[EM] The Symmetry and complexity of elections

[Rob Lanphier]

This is a great analysis.  In fairness to Saari, though, he is pointing
out a small, but legitimate problem.  We may be better off grudgingly
conceding that point, since voter cycles are the bane of any electoral
reformer's existence. It's not theoretically obvious why they would exist
in the first place, and yet we can't just write them off. 

You make a good case for why the "cyclical consistency" argument is
fallacious.  However, the problem with making a big of a deal about it is
that many pairwise methods are forced to come up with tenuous tiebreakers
in the face of cycles involving large numbers of voters.  It's not the
strongest card to play (though I did giggle when I read "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?".  Touche.) 

The thing we should point out is that the cure is worse than the disease. 
Examples involving cycles are very difficult to construct a "story" around
(unless they involve matters widely acknowledged as being purely a matter
of taste, such as ice cream flavors, where there are no known correlations
between various candidates). However, examples involving cases where Borda
breaks down are very easy to construct plausible stories around
(involving, say, choices for President involving Gore, Bush, McCain and
Bradley, for instance, where there are easy generalizations to be made
about likely correlations and cross-support).  

I'd do this myself, but I've posted enough tonight already!  :)

Rob

On Wed, 8 Mar 2000, Norman Petry wrote:

> Date: Wed, 8 Mar 2000 01:42:05 -0600
> From: Norman Petry <npetry at cableregina.com>
> Reply-To: election-methods-list at eskimo.com
> To: Election Methods List <election-methods-list at eskimo.com>
> Subject: [EM] The Symmetry and complexity of elections
> Resent-Date: Tue, 7 Mar 2000 23:44:19 -0800
> Resent-From: election-methods-list at eskimo.com
> 
> 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
> 
> 
> 

Rob Lanphier
robla at eskimo.com
http://www.eskimo.com/~robla