[EM] Democratic symmetry

[Craig Carey]

At 21:50 06.03.00, Rob Lanphier wrote:
 >To whom it may concern,
 >
...
 >The method that Dr. Saari purports 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.
 >

Standard(?) Borda would actually pass that test. The Economist article was
  quoted as saying: "Also if voters are not familiar with all candidates, and
  do not rank them all, the unassigned points must be divided up evenly
  between the unranked candidates.".

That "Majority Winner Criterion" test could be failed if the 1st weight was
  less for a candidate because the votes held less preferences. But it seems
  that the method described to the reporter may have had weights like these:
  (5,4,3,2,1,0) for ABCDE, (5,4,3,2,0.5,0.5) for ABCD,
  (5,4,3,1,1,1) for ABC, (5,4,1.5,1.5,1.5,1.5) for AB,
  (5,2,2,2,2,2) for (A). It seems that Mr Saari wasn't explaining a Borda
  method variant that failed that "Majority Winner" test.

  Power of (5,2,2,2,2,2):
   (method 1): Min((5-5)+5*(5-2),(5-2)+5*(2-2)) = Min(15,3) = 3
   (method 2): Min wrt. 2<=u<=5: |5-u|+5*|2-u| = (5-u)+5*(u-2) = 4u+5 = 13

  Power of (5,4,3,2,1,0):
   (method 1): Min(6*5-(5+4+3+2+1+0),(5+4+3+2+1+0)-6*0) = Min(15,15) = 15
   (method 2): Min wrt. 0<=u<=5: |5-u|+|4-u|+|3-u|+|2-u|+|1-u|+|0-u|
              {u=2.5 by symmetry}:  = 2*(5/2+3/2+1/2) = 9

  Since 3 <> 15 and since 13 <> 9, I retract my comment that the Borda method
  is a fair method. There is a clear hint that a voters can get 5 times the
  influence of other voters, in small elections. 


----

 >Dr. Saari deserves a pat on the back for persisting in the face of Kenn[e]th
 >Arrow's famous theorem, which suggests that there is no perfect voting
 >system.  Many in Dr. Saari's field have used this theorem as an excuse for
 >not finding better alternatives to first-past-the-post; Dr. Saari has
 >rightly chosen to question the basis of the theorem by pointing out
 >problems with it (for instance, the Independence from Irrelevant
 >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.

They are inconsistent too.

--

I too will write some notes about the LIIAC criteria copied from
  Mr B. Cretney's website:
   http://www.fortunecity.com/meltingpot/harrow/124/criteria.html

Mr Blake Cretney's site says: "Local Independence from Irrelevant
  Alternatives Criterion (LIIAC): If an election produces winner X, and a
  new alternative is added (Y), and Y is not in the Smith set, the new
  election must also produce winner X.
  Smith set: The smallest non-empty set of alternatives such that every
  alternative in the set pair-wise beats every alternative outside the set."

Mr Cretney's definition sets out a property that is not desirable to a
  mentionable degree. Specifically in the 3 paper election with papers
  (a:AB + b:B + c:C), adding candidate C to (a:A + b:B) lets the
  "criteria" wrongly claim that B should wins in the region (a<b)(a+b<c),
  and it wrongly says A should win in (b<a)(a+b<c). More interestingly,
  adding candidate A to ((a+b):B + c:C) lets the criteria say that B
  wins in this region (a<b)(2b<a+c)(c<a+b). LIIAC mandates a bad property
  of STV, in election methods. The problem that LIIAC can be seen caught
  trying to impose, is this alteration: (C{B+)-(AB{C+) [A 1st preference
  for C sees C lose with less than 50% of the vote, but it can win [e.g.
  in STV] when none of the preferences are for candidate C].

So, Arrow seems to have not obstructed progress (if LIIAC matches up with
  Arrows, and maybe it doesn't).

...
 >[1]  Election Methods Mailing list: http://www.eskimo.com/~robla/em
 >     Past discussions of Dr. Saari's work:
 >     
...
[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

... [some quick observations by Mr Catchpole have been deleted]

 >
 >    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

You got Mr Ossipoff hinting at a weightless idea there. Candidates that
  are not named with a preference, have the same constant added to their
  sums. The last step, just before finding the winners, is to compare the
  sum and find the largest, which is insensitive to constants added to all.