[EM] Markus: Why Benham, Woodall, and IRV meet CD

[Michael Ossipoff]

*Markus--*

*(It's writing boldface, not inteded by me)*

*First I'll repeat my CD definition, below. Then, below it, I'll tell why
Benham, Woodall, and IRV pass CD.*


*Supporting definitions:*

1. The A voters are the voters who prefer candidate A to everyone else. The
B voters are the voters who prefer candidate B to everyone else. The C
voters are the voters who prefer C to everyone else.

2. A particular voter votes sincerely if s/he doesn't falsify a preference,
or fail to vote a felt preference that the balloting system in use would
have allowed hir to vote in addition to the preferences that s/he actually
votes.

*Premise:*

1. There are 3 candidates: A, B, and C.

2. The A voters and the B voters, combined, add up to more than half of the
voters in the election.

3. The A voters and the B voters all prefer both A and B to C.

4. The A voters are more numerous than are the B voters.

5. Voting is sincere, except that the B voters refuse to vote A over anyone.

6. Candidate A would be the unique winner under sincere voting (...in other
words, if the B voters voted sincerely, as do all the other voters).

7. The C voters are indifferent between A and B, and vote neither over the
other.

*Requirement:*

B doesn't win.

[end of CD definition]


------------------------------

In the chicken dilemma scenario described in the premise of the Chicken
Dilemma Criterion (CD) defined above, if B won, then the B voters would
have successfully taken advantage of the A voters' co-operativeness. The A
voters wanted to vote both A and B over the candidates disliked by both the
A voters and B voters. Thereby they helped {A,B} against worse candidates.
But, with methods that fail CD, the message is "You help, you lose".
------------------------------

*Some methods that pass the Chicken Dilemma Criterion:*
ICT, Symmetrical ICT
<http://wiki.electorama.com/wiki/Symmetrical_ICT>,
MMPO<http://wiki.electorama.com/wiki/MMPO>,
MDDTR, IRV <http://wiki.electorama.com/wiki/IRV>, Benham's
method<http://wiki.electorama.com/w/index.php?title=Benham%27s_method&action=edit&redlink=1>,
Woodall's method <http://wiki.electorama.com/wiki/Woodall%27s_method>


Why Benham passes CD:

I forgot to say it in my definition, but the ordering of the candidates, in
terms of their numbers voters who consider them favorite, is: C>A>B. That
will be added to the premise of CD.

So, B is the favorite of the fewest.

A is the sincere CW. A would be the voted CW under sincere voting, and
would thereby win. But, by CD's premise, the B voters refuse to vote A over
anyone.

In Benham, that results in a cycle: A>B>C>A. Since there's no initial CW,
Benham does IRV till there is one. B is favorite of fewest, and is
immedately eliminated by IRV. That leaves C>A. Now C is the uneliminated
candidate who beats each of the other uneliminated candidates. C wins. B
doesn't win. CD's requirement is satisfied.

Why Woodall meets CD:

As before, with the B voters refusing to vote A overf anyone (as specified
in CDs premise), there's a cycle: A>B>C>A.

There are 3 candidates in the Smith set. Since there isn't yet only one
candidate in the Smith set, Woodall does IRV.

B is favorite of fewest, and therefore is immediately eliminated in IRV.

The B voters' rankings don't incude anyone but B, and so they have no
further effect in the election.

Because there are more C voters than A voters, then IRV next eliminates A.
Then,only one member of the initial Smith set remains uneliminated: C. C
wins. B doesn't win. CD's requirement is satisfied. Woodall meets CD.

Why IRV meets CD:

B, as the favorite of fewest, is immediately eliminated. B doesn't win.
CD's requirement is satisfied. IRV meets CD.

Michael Ossipoff
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