# Saari and Cyclic Ambiguities

Steve Barney barnes992001 at yahoo.com
Fri Mar 29 12:11:21 PST 2002

```When this profile is "decomposed" by the simplified method described by Alex
Small, you get:

5-5=0: ABCDE
6-5=1: BCDEA
7-5=2: CDEAB
8-5=3: DEABC
9-5=4: EABCD

According to Saari's theory, if the plurality and anti-plurality methods agree
with each other, then all positional methods agree on the outcome. If the
profile is essentially a "Basic profile" (dominated by the "Basic" components),
then all pairwise and positional methods agree on both the outcome and
normalized tally [see note 1, below]. (The anti-plurality vote is where you
vote against the worst candidate, or, equivalently, give 1 point to all but
your last place candidate.) For the sake of simplicity, let me use a similar 3
candidate profile to illustrate:

0:ABC
1:BCA
2:CAB

In this case, Plurality Vote and Anti-Plurality Vote disagree. Also, the
Pairwise Vote (Condorcet's method) disagrees with the Borda Count.

C>A~B Borda Count (2,1,0)
C>A>B Pairwise Vote

C>B>A Plurality Vote (1,0,0)
C>A>B Anti-Plurality Vote (1,1,0)

C>B>A Instant Runoff Vote

Therefore, this is NOT a "Basic" profile, according to Saari's strict
definition. For the full explanation and math, read the source cited below. I
would be happy to discuss it further - time permitting.

References:

Note 1. "An ideal starting point for a decomposition is with profiles which
achieve a major objective of choice theory { they admit no conflict.
Surprisingly, this profile subspace exists (but, it does not include unanimity
profiles). To underscore their central role, I call them Basic profiles. As
shown, the rankings and the (normalized) tallies of all positional methods and
pairwise outcomes agree with a Basic profile. This (two-dimensional) profile
subspace, then, allows no conflict or problems among procedures and their
derived methods. Basic profiles liberate us from the above difficulties where
election outcomes change with the choice of the procedures or as candidates
leave or enter."
--"EXPLAINING ALL THREE-ALTERNATIVE VOTING OUTCOMES," DONALD G. SAARI, pg 2,
<http://www.math.nwu.edu/~d_saari/vote/triple.pdf>.

SB

--- Michael Rouse <mrouse at cdsnet.net> wrote:
> Here's a cyclic ambiguity (a five-way circular tie) to try with various
> voting systems:
> 5: ABCDE
> 6: BCDEA
> 7: CDEAB
> 8: DEABC
> 9: EABCD
[...]

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