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<DIV><FONT face=Arial size=2>Hi Dave, Adam and others,</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>Perhaps I wrote too much to be convincing of
anything. Let me try again and focus on my primary interest.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>My primary interest is in recognizing the existing
of "degrees" of strength of a winner. Condorcet does this too, but not quite as
completely. That is, Condorcet also is incomplete (when pairwise cycles exist),
and requires refined measures to determine a winner, and those refinements can
require further refinements! My interest is to show that perhaps Condorcet's
Winner could use some refining itself.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>I think it is useful to attempt to label the
strength of a candidate in an election, even if
it isn't unconditionally used to pick the
winner.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>When I first started learning about
EMs over two years ago Craig Carey demanded from me to
define what I meant by a majority among 3 or more candidates (He claimed it
didn't exist). Whatever Craig's funny ideas are, it is a worthy
question.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>At the time, I though it was obvious - A
majority is more than half the votes, but of course that idea is of limited
value by itself among 3 or more candidates since there may be no candidate that
passes the test. Still this is a reasonable test since not all methods
satisfy it:</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2><STRONG>1. A S<FONT size=3><FONT
face="Times New Roman">overeign</FONT></FONT> Majority (SM) - If a candidate
exists that is never below first place among any subset of competitors. (Also
more simply defined If a candidate exists that has more plurality votes
than all others combined.)</STRONG></FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>This rule is certainly a good test of an election
method, and two well known methods can break it - Approval and
Borda.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>Examples: </FONT></DIV>
<DIV><FONT face=Arial size=2> Approval: [XY] means a vote for
both X and Y, but prefer X.</FONT></DIV>
<DIV><FONT face=Arial size=2> [AC]=51%,
[BC]=49% - 51% prefer A, but C wins.</FONT></DIV>
<DIV><FONT face=Arial size=2> Analysis: Foolish
approval "overvoting" can hide a sovereign-majority
preference.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2> Borda:</FONT></DIV>
<DIV>
<DIV><FONT face=Arial size=2> Election 1:
AC=65%, CA=35%</FONT></DIV>
<DIV>
<DIV><FONT face=Arial
size=2>
A=2*65+1*35=165</FONT></DIV>
<DIV><FONT face=Arial
size=2>
C=1*65+2*35=135</FONT></DIV><FONT face=Arial
size=2> Election 2: ACB=65%, CBA=35%
(Same preference between A and C)</FONT></DIV></DIV>
<DIV>
<DIV><FONT face=Arial size=2>
C=2*65+3*35=235</FONT></DIV><FONT face=Arial
size=2>
A=3*65+1*35=230</FONT></DIV>
<DIV>
<DIV><FONT face=Arial size=2>
B=1*65+2*35=135</FONT></DIV></DIV>
<DIV>
<DIV><FONT face=Arial size=2> Analysis: Introducing a losing
candidate hides a top-majority preference!</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>This tells me that if I'm interested in supporting
a "Sovereign-Majority" (SM) criterion, I don't want to support Approval or
Borda.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>Then we have another lower criterion that can be
applied if there is no sovereign-majority - Condorcet's winner.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2><STRONG>3. A Condorcet Majority (CM) - If a
candidate exists that is always above last place among all pairwise subset
elections with competitors. ("Above-last" is equivalent to saying "first" among
pairwise comparisons.)</STRONG></FONT></DIV><FONT face=Arial
size=2><STRONG></STRONG></FONT></DIV>
<DIV><FONT face=Arial size=2><STRONG></STRONG></FONT> </DIV>
<DIV><FONT face=Arial size=2>I call this "Concercet Majority" because it defines
a unique winner, in parallel how the SM works. Two candidates can never
simultaneously satisfy this criterion, although in some elections (with cyclic
preferences), there may again be no candidate that satisfies it.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>Whether I want to use this rule for picking a
winner, it is a real measure of a candidate. AND we all know that IRV can break
this rule and pick a different candidate sometimes.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>So, if I'm interesting in ALWAYS picking the CM
winner, then IRV is not a method I want to support.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>I misnumbered the CM as (3) above because there is
an intermediate criterion which I chose to call the "Fresh Egg
Majority".</FONT></DIV><FONT face=Arial size=2></FONT>
<DIV><FONT face=Arial size=2></FONT><FONT face=Arial size=2></FONT><FONT
face=Arial size=2></FONT><FONT face=Arial size=2></FONT><FONT face=Arial
size=2></FONT><FONT face=Arial size=2></FONT><FONT face=Arial
size=2></FONT><FONT face=Arial size=2></FONT><FONT face=Arial
size=2></FONT><FONT face=Arial size=2></FONT><FONT face=Arial
size=2></FONT><FONT face=Arial size=2></FONT><BR><FONT face=Arial
size=2><STRONG>2. A Fresh Egg Majority (FEM) - If a candidate exists that is
always above last place along all subset elections with
competitors.</STRONG></FONT></DIV>
<DIV><STRONG><FONT face=Arial size=2></FONT></STRONG> </DIV>
<DIV><FONT face=Arial size=2>Looking at FEM is similar to CM, except it is a
more strict requirement. Sometimes a CM winner won't exist, but more often a FEM
winner won't exist.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>On the surface this would appear to be
back-stepping, being less decisive than CM, less useful in practice, however I
choose to see it as being more careful, more demanding, and choosing to
recognize the difference between a candidate that can merely do well in pairwise
contests, and one that also does well in the plurality set and other
subsets.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>This criterion suggests to me that the Condorcet
Majority definition, however nice, is not the end-all-be-all rule for picking a
winner, and that it is reasonable to consider election methods that
don't always pick the Condorcet winner.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>As I said before an interesting result of the FEM
definition is that when it exists, both Condorcet AND IRV must agree on this
winner.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>I don't think I should have to define FEM as a
concept that is meaningful for recognizing candidate's strength, although I
admit many may not consider it useful for defining a better method.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>For me it is interesting to me to define these. I
call them "majorities" but only the first (SM) REALLY qualifies as a true
(50%+1) majority. The FEM and CM criteria succeed by defining a unique winner
sometimes, and uncertainty elsewhere. These are perhaps best
called pseudo-majorities for being a pass/fail test that can have at most
one winner. Condorcet supporters might talk of "strongest majority", although
I'd call that phrase as overstepping their case since a measure of what is
strongest can be subjective.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>Obviously other such criteria may be named, for
example: </FONT></DIV>
<DIV><FONT face=Arial size=2><BR><STRONG>1+1/2. A Noble Majority (NM) - If
a candidate exists that is always above the average vote along all
subset elections with competitors. (Average =
total_votes/subset_candidates)</STRONG></FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>Avoiding last place (FEM) is a good sign of a
winner, but ALWAYS being above the middle is even stronger. This criterion is
between SM and FEM. It occurs more often than SM but less often than
FEM.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>I'm note sure if this one has any use, but it is an
attractive criterion to satisfy.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>I think it is useful for recognizing some winners
will be stronger than others, AND it may be useful to grade winners in this way.
Condorcet is the method that first demanded this complexity of giving degrees of
winners, and there is still disagreement in the hardest cases. I am merely
continuing his story and seeing what Condorcet chose to
ignore.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>If I have a point, it is to suggest that Condorcet
may be more negligent than supporters wish to admit, and even if everyone agrees
that Condorcet picks a best winner, I'd like recognition when a pairwise winner
is picked without core support of plurality counts.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>Thanks for listening.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>Sincerely,</FONT></DIV>
<DIV><FONT face=Arial size=2>Tom Ruen</FONT></DIV>
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