't be
confused with typical expected behavior.  And when comparing worst-case
behavior with another system, it's important to compare apples with
apples.

As far as nightmare scenarios go, the one above doesn't seem all that
bad.  If we assume the A voters are voting approval for B due to a total
lack of strategic information, then B must hold a normalized utility
higher than 0.5 for all of these voters.  Thus the winner has a utility
of > 0.5, and a utility deficit compared to A of < 0.5.  This compares
favorably with Borda, and is about the best one can expect from any
system.

If considering extreme manipulation of polling info, then things could
of course be worse.  If the A voters are persuaded that C is a real
threat under sincere AV, and that B is the sincere Condorcet winner,
then they might approve of B even if B's utility is considerably lower
than 0.5.  But in that case, how does Borda compare?

If the A voters are persuaded that C is the likely Borda winner, and
that B is a more viable challenge to C, then the A voters have incentive
to vote B>A>C while the the C voters will either vote C>B>A (or C>A>B if
they expect strategic behavior from the other voters).

In the case of Approval, all 9,999 A voters must wrongly approve of B. 
Under Borda, only 5,000 voters need to miscalculate:

Sincere preferences:
------------
9,999 A>B>C
1 C>B>A

Actual votes:
-----------
4,999 A>B>C
5,000 B>A>C
1 C>B>A

As with the manipulated-poll approval example, B could hold fairly low
utility for the sincere A voters, and thus be a low-utility winner
overall when compared to A

Bart