Democratic symmetry (fwd)

[DEMOREP1 at aol.com]

Prof. Saari wrote (9 Mar 2000)--

    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.)
---
D- I mention again- adding or subtracting votes is commonly called election 
fraud (aka stuffing the ballot box and/or vote stealing).
---
Prof. Saari wrote (10 Mar 2000)--

3 ABC, 4 ACB, 4 BCA, 3 CBA leads to a plurality ranking of
A B C with a 7 : 4 : 3 tally.  Now, investigate the reversed
ranking (i.e., now give points to the bottom ranked candidate).
The tally is precisely the same.  A B C with 7:4:3.  

D- 
7 AB 7
7 BC 7
7 CA 7

Which proves what ???
---
Prof. Saari wrote more-

Let me illustrate with my favorite example.  

6 prefer Milk>Wine>Beer, 5 have Beer>Wine>Milk, 4 have Wine>Beer>Milk.

The plurality outcome is MBW.  The pairwise outcome is the exact opposite
of WBM.  Why?  Well, notice that the first two preferences reverse one
another.  So, as far as the pairwise vote is considered, the real data
set is that [5 MWB, 5 BWM] cancels leaving a tie, so we are left with
1 MWB (what is left over from the 6) and 4 WBM.  Here, the outcome is
the natural WBM.  So, the main difference between the pairwise and the
plurality vote in this example is that the pairwise vote treats [5 MWB, 5 BWM]
as a tie, while the plurality vote treats it as giving M = B (> W) each
five points.
--
D- 
6 MW 9
5 BW 10
9 BM 6

W happens to be a Condorcet Winner (CW)
B beats M
Thus WBM

Condorcet uses ALL of the votes.
---
Prof. Saari wrote more-

ROTATIONAL SYMMETRY

For reversal symmetry, start with a disk. 

***
    As a final comment, even a *trace* of rotational symmetry can cause
difficulties.  For instance, take a ranking wheel with six candidates
and the initial ranking ABCDEF.  Now, only rotate the wheel three times
(rather than the full six) to create a profile
    ABCDEF, BCDEFA, CDEFAB
It is easy to argue that C probably is the favorite of these voters, and
F (who is bottom ranked so often) is the least favored. In any case, 
it is clear that C, D, E are ranked above F. 
---
D-  
F obviously is the clone of E on ALL ballots;
E obviously is the clone of D on ALL ballots; and
D obviously is the clone of C on ALL ballots

leaving A, B and C.

Generally speaking it appears that Prof. Saari has mystified the existance of 
clones with some new verbiage- data symmetry, reflection symmetry, rotational 
symmetry.

Anybody yet slog thru the Economic Theory, pp. 1-103 article ???

I note again- 
majority > minority
and
minority < majority
(i.e. NO symmetry).