[EM] "margins vs. winning votes" and Woodall's "Plurality" criterion

[Chris Benham]

Russ,
While  I like Blake Cretney's  anti-WV argument, given here

http://condorcet.org/rp/inc.shtml

it doesn't tell the whole story. One big ace that WV has over Margins is 
that it meets Woodall's "Plurality" criterion/property.

"Plurality:  if some candidate x has more first-preference votes than 
some other candidate y has votes in total, then x's probability
of election must be greater than y's."

In his papers, Woodall likes to economise on axioms; so doesn't include 
the common-sense axiom that a ballot that leaves one candidate
unranked should be treated/regarded  the same as a ballot that ranks 
this candidate last and all the other candidates the same.
His  "votes in total" refers to explicit rankings in any position.

Incorporating this axiom, a "version" of the Plurality criterion  I like is

"If  some candidate x is ranked in first-place on more ballots than 
candidate y is ranked above equal-last, then y can't win".

A Woodall example:
11  ab
07  b
12  c

C has more first-preference votes than A has above-last-place votes, so 
A is barred by C's  "plurality" over A.
A>B 11-7  (m3)
B>C 18-12 (m6)
C>A 12-11 (m1)
Margins elects A, while WV elects B.

It took me a long time before I could see any particular point to this 
criterion, but now I rate it very highly.
Obviously if candidate x has a "plurality" over candidate y, then x 
pairwise beats y. If y is elected, then those voters that prefer x to y
will have a virtually  *unanswerable* complaint:  "How can you justify x 
losing to y? Maybe x losing to some z can be justified, but not to y!"

If some pairwise method that fails Plurality has just replaced FPP, and  
x's supporters are not great fans of the new method and x comes FP-first,
then it is easy to imagine that a riot could be fomented.

Unfortunately, "Plurality" used as a "social choice function" (SCF) 
fails Clone Independence and  monotonicity (Woodall's  "mono-raise").
That means that if we put a rule in front of Margins that candidates 
barred by plurality aren't allowed to win, then we'll have those failings.

Also of some value in my book is Steve Eppley's  "Minimal Defense" 
criterion that has recently been discussed by Kevin Venzke.

> "Minimal Defense" says that if more than half of the voters rank A 
> above B, and don't rank B above anybody, then B must be elected with 
> 0% probability. This implies Mike Ossipoff's SDSC. 

Margins fails MD, while WV meets it. 

Margins meets Woodall's  "Symmetric Completion" criterion (and therefore 
"No Zero-Information Strategy"),   while   WV   doesn't.

> Oh, by the way, I would *not* allow equal rankings. Why not? I just 
> don't like them. They strike me as an unnecessary complication and 
> little more than a way to game the system. 

I  think an ideal method in an ideal world should allow them; but I can 
see that voters are not likely to be enthusiastic or to see any great 
point, and that they could be untidy
from the practical point-of-view. Also with paper ballots, there could 
maybe be a theoretical possibility or suspicion that some extra "1"s  
could be added after the ballots have
been cast.

Chris Benham