[EM] The Symmetry and complexity of elections

[Norman Petry]

List members interested in reading some nonsense about the merits of Borda's
method might be interested in:

http://www.colorado.edu/education/DMP/voting_b.html

In this paper, Saari argues against Condorcet's method by providing the
following "bad example":

ABC 29
BAC 28

Condorcet:

AB20, AC48, BA28, BC48, CA0, CB0 --> B>A>C

Borda:

A68, B76, C0 --> B>A>C

so far, so good.  He then introduces what he considers a group of "confused
and irrational" voters with the following (cyclic) preferences:

ABC 9
BCA 9
CAB 9

When these voters are added to the election the results are:

Condorcet:

AB38, AC66, BA37, BC66, CA18, CB9 --> A>B>C (changed)

Borda:

A95, B103, C27 --> B>A>C (same as before)

He then states:

"This Condorcet addition has no impact on the weighted voting rankings, but
the resulting pairwise cycle changes the pairwise outcomes. It has to; the
pairwise vote treats these new voters as being confused and irrational with
cyclic preferences!"

He seems to think we should be surprised that these additional voters
changed the result from B to A under Condorcet.  Well, let's see... is it
surprising that when you add 27 voters to an election who express a 2:1
preference for A over B that the winner might change from B to A?  It is not
the "pairwise vote" that treats these additional voters as "confused and
irrational", it is Borda's method, which eliminates any impact these voters
have on the borda scores (adding 27 to each candidate's total).

The only one confused here is Saari.  This "bad example" for Condorcet
actually demonstrates how deeply flawed the Borda count is.  Saari seems to
think that a good method should discount the effect of the addition of a
contrived group of ballots which happen to form a symmetric voting cycle.
Presumably, such sets of ballots represent a kind of "noise" input which the
counting system should cancel, if possible.  He fallaciously assumes that a
group of voters with preferences like these:

ABC 9
BCA 9
CAB 9

provide no useful information, but this is clearly false.  In the first
place, voting cycles only are meaningful in connection with an actual
*outcome*, so until these additional ballots are aggregated with the
original ballots the issue of cycles is irrelevant.  Far from providing no
useful information, these voters are unequivocally stating that A>B 18:9,
B>C 18:9, and C>A 18:9.  If one of these 3 candidates was eliminated, for
example, would we be able to determine with certainty which of the remaining
two should be chosen?  How is this possible if these ballots contain no
information?  Saari's example actually demonstrates how deeply flawed
Borda's method is, precisely because it cannot distinguish the following
group of "confused and irrational" voters from the previous group:

ACB 9
CBA 9
BAC 9

Even if we entertain the dubious idea of symmetric cycles of voters in
isolation from actual outcomes, there is important information contained in
the *direction* of a cycle which is ignored by the Borda count.

There is clearly something wrong with the state of the art in Social Choice
theory when work of such poor quality is still taken seriously by academics.


-- Norm Petry